3947:
3770:
3411:
2499:
3449:
and histograms: Histograms are frequently encountered in applications of real-life problems. Most observations are usually available under the form of nonnegative vectors of counts, which, if normalized, yield histograms of frequencies. It has been shown that the following family of squared metrics,
8764:
including repetitions. Where sampling points lie dense, the true density function must take large values. A simple density estimate is possible by counting the number of samples in each cell of a grid, and plotting the resulting histogram, which yields a piecewise constant density estimate. A better
7750:
which states that every minimizer function in an RKHS can be written as a linear combination of the kernel function evaluated at the training points. This is a practically useful result as it effectively simplifies the empirical risk minimization problem from an infinite dimensional to a finite
2620:
2349:
8961:
Applications of p.d. kernels in various other branches of mathematics are in multivariate integration, multivariate optimization, and in numerical analysis and scientific computing, where one studies fast, accurate and adaptive algorithms ideally implemented in high-performance computing
4779:. The original object of Mercer’s paper was to characterize the kernels which are definite in the sense of Hilbert, but Mercer soon found that the class of such functions was too restrictive to characterize in terms of determinants. He therefore defined a continuous real symmetric kernel
4459:
3139:
5363:
Positive-definite kernels provide a framework that encompasses some basic
Hilbert space constructions. In the following we present a tight relationship between positive-definite kernels and two mathematical objects, namely reproducing Hilbert spaces and feature maps.
3774:
2209:
3586:
3582:
1094:
1549:
2361:
3046:
4054:
Positive-definite kernels, as defined in (1.1), appeared first in 1909 in a paper on integral equations by James Mercer. Several other authors made use of this concept in the following two decades, but none of them explicitly used kernels
949:
4038:
1711:
5989:
As stated earlier, positive definite kernels can be constructed from inner products. This fact can be used to connect p.d. kernels with another interesting object that arises in machine learning applications, namely the feature map. Let
1873:
6413:, by the reproducing property. This suggests a new look at p.d. kernels as inner products in appropriate Hilbert spaces, or in other words p.d. kernels can be viewed as similarity maps which quantify effectively how similar two points
4707:
9097:
Krein. M (1949/1950). "Hermitian-positive kernels on homogeneous spaces I and II" (in
Russian), Ukrain. Mat. Z. 1(1949), pp. 64–98, and 2(1950), pp. 10–59. English translation: Amer. Math. Soc. Translations Ser. 2, 34 (1963), pp.
4231:
1937:
2911:
8882:
5001:
2726:
2507:
2220:
332:
6930:
7150:
4257:
643:
186:
472:
7061:
6411:
9052:
Mercer, J. (1909). “Functions of positive and negative type and their connection with the theory of integral equations”. Philosophical
Transactions of the Royal Society of London, Series A 209, pp. 415–446.
5498:
3134:
2076:
1981:
822:
7722:
1206:
8762:
2123:
768:
404:
6619:
7204:
3473:
1326:
5780:
8658:
1272:
1154:
588:
6180:
5325:
7941:
1437:
7316:
5720:
5327:. This property is called the reproducing property of the kernel and turns out to have importance in the solution of boundary-value problems for elliptic partial differential equations.
2782:
1377:
712:
523:, and positive semi-definite (p.s.d.) kernels, which do not impose this condition. Note that this is equivalent to requiring that every finite matrix constructed by pairwise evaluation,
9070:
Young, W. H. (1909). "A note on a class of symmetric functions and on a theorem required in the theory of integral equations", Philos. Trans. Roy.Soc. London, Ser. A, 209, pp. 415–446.
6872:
3406:{\displaystyle \sum _{i,j=1}^{n}c_{i}c_{j}(x_{i},x_{j})_{H}=\left(\sum _{i=1}^{n}c_{i}x_{i},\sum _{j=1}^{n}c_{j}x_{j}\right)_{H}=\left\|\sum _{i=1}^{n}c_{i}x_{i}\right\|_{H}^{2}\geq 0}
5585:
3942:{\displaystyle \psi _{H_{1}}=\sum _{i}\left|{\sqrt {\theta _{i}}}-{\sqrt {\theta _{i}'}}\right|,\psi _{H_{2}}=\sum _{i}\left|{\sqrt {\theta _{i}}}-{\sqrt {\theta _{i}'}}\right|^{2},}
5642:
1554:
986:
8159:
1716:
3447:
521:
7258:
6047:
5439:
7396:
4588:
7450:
5006:
At about the same time W. H. Young, motivated by a different question in the theory of integral equations, showed that for continuous kernels condition (1.1) is equivalent to
2115:
1004:
3765:{\displaystyle \psi _{\chi ^{2}}=\sum _{i}{\frac {(\theta _{i}-\theta _{i}')^{2}}{\theta _{i}+\theta _{i}'}},\quad \psi _{TV}=\sum _{i}\left|\theta _{i}-\theta _{i}'\right|,}
8706:
7854:
7340:
6954:
6791:
6268:
5836:
4901:, and he proved that (1.1) is a necessary and sufficient condition for a kernel to be of positive type. Mercer then proved that for any continuous p.d. kernel the expansion
4127:
1442:
210:
135:
7523:
4548:
4109:
2494:{\displaystyle K(\mathbf {x} ,\mathbf {y} )=e^{-{\frac {\|\mathbf {x} -\mathbf {y} \|^{2}}{2\sigma ^{2}}}},\quad \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{d},\sigma >0}
8451:
6663:
6202:
follows from the p.d. property of the inner product. On the other hand, every p.d. kernel, and its corresponding RKHS, have many associated feature maps. For example: Let
6079:
4579:
3952:
8788:
8478:
5204:
5087:
5039:
4847:
5171:
4742:
8389:
8058:
7571:
6704:
2950:
2938:
7597:
4904:
830:
6486:
6294:
5888:
5862:
5544:
5145:
4812:
4777:
8022:
7995:
7968:
7477:
3468:
2008:
8418:
8315:
8286:
8217:
8188:
7790:
6730:
6226:
220:
9134:
Schaback, R. and
Wendland, H. (2006). "Kernel Techniques: From Machine Learning to Meshless Methods", Cambridge University Press, Acta Numerica (2006), pp. 1–97.
8783:
8682:
8558:
8538:
8518:
8498:
8257:
8237:
8123:
8103:
7830:
7810:
7617:
7422:
7360:
6585:
6561:
6538:
6518:
6451:
6431:
6200:
6099:
6008:
5984:
5964:
5926:
5669:
5518:
5405:
5385:
5244:
5224:
5110:
4867:
4515:
4495:
3071:
2028:
4899:
61:. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in
1878:
2787:
6964:
9125:
Berg, C., Christensen, J. P. R., and Ressel, P. (1984). "Harmonic
Analysis on Semigroups". Number 100 in Graduate Texts in Mathematics, Springer Verlag.
5942:
Every reproducing kernel is positive-definite, and every positive definite kernel defines a unique RKHS, of which it is the unique reproducing kernel.
2615:{\displaystyle K(\mathbf {x} ,\mathbf {y} )=e^{-\alpha \|\mathbf {x} -\mathbf {y} \|},\quad \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{d},\alpha >0}
7404:
On the other hand, n.d. kernels can be identified with a subfamily of p.d. kernels known as infinitely divisible kernels. A nonnegative-valued kernel
2628:
2344:{\displaystyle K(\mathbf {x} ,\mathbf {y} )=(\mathbf {x} ^{T}\mathbf {y} +r)^{n},\quad \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{d},r\geq 0,n\geq 1}
9061:
Hilbert, D. (1904). "Grundzuge einer allgemeinen
Theorie der linearen Integralgleichungen I", Gott. Nachrichten, math.-phys. K1 (1904), pp. 49–91.
8942:
In the literature on computer experiments and other engineering experiments, one increasingly encounters models based on p.d. kernels, RBFs or
7742:
Positive-definite kernels, through their equivalence with reproducing kernel
Hilbert spaces (RKHS), are particularly important in the field of
6887:
6877:
One link between distances and p.d. kernels is given by a particular kind of kernel, called a negative definite kernel, and defined as follows
7342:. At the same time each distance does not correspond necessarily to a n.d. kernel. This is only true for Hilbertian distances, where distance
9024:
8986:
7082:
5343:
7622:
8955:
600:
143:
409:
8912:
6299:
6104:
5330:
Another line of development in which p.d. kernels played a large role was the theory of harmonics on homogeneous spaces as begun by
6488:. Moreover, through the equivalence of p.d. kernels and its corresponding RKHS, every feature map can be used to construct a RKHS.
5444:
3080:
8991:
5358:
2033:
8916:
4454:{\displaystyle \int _{a}^{b}\int _{a}^{b}K(s,t)x(s)x(t)\ \mathrm {d} s\,\mathrm {d} t=\sum {\frac {1}{\lambda _{n}}}\left^{2},}
1945:
773:
477:
In probability theory, a distinction is sometimes made between positive-definite kernels, for which equality in (1.1) implies
2204:{\displaystyle K(\mathbf {x} ,\mathbf {y} )=\mathbf {x} ^{T}\mathbf {y} ,\quad \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{d}}
992:. In the rest of this article we assume real-valued functions, which is the common practice in applications of p.d. kernels.
1159:
8420:
can be obtained by using kernel interpolation with the covariance kernel, ignoring the probabilistic background completely.
8904:
8711:
6497:
717:
353:
90:
9153:
6590:
8947:
8932:
7155:
4582:
2355:
1277:
591:
54:
5728:
8563:
7743:
1213:
1101:
526:
9116:
Rosasco, L. and Poggio, T. (2015). "A Regularization Tour of
Machine Learning – MIT 9.520 Lecture Notes" Manuscript.
5249:
8908:
7859:
1387:
7267:
4116:
46:
9040:
5685:
2737:
1333:
652:
7264:
The parallel between n.d. kernels and distances is in the following: whenever a n.d. kernel vanishes on the set
4115:
seem not to have been aware of the study of p.d. kernels). Mercer’s work arose from
Hilbert’s paper of 1904 on
9171:
6797:
50:
5549:
597:
In mathematical literature, kernels are usually complex-valued functions. That is, a complex-valued function
9176:
9143:
Haaland, B. and Qian, P. Z. G. (2010). "Accurate emulators for large-scale computer experiments", Ann. Stat.
5590:
957:
8391:. Such a kernel exists and is positive-definite under weak additional assumptions. Now a good estimate for
8128:
3577:{\displaystyle \psi _{JD}=H\left({\frac {\theta +\theta '}{2}}\right)-{\frac {H(\theta )+H(\theta ')}{2}},}
3418:
480:
86:
6013:
5410:
1089:{\displaystyle (K_{i})_{i\in \mathbb {N} },\ \ K_{i}:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }
9079:
Moore, E.H. (1916). "On properly positive
Hermitian matrices", Bull. Amer. Math. Soc. 23, 59, pp. 66–67.
7365:
7209:
7427:
1544:{\displaystyle (K_{i})_{i=1}^{n},\ \ K_{i}:{\mathcal {X}}_{i}\times {\mathcal {X}}_{i}\to \mathbb {R} }
2091:
6564:
8687:
7835:
7321:
6935:
6737:
6231:
5787:
191:
116:
8971:
7747:
7482:
4520:
4058:
4049:
8427:
6627:
6052:
4557:
8981:
8920:
8907:. Some of the popular meshfree methods are closely related to positive-definite kernels (such as
8456:
7529:
3041:{\displaystyle K(x,y)=\operatorname {sinc} (\alpha (x-y)),\quad x,y\in \mathbb {R} ,\alpha >0}
138:
102:
66:
17:
6500:. In this section we discuss parallels between their two respective ingredients, namely kernels
5932:
Now the connection between positive definite kernels and RKHS is given by the following theorem
5176:
5044:
5009:
4817:
944:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}\xi _{i}{\overline {\xi }}_{j}K(x_{i},x_{j})\geq 0}
5150:
4712:
1706:{\displaystyle K((x_{1},\dots ,x_{n}),(y_{1},\dots ,y_{n}))=\prod _{i=1}^{n}K_{i}(x_{i},y_{i})}
9020:
8976:
8958:. Here one often uses implicit surface models to approximate or interpolate point cloud data.
8951:
8320:
8027:
7535:
6668:
2916:
2214:
1868:{\displaystyle K((x_{1},\dots ,x_{n}),(y_{1},\dots ,y_{n}))=\sum _{i=1}^{n}K_{i}(x_{i},y_{i})}
989:
98:
82:
58:
7576:
6101:
the feature space. It is easy to see that every feature map defines a unique p.d. kernel by
5206:. Moore was interested in generalization of integral equations and showed that to each such
8900:
8894:
6456:
6273:
5867:
5841:
5523:
5115:
4782:
4747:
94:
74:
62:
9088:
Moore, E.H. (1935). "General Analysis, Part I", Memoirs Amer. Philos. Soc. 1, Philadelphia.
8000:
7973:
7946:
7455:
5671:
is called a reproducing kernel Hilbert space if the evaluation functionals are continuous.
5338:
and S. Ito. The most comprehensive theory of p.d. kernels in homogeneous spaces is that of
4702:{\displaystyle J(x)=\int _{a}^{b}\int _{a}^{b}K(s,t)x(s)x(t)\ \mathrm {d} s\;\mathrm {d} t}
3453:
1986:
8394:
8291:
8262:
8193:
8164:
7766:
4112:
2941:
70:
38:
31:
6709:
6205:
5349:
In probability theory, p.d. kernels arise as covariance kernels of stochastic processes.
8768:
8667:
8543:
8523:
8503:
8483:
8242:
8222:
8108:
8063:
7815:
7795:
7602:
7407:
7345:
6570:
6546:
6523:
6503:
6436:
6416:
6185:
6084:
5993:
5969:
5949:
5911:
5654:
5503:
5390:
5370:
5331:
5229:
5209:
5095:
4852:
4500:
4480:
4226:{\displaystyle f(s)=\varphi (s)-\lambda \int _{a}^{b}K(s,t)\varphi (t)\ \mathrm {d} t.}
3056:
2013:
1932:{\displaystyle {\mathcal {X}}={\mathcal {X}}_{1}\times \dots \times {\mathcal {X}}_{n}}
78:
4872:
2906:{\displaystyle K(x,y)=\|x-y\|_{2}^{k-{\frac {d}{2}}}B_{k-{\frac {d}{2}}}(\|x-y\|_{2})}
9165:
8996:
7737:
4551:
4033:{\displaystyle K(\theta ,\theta ')=e^{-\alpha \psi (\theta ,\theta ')},\alpha >0.}
3074:
2732:
8877:{\displaystyle f(x)={\frac {1}{n}}\sum _{i=1}^{n}K\left({\frac {x-x_{i}}{h}}\right)}
4585:
of (1.2). Hilbert defined a “definite” kernel as one for which the double integral
5675:
Every RKHS has a special function associated to it, namely the reproducing kernel:
5335:
4996:{\displaystyle K(s,t)=\sum _{n}{\frac {\psi _{n}(s)\psi _{n}(t)}{\lambda _{n}}}}
7759:
There are several different ways in which kernels arise in probability theory.
2721:{\displaystyle K(x,y)=e^{-\alpha |x-y|},\quad x,y\in \mathbb {R} ,\alpha >0}
9107:
Loève, M. (1960). "Probability theory", 2nd ed., Van Nostrand, Princeton, N.J.
5339:
8765:
estimate can be obtained by using a nonnegative translation invariant kernel
7763:
Nondeterministic recovery problems: Assume that we want to find the response
5898:
The following result shows equivalence between RKHS and reproducing kernels:
9154:
Fast radial basis function interpolation via preconditioned Krylov iteration
8317:
will often show similar behaviour. This is described by a covariance kernel
327:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}c_{i}c_{j}K(x_{i},x_{j})\geq 0}
7532:, where the first constraint on the distance function is loosened to allow
5342:
which includes as special cases the work on p.d. functions and irreducible
9043:". In Ghahramani, Z. and Cowell, R., editors, Proceedings of AISTATS 2005.
5092:
E.H. Moore initiated the study of a very general kind of p.d. kernel. If
7398:
6925:{\displaystyle \psi :{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }
9041:
Hilbertian metrics and positive definite kernels on probability measures
8943:
7318:, and is zero only on this set, then its square root is a distance for
7145:{\displaystyle n\in \mathbb {N} ,x_{1},\dots ,x_{n}\in {\mathcal {X}},}
5928:
induces a unique RKHS, and every RKHS has a unique reproducing kernel.
3470:-square, Total Variation, and two variations of the Hellinger distance:
6543:
Here by a distance function between each pair of elements of some set
8664:
Density estimation by kernels: The problem is to recover the density
638:{\displaystyle K:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {C} }
181:{\displaystyle K:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }
6496:
Kernel methods are often compared to distance based methods such as
467:{\displaystyle n\in \mathbb {N} ,c_{1},\dots ,c_{n}\in \mathbb {R} }
8560:
is identical to the above one, but with a modified kernel given by
7056:{\displaystyle \sum _{i,j=1}^{n}c_{i}c_{j}\psi (x_{i},x_{j})\leq 0}
5353:
Connection with reproducing kernel Hilbert spaces and feature maps
8950:. Other types of applications that boil down to data fitting are
6406:{\displaystyle (\Phi (x),\Phi (y))_{F}=(K_{x},K_{y})_{H}=K(x,y)}
3949:
can be used to define p.d. kernels using the following formula
590:, has either entirely positive (p.d.) or nonnegative (p.s.d.)
9019:. Providence, RI: American Mathematical Soc. pp. 45–47.
5644:. We first define a reproducing kernel Hilbert space (RKHS):
5493:{\displaystyle (\cdot ,\cdot )_{H}:H\times H\to \mathbb {R} }
3129:{\displaystyle (\cdot ,\cdot )_{H}:H\times H\to \mathbb {R} }
137:
be a nonempty set, sometimes referred to as the index set. A
8749:
8693:
8146:
7841:
7374:
7327:
7300:
7134:
6941:
6909:
6899:
6606:
6596:
5173:“positive Hermitian matrices” if they satisfy (1.1) for all
2057:
2040:
1968:
1952:
1918:
1895:
1884:
1522:
1505:
1397:
1073:
1063:
755:
622:
612:
391:
197:
165:
155:
122:
2088:
Common examples of p.d. kernels defined on Euclidean space
2071:{\displaystyle {\mathcal {X}}_{0}\times {\mathcal {X}}_{0}}
8024:
but rather a realization of a real-valued random variable
6567:
defined on that set, i.e. any nonnegative-valued function
30:"Kernel function" redirects here. Not to be confused with
7856:, provided that we have a sample of input-response pairs
5894:
The latter property is called the reproducing property.
1976:{\displaystyle {\mathcal {X}}_{0}\subset {\mathcal {X}}}
817:{\displaystyle \xi _{1},\dots ,\xi _{n}\in \mathbb {C} }
714:
and positive definite if for every finite set of points
8540:
there, then the problem of finding a good estimate for
7362:
is called Hilbertian if one can embed the metric space
7717:{\displaystyle d(x,y)={\sqrt {K(x,x)-2K(x,y)+K(y,y)}}}
7212:
1201:{\displaystyle \lambda _{1},\dots ,\lambda _{n}\geq 0}
9017:
Expansions in eigenfunctions of selfadjoint operators
8791:
8771:
8714:
8690:
8670:
8566:
8546:
8526:
8506:
8486:
8459:
8430:
8397:
8323:
8294:
8265:
8245:
8225:
8196:
8167:
8131:
8111:
8066:
8030:
8003:
7976:
7949:
7862:
7838:
7818:
7798:
7769:
7625:
7605:
7579:
7538:
7485:
7458:
7430:
7410:
7368:
7348:
7324:
7270:
7158:
7085:
6967:
6938:
6890:
6800:
6740:
6712:
6671:
6630:
6593:
6573:
6549:
6526:
6506:
6459:
6439:
6419:
6302:
6276:
6234:
6208:
6188:
6107:
6087:
6055:
6016:
5996:
5972:
5952:
5914:
5870:
5844:
5790:
5731:
5688:
5657:
5593:
5552:
5526:
5506:
5447:
5413:
5393:
5373:
5252:
5232:
5212:
5179:
5153:
5118:
5098:
5047:
5012:
4907:
4875:
4855:
4820:
4785:
4750:
4715:
4591:
4560:
4523:
4503:
4483:
4260:
4130:
4061:
3955:
3777:
3589:
3476:
3456:
3421:
3142:
3083:
3059:
2953:
2919:
2790:
2740:
2631:
2510:
2364:
2223:
2126:
2094:
2036:
2016:
1989:
1948:
1881:
1719:
1557:
1445:
1390:
1336:
1280:
1216:
1162:
1104:
1007:
960:
833:
776:
720:
655:
603:
529:
483:
412:
356:
223:
194:
146:
119:
57:
in the early 20th century, in the context of solving
8889:
Numerical solution of partial differential equations
8757:{\displaystyle x_{1},\dots ,x_{n}\in {\mathcal {X}}}
8060:. The goal is to get information about the function
763:{\displaystyle x_{1},\dots ,x_{n}\in {\mathcal {X}}}
399:{\displaystyle x_{1},\dots ,x_{n}\in {\mathcal {X}}}
8899:One of the greatest application areas of so-called
8500:, such that the noise is independent for different
6614:{\displaystyle {\mathcal {X}}\times {\mathcal {X}}}
4814:to be of positive type (i.e. positive-definite) if
8876:
8777:
8756:
8700:
8676:
8652:
8552:
8532:
8512:
8492:
8472:
8445:
8412:
8383:
8309:
8280:
8251:
8231:
8211:
8182:
8153:
8117:
8097:
8052:
8016:
7989:
7962:
7935:
7848:
7824:
7804:
7784:
7716:
7611:
7591:
7565:
7517:
7471:
7444:
7416:
7390:
7354:
7334:
7310:
7252:
7199:{\displaystyle c_{1},\dots ,c_{n}\in \mathbb {R} }
7198:
7144:
7055:
6948:
6924:
6866:
6785:
6724:
6698:
6657:
6613:
6579:
6555:
6532:
6512:
6480:
6445:
6425:
6405:
6288:
6262:
6220:
6194:
6174:
6093:
6073:
6041:
6002:
5978:
5966:, it is possible to build an associated RKHS with
5958:
5920:
5882:
5856:
5830:
5774:
5714:
5663:
5636:
5579:
5538:
5512:
5492:
5433:
5399:
5379:
5319:
5238:
5218:
5198:
5165:
5139:
5104:
5081:
5033:
4995:
4893:
4861:
4841:
4806:
4771:
4736:
4701:
4573:
4542:
4509:
4489:
4453:
4225:
4103:
4032:
3941:
3764:
3576:
3462:
3441:
3405:
3128:
3065:
3040:
2932:
2905:
2776:
2720:
2614:
2493:
2343:
2203:
2109:
2070:
2022:
2002:
1975:
1931:
1867:
1705:
1543:
1431:
1371:
1321:{\displaystyle a_{1},\dots ,a_{n}\in \mathbb {N} }
1320:
1266:
1200:
1148:
1088:
980:
943:
816:
762:
706:
637:
582:
515:
466:
398:
326:
204:
180:
129:
9156:". SIAM J. Scient. Computing 29/5, pp. 1876–1899.
7943:given by observation or experiment. The response
5775:{\displaystyle K_{x}(\cdot )\in H,\forall x\in X}
8653:{\displaystyle K(x,y)=E+\sigma ^{2}\delta _{xy}}
7424:is said to be infinitely divisible if for every
1344:
1267:{\displaystyle K_{1}^{a_{1}}\dots K_{n}^{a_{n}}}
1149:{\displaystyle \sum _{i=1}^{n}\lambda _{i}K_{i}}
583:{\displaystyle \mathbf {K} _{ij}=K(x_{i},x_{j})}
8785:, with total integral equal to one, and define
8125:in the deterministic setting. For two elements
6932:is called a negative definite (n.d.) kernel on
6175:{\displaystyle K(x,y)=(\Phi (x),\Phi (y))_{F}.}
5320:{\displaystyle f\in H,f(y)=(f,K(\cdot ,y))_{H}}
188:is called a positive-definite (p.d.) kernel on
7936:{\displaystyle (x_{i},f_{i})=(x_{i},f(x_{i}))}
6081:is called a feature map. In this case we call
1432:{\displaystyle ({\mathcal {X}}_{i})_{i=1}^{n}}
8919:). These methods use radial basis kernel for
8684:of a multivariate distribution over a domain
7528:Another link is that a p.d. kernel induces a
7311:{\displaystyle \{(x,x):x\in {\mathcal {X}}\}}
4111:, i.e. p.d. functions (indeed M. Mathias and
8:
7305:
7271:
4537:
4524:
2891:
2878:
2825:
2812:
2563:
2547:
2418:
2401:
27:Generalization of a positive-definite matrix
9152:Gumerov, N. A. and Duraiswami, R. (2007). "
5936:
5902:
5715:{\displaystyle K:X\times X\to \mathbb {R} }
2777:{\displaystyle W_{2}^{k}(\mathbb {R} ^{d})}
1372:{\displaystyle K=\lim _{n\to \infty }K_{n}}
707:{\displaystyle K(x,y)={\overline {K(y,x)}}}
4690:
500:
8913:Reproducing kernel particle method (RKPM)
8858:
8845:
8832:
8821:
8807:
8790:
8770:
8748:
8747:
8738:
8719:
8713:
8692:
8691:
8689:
8669:
8641:
8631:
8565:
8545:
8525:
8505:
8485:
8464:
8458:
8429:
8396:
8322:
8293:
8264:
8244:
8224:
8195:
8166:
8145:
8144:
8130:
8110:
8083:
8065:
8041:
8029:
8008:
8002:
7981:
7975:
7954:
7948:
7921:
7902:
7883:
7870:
7861:
7840:
7839:
7837:
7817:
7797:
7768:
7647:
7624:
7604:
7578:
7537:
7509:
7499:
7484:
7463:
7457:
7438:
7437:
7429:
7409:
7373:
7372:
7367:
7347:
7326:
7325:
7323:
7299:
7298:
7269:
7238:
7228:
7217:
7211:
7192:
7191:
7182:
7163:
7157:
7133:
7132:
7123:
7104:
7093:
7092:
7084:
7038:
7025:
7009:
6999:
6989:
6972:
6966:
6940:
6939:
6937:
6918:
6917:
6908:
6907:
6898:
6897:
6889:
6867:{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}
6799:
6739:
6711:
6670:
6629:
6605:
6604:
6595:
6594:
6592:
6572:
6548:
6525:
6505:
6458:
6438:
6418:
6376:
6366:
6353:
6337:
6301:
6275:
6254:
6233:
6207:
6187:
6163:
6106:
6086:
6054:
6049:the corresponding inner product. Any map
6033:
6015:
5995:
5971:
5951:
5913:
5869:
5843:
5804:
5789:
5736:
5730:
5708:
5707:
5687:
5656:
5604:
5592:
5573:
5572:
5557:
5551:
5525:
5505:
5486:
5485:
5464:
5446:
5427:
5426:
5412:
5392:
5372:
5311:
5251:
5231:
5211:
5184:
5178:
5152:
5117:
5097:
5058:
5046:
5011:
4985:
4965:
4946:
4939:
4933:
4906:
4874:
4854:
4819:
4784:
4749:
4714:
4691:
4682:
4631:
4626:
4616:
4611:
4590:
4565:
4559:
4531:
4522:
4502:
4482:
4442:
4429:
4428:
4401:
4391:
4386:
4368:
4359:
4345:
4344:
4336:
4285:
4280:
4270:
4265:
4259:
4212:
4173:
4168:
4129:
4060:
3986:
3954:
3930:
3914:
3908:
3897:
3891:
3879:
3864:
3859:
3836:
3830:
3819:
3813:
3802:
3787:
3782:
3776:
3745:
3732:
3717:
3701:
3681:
3668:
3656:
3643:
3630:
3620:
3614:
3599:
3594:
3588:
3530:
3500:
3481:
3475:
3455:
3433:
3428:
3424:
3423:
3420:
3391:
3386:
3375:
3365:
3355:
3344:
3325:
3314:
3304:
3294:
3283:
3270:
3260:
3250:
3239:
3220:
3210:
3197:
3184:
3174:
3164:
3147:
3141:
3122:
3121:
3100:
3082:
3058:
3022:
3021:
2952:
2924:
2918:
2894:
2863:
2856:
2840:
2833:
2828:
2789:
2765:
2761:
2760:
2750:
2745:
2739:
2702:
2701:
2678:
2664:
2657:
2630:
2594:
2590:
2589:
2580:
2572:
2558:
2550:
2540:
2525:
2517:
2509:
2473:
2469:
2468:
2459:
2451:
2436:
2421:
2412:
2404:
2398:
2394:
2379:
2371:
2363:
2311:
2307:
2306:
2297:
2289:
2279:
2264:
2258:
2253:
2238:
2230:
2222:
2195:
2191:
2190:
2181:
2173:
2164:
2158:
2153:
2141:
2133:
2125:
2101:
2097:
2096:
2093:
2062:
2056:
2055:
2045:
2039:
2038:
2035:
2015:
1994:
1988:
1967:
1966:
1957:
1951:
1950:
1947:
1923:
1917:
1916:
1900:
1894:
1893:
1883:
1882:
1880:
1856:
1843:
1830:
1820:
1809:
1790:
1771:
1752:
1733:
1718:
1694:
1681:
1668:
1658:
1647:
1628:
1609:
1590:
1571:
1556:
1537:
1536:
1527:
1521:
1520:
1510:
1504:
1503:
1493:
1474:
1463:
1453:
1444:
1423:
1412:
1402:
1396:
1395:
1389:
1363:
1347:
1335:
1314:
1313:
1304:
1285:
1279:
1256:
1251:
1246:
1231:
1226:
1221:
1215:
1186:
1167:
1161:
1140:
1130:
1120:
1109:
1103:
1082:
1081:
1072:
1071:
1062:
1061:
1052:
1033:
1032:
1025:
1015:
1006:
972:
962:
959:
926:
913:
897:
887:
880:
870:
859:
849:
838:
832:
810:
809:
800:
781:
775:
754:
753:
744:
725:
719:
677:
654:
631:
630:
621:
620:
611:
610:
602:
571:
558:
536:
531:
528:
488:
482:
460:
459:
450:
431:
420:
419:
411:
390:
389:
380:
361:
355:
309:
296:
280:
270:
260:
249:
239:
228:
222:
196:
195:
193:
174:
173:
164:
163:
154:
153:
145:
121:
120:
118:
7619:, we can define a distance function as:
7452:there exists a positive-definite kernel
9007:
5946:Thus, given a positive-definite kernel
5580:{\displaystyle e_{x}:H\to \mathbb {R} }
5112:is an abstract set, he calls functions
4497:is a continuous real symmetric kernel,
3077:, then its corresponding inner product
5637:{\displaystyle f\mapsto e_{x}(f)=f(x)}
4249:In particular, Hilbert had shown that
2947:Kernel generating Paley–Wiener space:
1551:a sequence of p.d. kernels, then both
981:{\displaystyle {\overline {\xi }}_{j}}
9015:Berezanskij, Jurij MakaroviÄŤ (1968).
8987:Positive-definite function on a group
8917:smoothed-particle hydrodynamics (SPH)
8909:meshless local Petrov Galerkin (MLPG)
8219:will not be uncorrelated, because if
8154:{\displaystyle x,y\in {\mathcal {X}}}
7:
8259:the random experiments described by
6958:
4251:
4121:
3450:respectively Jensen divergence, the
3442:{\displaystyle \mathbb {R} _{+}^{d}}
516:{\displaystyle c_{i}=0\;(\forall i)}
214:
9039:Hein, M. and Bousquet, O. (2005). "
7599:. Given a positive-definite kernel
7253:{\textstyle \sum _{i=1}^{n}c_{i}=0}
6042:{\displaystyle (\cdot ,\cdot )_{F}}
5682:: Reproducing kernel is a function
5500:the corresponding inner product on
5434:{\displaystyle f:X\to \mathbb {R} }
7751:dimensional optimization problem.
7391:{\displaystyle ({\mathcal {X}},d)}
6321:
6306:
6235:
6147:
6132:
6056:
5760:
4849:for all real continuous functions
4692:
4683:
4430:
4346:
4337:
4213:
3136:is a p.d. kernel. Indeed, we have
1354:
504:
25:
8424:Assume now that a noise variable
7445:{\displaystyle n\in \mathbb {N} }
6182:Indeed, positive definiteness of
5246:of functions such that, for each
2942:Bessel function of the third kind
18:Positive-definite kernel function
8992:Reproducing kernel Hilbert space
8903:is in the numerical solution of
5359:Reproducing kernel Hilbert space
5003:holds absolutely and uniformly.
2581:
2573:
2559:
2551:
2526:
2518:
2460:
2452:
2413:
2405:
2380:
2372:
2298:
2290:
2265:
2254:
2239:
2231:
2182:
2174:
2165:
2154:
2142:
2134:
2110:{\displaystyle \mathbb {R} ^{d}}
532:
7755:Kernels in probabilistic models
3696:
3008:
2688:
2571:
2450:
2288:
2172:
8801:
8795:
8701:{\displaystyle {\mathcal {X}}}
8621:
8618:
8612:
8603:
8597:
8591:
8582:
8570:
8453:, with zero mean and variance
8440:
8434:
8407:
8401:
8378:
8375:
8369:
8360:
8354:
8348:
8339:
8327:
8304:
8298:
8275:
8269:
8206:
8200:
8177:
8171:
8092:
8089:
8076:
8070:
8047:
8034:
7930:
7927:
7914:
7895:
7889:
7863:
7849:{\displaystyle {\mathcal {X}}}
7779:
7773:
7709:
7697:
7688:
7676:
7664:
7652:
7641:
7629:
7554:
7542:
7506:
7492:
7385:
7369:
7335:{\displaystyle {\mathcal {X}}}
7286:
7274:
7044:
7018:
6949:{\displaystyle {\mathcal {X}}}
6914:
6858:
6846:
6837:
6825:
6816:
6804:
6786:{\displaystyle d(x,y)=d(y,x),}
6777:
6765:
6756:
6744:
6687:
6675:
6646:
6634:
6475:
6463:
6400:
6388:
6373:
6346:
6334:
6330:
6324:
6315:
6309:
6303:
6263:{\displaystyle \Phi (x)=K_{x}}
6244:
6238:
6160:
6156:
6150:
6141:
6135:
6129:
6123:
6111:
6065:
6030:
6017:
5831:{\displaystyle (f,K_{x})=f(x)}
5825:
5819:
5810:
5791:
5748:
5742:
5704:
5631:
5625:
5616:
5610:
5597:
5569:
5482:
5461:
5448:
5423:
5308:
5304:
5292:
5280:
5274:
5268:
5134:
5122:
5076:
5064:
5022:
5016:
4977:
4971:
4958:
4952:
4923:
4911:
4888:
4876:
4830:
4824:
4801:
4789:
4760:
4754:
4725:
4719:
4676:
4670:
4664:
4658:
4652:
4640:
4601:
4595:
4425:
4419:
4413:
4407:
4330:
4324:
4318:
4312:
4306:
4294:
4206:
4200:
4194:
4182:
4155:
4149:
4140:
4134:
4098:
4086:
4077:
4065:
4013:
3996:
3976:
3959:
3653:
3623:
3562:
3551:
3542:
3536:
3382:
3336:
3217:
3190:
3118:
3097:
3084:
3002:
2999:
2987:
2981:
2969:
2957:
2900:
2875:
2806:
2794:
2771:
2756:
2679:
2665:
2647:
2635:
2530:
2514:
2384:
2368:
2276:
2249:
2243:
2227:
2146:
2130:
1862:
1836:
1799:
1796:
1764:
1758:
1726:
1723:
1700:
1674:
1637:
1634:
1602:
1596:
1564:
1561:
1533:
1460:
1446:
1409:
1391:
1351:
1078:
1022:
1008:
932:
906:
695:
683:
671:
659:
627:
577:
551:
510:
501:
315:
289:
205:{\displaystyle {\mathcal {X}}}
170:
130:{\displaystyle {\mathcal {X}}}
91:partial differential equations
1:
7792:of an unknown model function
7518:{\displaystyle K=(K_{n})^{n}}
5407:a Hilbert space of functions
4543:{\displaystyle \{\psi _{n}\}}
4104:{\displaystyle K(x,y)=f(x-y)}
1001:For a family of p.d. kernels
53:. It was first introduced by
41:, a branch of mathematics, a
8948:response surface methodology
8933:Stinespring dilation theorem
8927:Stinespring dilation theorem
8446:{\displaystyle \epsilon (x)}
6658:{\displaystyle d(x,y)\geq 0}
6074:{\displaystyle \Phi :X\to F}
4574:{\displaystyle \lambda _{n}}
1379:is p.d. if the limit exists.
967:
892:
699:
8473:{\displaystyle \sigma ^{2}}
7997:is not a fixed function of
7744:statistical learning theory
7732:Kernels in machine learning
5346:of locally compact groups.
4117:Fredholm integral equations
1439:is a sequence of sets, and
59:integral operator equations
9193:
8930:
8892:
7746:because of the celebrated
7735:
5546:the evaluation functional
5356:
5334:in 1929, and continued by
5199:{\displaystyle x_{i}\in E}
5082:{\displaystyle x\in L^{1}}
5034:{\displaystyle J(x)\geq 0}
4842:{\displaystyle J(x)\geq 0}
4552:orthonormal eigenfunctions
4047:
47:positive-definite function
29:
7401:into some Hilbert space.
5986:as a reproducing kernel.
5908:Every reproducing kernel
5226:there is a Hilbert space
5166:{\displaystyle E\times E}
4737:{\displaystyle J(x)>0}
4581:’s are the corresponding
45:is a generalization of a
8384:{\displaystyle K(x,y)=E}
8053:{\displaystyle Z(x_{i})}
7566:{\displaystyle d(x,y)=0}
6699:{\displaystyle d(x,y)=0}
6010:be a Hilbert space, and
4550:is a complete system of
2933:{\displaystyle B_{\nu }}
2083:Examples of p.d. kernels
770:and any complex numbers
51:positive-definite matrix
43:positive-definite kernel
7592:{\displaystyle x\neq y}
6884:: A symmetric function
5344:unitary representations
1983:. Then the restriction
996:Some general properties
87:boundary-value problems
75:complex function-theory
8878:
8837:
8779:
8758:
8708:, from a large sample
8702:
8678:
8654:
8554:
8534:
8514:
8494:
8474:
8447:
8414:
8385:
8311:
8282:
8253:
8233:
8213:
8184:
8155:
8119:
8099:
8054:
8018:
7991:
7964:
7937:
7850:
7826:
7806:
7786:
7718:
7613:
7593:
7567:
7519:
7473:
7446:
7418:
7392:
7356:
7336:
7312:
7262:
7254:
7233:
7200:
7146:
7057:
6994:
6950:
6926:
6868:
6787:
6726:
6700:
6659:
6615:
6581:
6557:
6534:
6514:
6482:
6481:{\displaystyle K(x,y)}
6453:are through the value
6447:
6427:
6407:
6290:
6289:{\displaystyle x\in X}
6264:
6222:
6196:
6176:
6095:
6075:
6043:
6004:
5980:
5960:
5922:
5896:
5884:
5883:{\displaystyle x\in X}
5858:
5857:{\displaystyle f\in H}
5832:
5776:
5716:
5673:
5665:
5638:
5581:
5540:
5539:{\displaystyle x\in X}
5514:
5494:
5435:
5401:
5381:
5321:
5240:
5220:
5200:
5167:
5141:
5140:{\displaystyle K(x,y)}
5106:
5083:
5035:
4997:
4895:
4863:
4843:
4808:
4807:{\displaystyle K(s,t)}
4773:
4772:{\displaystyle x(t)=0}
4738:
4703:
4575:
4544:
4511:
4491:
4455:
4227:
4105:
4034:
3943:
3766:
3578:
3464:
3443:
3407:
3360:
3299:
3255:
3169:
3130:
3067:
3042:
2934:
2907:
2778:
2722:
2616:
2495:
2345:
2205:
2111:
2078:is also a p.d. kernel.
2072:
2024:
2004:
1977:
1933:
1869:
1825:
1707:
1663:
1545:
1433:
1373:
1322:
1268:
1202:
1150:
1125:
1090:
982:
945:
875:
854:
818:
764:
708:
639:
584:
517:
468:
400:
328:
265:
244:
206:
182:
131:
8931:Further information:
8893:Further information:
8884:as a smooth estimate.
8879:
8817:
8780:
8759:
8703:
8679:
8655:
8555:
8535:
8515:
8495:
8475:
8448:
8415:
8386:
8312:
8283:
8254:
8234:
8214:
8185:
8161:the random variables
8156:
8120:
8100:
8055:
8019:
8017:{\displaystyle x_{i}}
7992:
7990:{\displaystyle x_{i}}
7965:
7963:{\displaystyle f_{i}}
7938:
7851:
7827:
7807:
7787:
7736:Further information:
7719:
7614:
7594:
7568:
7520:
7474:
7472:{\displaystyle K_{n}}
7447:
7419:
7393:
7357:
7337:
7313:
7255:
7213:
7201:
7147:
7058:
6968:
6951:
6927:
6879:
6869:
6788:
6727:
6701:
6660:
6616:
6582:
6558:
6535:
6515:
6492:Kernels and distances
6483:
6448:
6428:
6408:
6291:
6265:
6223:
6197:
6177:
6096:
6076:
6044:
6005:
5981:
5961:
5923:
5885:
5859:
5833:
5777:
5717:
5677:
5666:
5646:
5639:
5582:
5541:
5515:
5495:
5436:
5402:
5382:
5357:Further information:
5322:
5241:
5221:
5201:
5168:
5142:
5107:
5084:
5036:
4998:
4896:
4864:
4844:
4809:
4774:
4739:
4704:
4576:
4545:
4512:
4492:
4456:
4228:
4106:
4035:
3944:
3767:
3579:
3465:
3463:{\displaystyle \chi }
3444:
3408:
3340:
3279:
3235:
3143:
3131:
3068:
3043:
2935:
2908:
2779:
2723:
2617:
2496:
2346:
2206:
2112:
2073:
2025:
2005:
2003:{\displaystyle K_{0}}
1978:
1934:
1870:
1805:
1708:
1643:
1546:
1434:
1374:
1323:
1269:
1203:
1151:
1105:
1091:
983:
946:
855:
834:
819:
765:
709:
640:
585:
518:
469:
401:
329:
245:
224:
207:
183:
132:
8946:. One such topic is
8789:
8769:
8712:
8688:
8668:
8564:
8544:
8524:
8504:
8484:
8457:
8428:
8413:{\displaystyle Z(x)}
8395:
8321:
8310:{\displaystyle Z(y)}
8292:
8281:{\displaystyle Z(x)}
8263:
8243:
8223:
8212:{\displaystyle Z(y)}
8194:
8183:{\displaystyle Z(x)}
8165:
8129:
8109:
8064:
8028:
8001:
7974:
7947:
7860:
7836:
7816:
7796:
7785:{\displaystyle f(x)}
7767:
7623:
7603:
7577:
7536:
7483:
7456:
7428:
7408:
7366:
7346:
7322:
7268:
7210:
7156:
7083:
6965:
6936:
6888:
6798:
6738:
6710:
6669:
6628:
6591:
6571:
6547:
6524:
6504:
6457:
6437:
6417:
6300:
6274:
6232:
6206:
6186:
6105:
6085:
6053:
6014:
5994:
5970:
5950:
5912:
5868:
5842:
5788:
5729:
5686:
5655:
5591:
5550:
5524:
5504:
5445:
5411:
5391:
5371:
5250:
5230:
5210:
5177:
5151:
5116:
5096:
5045:
5010:
4905:
4873:
4853:
4818:
4783:
4748:
4713:
4589:
4558:
4521:
4501:
4481:
4258:
4128:
4119:of the second kind:
4059:
3953:
3775:
3587:
3474:
3454:
3419:
3140:
3081:
3057:
2951:
2917:
2788:
2738:
2629:
2508:
2362:
2221:
2124:
2092:
2034:
2014:
1987:
1946:
1879:
1875:are p.d. kernels on
1717:
1555:
1443:
1388:
1334:
1278:
1214:
1160:
1102:
1005:
958:
831:
774:
718:
653:
601:
527:
481:
410:
354:
221:
192:
144:
117:
8972:Covariance function
8520:and independent of
7748:representer theorem
6725:{\displaystyle x=y}
6221:{\displaystyle F=H}
5940: —
5906: —
4636:
4621:
4396:
4290:
4275:
4178:
3922:
3844:
3753:
3689:
3651:
3438:
3415:Kernels defined on
3396:
2851:
2755:
1479:
1428:
1263:
1238:
105:, and other areas.
8982:Integral transform
8938:Other applications
8874:
8775:
8754:
8698:
8674:
8650:
8550:
8530:
8510:
8490:
8470:
8443:
8410:
8381:
8307:
8278:
8249:
8229:
8209:
8180:
8151:
8115:
8095:
8050:
8014:
7987:
7960:
7933:
7846:
7822:
7802:
7782:
7714:
7609:
7589:
7563:
7515:
7469:
7442:
7414:
7388:
7352:
7332:
7308:
7250:
7196:
7142:
7053:
6946:
6922:
6864:
6783:
6722:
6696:
6655:
6611:
6577:
6553:
6530:
6510:
6478:
6443:
6423:
6403:
6286:
6260:
6218:
6192:
6172:
6091:
6071:
6039:
6000:
5976:
5956:
5938:
5918:
5904:
5880:
5854:
5828:
5772:
5712:
5661:
5634:
5577:
5536:
5510:
5490:
5431:
5397:
5377:
5317:
5236:
5216:
5196:
5163:
5137:
5102:
5079:
5031:
4993:
4938:
4891:
4859:
4839:
4804:
4769:
4734:
4699:
4622:
4607:
4571:
4540:
4507:
4487:
4451:
4382:
4276:
4261:
4223:
4164:
4101:
4030:
3939:
3910:
3884:
3832:
3807:
3762:
3741:
3722:
3677:
3639:
3619:
3574:
3460:
3439:
3422:
3403:
3334:
3126:
3063:
3038:
2930:
2903:
2824:
2774:
2741:
2731:Kernel generating
2718:
2612:
2504:Laplacian kernel:
2491:
2341:
2201:
2107:
2068:
2020:
2000:
1973:
1929:
1865:
1703:
1541:
1459:
1429:
1408:
1369:
1358:
1318:
1264:
1242:
1217:
1198:
1146:
1086:
978:
941:
814:
760:
704:
635:
580:
513:
464:
396:
324:
202:
178:
139:symmetric function
127:
103:information theory
83:integral equations
67:probability theory
9026:978-0-8218-1567-0
8977:Integral equation
8956:computer graphics
8952:rapid prototyping
8868:
8815:
8778:{\displaystyle K}
8677:{\displaystyle f}
8553:{\displaystyle f}
8533:{\displaystyle Z}
8513:{\displaystyle x}
8493:{\displaystyle x}
8252:{\displaystyle y}
8232:{\displaystyle x}
8118:{\displaystyle f}
8098:{\displaystyle E}
7825:{\displaystyle x}
7805:{\displaystyle f}
7727:Some applications
7712:
7612:{\displaystyle K}
7417:{\displaystyle K}
7355:{\displaystyle d}
7077:
7076:
6580:{\displaystyle d}
6556:{\displaystyle X}
6533:{\displaystyle d}
6513:{\displaystyle K}
6498:nearest neighbors
6446:{\displaystyle y}
6426:{\displaystyle x}
6195:{\displaystyle K}
6094:{\displaystyle F}
6003:{\displaystyle F}
5979:{\displaystyle K}
5959:{\displaystyle K}
5921:{\displaystyle K}
5664:{\displaystyle H}
5513:{\displaystyle H}
5400:{\displaystyle H}
5380:{\displaystyle X}
5239:{\displaystyle H}
5219:{\displaystyle K}
5105:{\displaystyle E}
4991:
4929:
4862:{\displaystyle x}
4681:
4510:{\displaystyle x}
4490:{\displaystyle K}
4475:
4474:
4374:
4335:
4247:
4246:
4211:
3923:
3903:
3875:
3845:
3825:
3798:
3713:
3691:
3610:
3569:
3521:
3066:{\displaystyle H}
2871:
2848:
2443:
2354:Gaussian kernel (
2215:Polynomial kernel
2023:{\displaystyle K}
1488:
1485:
1343:
1047:
1044:
990:complex conjugate
970:
895:
702:
348:
347:
99:embedding problem
16:(Redirected from
9184:
9157:
9150:
9144:
9141:
9135:
9132:
9126:
9123:
9117:
9114:
9108:
9105:
9099:
9095:
9089:
9086:
9080:
9077:
9071:
9068:
9062:
9059:
9053:
9050:
9044:
9037:
9031:
9030:
9012:
8901:meshfree methods
8895:Meshfree methods
8883:
8881:
8880:
8875:
8873:
8869:
8864:
8863:
8862:
8846:
8836:
8831:
8816:
8808:
8784:
8782:
8781:
8776:
8763:
8761:
8760:
8755:
8753:
8752:
8743:
8742:
8724:
8723:
8707:
8705:
8704:
8699:
8697:
8696:
8683:
8681:
8680:
8675:
8659:
8657:
8656:
8651:
8649:
8648:
8636:
8635:
8559:
8557:
8556:
8551:
8539:
8537:
8536:
8531:
8519:
8517:
8516:
8511:
8499:
8497:
8496:
8491:
8479:
8477:
8476:
8471:
8469:
8468:
8452:
8450:
8449:
8444:
8419:
8417:
8416:
8411:
8390:
8388:
8387:
8382:
8316:
8314:
8313:
8308:
8287:
8285:
8284:
8279:
8258:
8256:
8255:
8250:
8239:is too close to
8238:
8236:
8235:
8230:
8218:
8216:
8215:
8210:
8189:
8187:
8186:
8181:
8160:
8158:
8157:
8152:
8150:
8149:
8124:
8122:
8121:
8116:
8104:
8102:
8101:
8096:
8088:
8087:
8059:
8057:
8056:
8051:
8046:
8045:
8023:
8021:
8020:
8015:
8013:
8012:
7996:
7994:
7993:
7988:
7986:
7985:
7969:
7967:
7966:
7961:
7959:
7958:
7942:
7940:
7939:
7934:
7926:
7925:
7907:
7906:
7888:
7887:
7875:
7874:
7855:
7853:
7852:
7847:
7845:
7844:
7831:
7829:
7828:
7823:
7811:
7809:
7808:
7803:
7791:
7789:
7788:
7783:
7723:
7721:
7720:
7715:
7713:
7648:
7618:
7616:
7615:
7610:
7598:
7596:
7595:
7590:
7572:
7570:
7569:
7564:
7524:
7522:
7521:
7516:
7514:
7513:
7504:
7503:
7478:
7476:
7475:
7470:
7468:
7467:
7451:
7449:
7448:
7443:
7441:
7423:
7421:
7420:
7415:
7397:
7395:
7394:
7389:
7378:
7377:
7361:
7359:
7358:
7353:
7341:
7339:
7338:
7333:
7331:
7330:
7317:
7315:
7314:
7309:
7304:
7303:
7259:
7257:
7256:
7251:
7243:
7242:
7232:
7227:
7205:
7203:
7202:
7197:
7195:
7187:
7186:
7168:
7167:
7151:
7149:
7148:
7143:
7138:
7137:
7128:
7127:
7109:
7108:
7096:
7071:
7062:
7060:
7059:
7054:
7043:
7042:
7030:
7029:
7014:
7013:
7004:
7003:
6993:
6988:
6959:
6955:
6953:
6952:
6947:
6945:
6944:
6931:
6929:
6928:
6923:
6921:
6913:
6912:
6903:
6902:
6873:
6871:
6870:
6865:
6792:
6790:
6789:
6784:
6731:
6729:
6728:
6723:
6705:
6703:
6702:
6697:
6664:
6662:
6661:
6656:
6621:which satisfies
6620:
6618:
6617:
6612:
6610:
6609:
6600:
6599:
6586:
6584:
6583:
6578:
6562:
6560:
6559:
6554:
6539:
6537:
6536:
6531:
6519:
6517:
6516:
6511:
6487:
6485:
6484:
6479:
6452:
6450:
6449:
6444:
6432:
6430:
6429:
6424:
6412:
6410:
6409:
6404:
6381:
6380:
6371:
6370:
6358:
6357:
6342:
6341:
6295:
6293:
6292:
6287:
6269:
6267:
6266:
6261:
6259:
6258:
6227:
6225:
6224:
6219:
6201:
6199:
6198:
6193:
6181:
6179:
6178:
6173:
6168:
6167:
6100:
6098:
6097:
6092:
6080:
6078:
6077:
6072:
6048:
6046:
6045:
6040:
6038:
6037:
6009:
6007:
6006:
6001:
5985:
5983:
5982:
5977:
5965:
5963:
5962:
5957:
5941:
5927:
5925:
5924:
5919:
5907:
5889:
5887:
5886:
5881:
5863:
5861:
5860:
5855:
5837:
5835:
5834:
5829:
5809:
5808:
5781:
5779:
5778:
5773:
5741:
5740:
5721:
5719:
5718:
5713:
5711:
5670:
5668:
5667:
5662:
5643:
5641:
5640:
5635:
5609:
5608:
5586:
5584:
5583:
5578:
5576:
5562:
5561:
5545:
5543:
5542:
5537:
5519:
5517:
5516:
5511:
5499:
5497:
5496:
5491:
5489:
5469:
5468:
5440:
5438:
5437:
5432:
5430:
5406:
5404:
5403:
5398:
5386:
5384:
5383:
5378:
5326:
5324:
5323:
5318:
5316:
5315:
5245:
5243:
5242:
5237:
5225:
5223:
5222:
5217:
5205:
5203:
5202:
5197:
5189:
5188:
5172:
5170:
5169:
5164:
5146:
5144:
5143:
5138:
5111:
5109:
5108:
5103:
5088:
5086:
5085:
5080:
5063:
5062:
5040:
5038:
5037:
5032:
5002:
5000:
4999:
4994:
4992:
4990:
4989:
4980:
4970:
4969:
4951:
4950:
4940:
4937:
4900:
4898:
4897:
4894:{\displaystyle }
4892:
4868:
4866:
4865:
4860:
4848:
4846:
4845:
4840:
4813:
4811:
4810:
4805:
4778:
4776:
4775:
4770:
4743:
4741:
4740:
4735:
4708:
4706:
4705:
4700:
4695:
4686:
4679:
4635:
4630:
4620:
4615:
4580:
4578:
4577:
4572:
4570:
4569:
4549:
4547:
4546:
4541:
4536:
4535:
4516:
4514:
4513:
4508:
4496:
4494:
4493:
4488:
4469:
4460:
4458:
4457:
4452:
4447:
4446:
4441:
4437:
4433:
4406:
4405:
4395:
4390:
4375:
4373:
4372:
4360:
4349:
4340:
4333:
4289:
4284:
4274:
4269:
4252:
4241:
4232:
4230:
4229:
4224:
4216:
4209:
4177:
4172:
4122:
4110:
4108:
4107:
4102:
4050:Mercer's theorem
4039:
4037:
4036:
4031:
4017:
4016:
4012:
3975:
3948:
3946:
3945:
3940:
3935:
3934:
3929:
3925:
3924:
3918:
3909:
3904:
3902:
3901:
3892:
3883:
3871:
3870:
3869:
3868:
3851:
3847:
3846:
3840:
3831:
3826:
3824:
3823:
3814:
3806:
3794:
3793:
3792:
3791:
3771:
3769:
3768:
3763:
3758:
3754:
3749:
3737:
3736:
3721:
3709:
3708:
3692:
3690:
3685:
3673:
3672:
3662:
3661:
3660:
3647:
3635:
3634:
3621:
3618:
3606:
3605:
3604:
3603:
3583:
3581:
3580:
3575:
3570:
3565:
3561:
3531:
3526:
3522:
3517:
3516:
3501:
3489:
3488:
3469:
3467:
3466:
3461:
3448:
3446:
3445:
3440:
3437:
3432:
3427:
3412:
3410:
3409:
3404:
3395:
3390:
3385:
3381:
3380:
3379:
3370:
3369:
3359:
3354:
3330:
3329:
3324:
3320:
3319:
3318:
3309:
3308:
3298:
3293:
3275:
3274:
3265:
3264:
3254:
3249:
3225:
3224:
3215:
3214:
3202:
3201:
3189:
3188:
3179:
3178:
3168:
3163:
3135:
3133:
3132:
3127:
3125:
3105:
3104:
3072:
3070:
3069:
3064:
3047:
3045:
3044:
3039:
3025:
2939:
2937:
2936:
2931:
2929:
2928:
2912:
2910:
2909:
2904:
2899:
2898:
2874:
2873:
2872:
2864:
2850:
2849:
2841:
2832:
2783:
2781:
2780:
2775:
2770:
2769:
2764:
2754:
2749:
2727:
2725:
2724:
2719:
2705:
2684:
2683:
2682:
2668:
2621:
2619:
2618:
2613:
2599:
2598:
2593:
2584:
2576:
2567:
2566:
2562:
2554:
2529:
2521:
2500:
2498:
2497:
2492:
2478:
2477:
2472:
2463:
2455:
2446:
2445:
2444:
2442:
2441:
2440:
2427:
2426:
2425:
2416:
2408:
2399:
2383:
2375:
2350:
2348:
2347:
2342:
2316:
2315:
2310:
2301:
2293:
2284:
2283:
2268:
2263:
2262:
2257:
2242:
2234:
2210:
2208:
2207:
2202:
2200:
2199:
2194:
2185:
2177:
2168:
2163:
2162:
2157:
2145:
2137:
2116:
2114:
2113:
2108:
2106:
2105:
2100:
2077:
2075:
2074:
2069:
2067:
2066:
2061:
2060:
2050:
2049:
2044:
2043:
2029:
2027:
2026:
2021:
2009:
2007:
2006:
2001:
1999:
1998:
1982:
1980:
1979:
1974:
1972:
1971:
1962:
1961:
1956:
1955:
1938:
1936:
1935:
1930:
1928:
1927:
1922:
1921:
1905:
1904:
1899:
1898:
1888:
1887:
1874:
1872:
1871:
1866:
1861:
1860:
1848:
1847:
1835:
1834:
1824:
1819:
1795:
1794:
1776:
1775:
1757:
1756:
1738:
1737:
1712:
1710:
1709:
1704:
1699:
1698:
1686:
1685:
1673:
1672:
1662:
1657:
1633:
1632:
1614:
1613:
1595:
1594:
1576:
1575:
1550:
1548:
1547:
1542:
1540:
1532:
1531:
1526:
1525:
1515:
1514:
1509:
1508:
1498:
1497:
1486:
1483:
1478:
1473:
1458:
1457:
1438:
1436:
1435:
1430:
1427:
1422:
1407:
1406:
1401:
1400:
1378:
1376:
1375:
1370:
1368:
1367:
1357:
1327:
1325:
1324:
1319:
1317:
1309:
1308:
1290:
1289:
1273:
1271:
1270:
1265:
1262:
1261:
1260:
1250:
1237:
1236:
1235:
1225:
1207:
1205:
1204:
1199:
1191:
1190:
1172:
1171:
1155:
1153:
1152:
1147:
1145:
1144:
1135:
1134:
1124:
1119:
1098:The conical sum
1095:
1093:
1092:
1087:
1085:
1077:
1076:
1067:
1066:
1057:
1056:
1045:
1042:
1038:
1037:
1036:
1020:
1019:
987:
985:
984:
979:
977:
976:
971:
963:
950:
948:
947:
942:
931:
930:
918:
917:
902:
901:
896:
888:
885:
884:
874:
869:
853:
848:
823:
821:
820:
815:
813:
805:
804:
786:
785:
769:
767:
766:
761:
759:
758:
749:
748:
730:
729:
713:
711:
710:
705:
703:
698:
678:
647:Hermitian kernel
644:
642:
641:
636:
634:
626:
625:
616:
615:
589:
587:
586:
581:
576:
575:
563:
562:
544:
543:
535:
522:
520:
519:
514:
493:
492:
473:
471:
470:
465:
463:
455:
454:
436:
435:
423:
405:
403:
402:
397:
395:
394:
385:
384:
366:
365:
342:
333:
331:
330:
325:
314:
313:
301:
300:
285:
284:
275:
274:
264:
259:
243:
238:
215:
211:
209:
208:
203:
201:
200:
187:
185:
184:
179:
177:
169:
168:
159:
158:
136:
134:
133:
128:
126:
125:
95:machine learning
63:Fourier analysis
21:
9192:
9191:
9187:
9186:
9185:
9183:
9182:
9181:
9172:Operator theory
9162:
9161:
9160:
9151:
9147:
9142:
9138:
9133:
9129:
9124:
9120:
9115:
9111:
9106:
9102:
9096:
9092:
9087:
9083:
9078:
9074:
9069:
9065:
9060:
9056:
9051:
9047:
9038:
9034:
9027:
9014:
9013:
9009:
9005:
8968:
8940:
8935:
8929:
8897:
8891:
8854:
8847:
8841:
8787:
8786:
8767:
8766:
8734:
8715:
8710:
8709:
8686:
8685:
8666:
8665:
8637:
8627:
8562:
8561:
8542:
8541:
8522:
8521:
8502:
8501:
8482:
8481:
8460:
8455:
8454:
8426:
8425:
8393:
8392:
8319:
8318:
8290:
8289:
8261:
8260:
8241:
8240:
8221:
8220:
8192:
8191:
8163:
8162:
8127:
8126:
8107:
8106:
8105:which replaces
8079:
8062:
8061:
8037:
8026:
8025:
8004:
7999:
7998:
7977:
7972:
7971:
7950:
7945:
7944:
7917:
7898:
7879:
7866:
7858:
7857:
7834:
7833:
7814:
7813:
7812:at a new point
7794:
7793:
7765:
7764:
7757:
7740:
7734:
7729:
7621:
7620:
7601:
7600:
7575:
7574:
7534:
7533:
7505:
7495:
7481:
7480:
7459:
7454:
7453:
7426:
7425:
7406:
7405:
7364:
7363:
7344:
7343:
7320:
7319:
7266:
7265:
7234:
7208:
7207:
7178:
7159:
7154:
7153:
7119:
7100:
7081:
7080:
7069:
7034:
7021:
7005:
6995:
6963:
6962:
6934:
6933:
6886:
6885:
6796:
6795:
6736:
6735:
6708:
6707:
6706:if and only if
6667:
6666:
6626:
6625:
6589:
6588:
6569:
6568:
6545:
6544:
6522:
6521:
6502:
6501:
6494:
6455:
6454:
6435:
6434:
6415:
6414:
6372:
6362:
6349:
6333:
6298:
6297:
6272:
6271:
6250:
6230:
6229:
6204:
6203:
6184:
6183:
6159:
6103:
6102:
6083:
6082:
6051:
6050:
6029:
6012:
6011:
5992:
5991:
5968:
5967:
5948:
5947:
5944:
5939:
5930:
5910:
5909:
5905:
5866:
5865:
5840:
5839:
5800:
5786:
5785:
5732:
5727:
5726:
5684:
5683:
5653:
5652:
5600:
5589:
5588:
5553:
5548:
5547:
5522:
5521:
5502:
5501:
5460:
5443:
5442:
5409:
5408:
5389:
5388:
5369:
5368:
5361:
5355:
5307:
5248:
5247:
5228:
5227:
5208:
5207:
5180:
5175:
5174:
5149:
5148:
5114:
5113:
5094:
5093:
5054:
5043:
5042:
5008:
5007:
4981:
4961:
4942:
4941:
4903:
4902:
4871:
4870:
4851:
4850:
4816:
4815:
4781:
4780:
4746:
4745:
4711:
4710:
4587:
4586:
4561:
4556:
4555:
4527:
4519:
4518:
4517:is continuous,
4499:
4498:
4479:
4478:
4467:
4397:
4381:
4377:
4376:
4364:
4256:
4255:
4239:
4126:
4125:
4057:
4056:
4052:
4046:
4005:
3982:
3968:
3951:
3950:
3893:
3890:
3886:
3885:
3860:
3855:
3815:
3812:
3808:
3783:
3778:
3773:
3772:
3728:
3727:
3723:
3697:
3664:
3663:
3652:
3626:
3622:
3595:
3590:
3585:
3584:
3554:
3532:
3509:
3502:
3496:
3477:
3472:
3471:
3452:
3451:
3417:
3416:
3371:
3361:
3339:
3335:
3310:
3300:
3266:
3256:
3234:
3230:
3229:
3216:
3206:
3193:
3180:
3170:
3138:
3137:
3096:
3079:
3078:
3055:
3054:
2949:
2948:
2920:
2915:
2914:
2890:
2852:
2786:
2785:
2759:
2736:
2735:
2653:
2627:
2626:
2588:
2536:
2506:
2505:
2467:
2432:
2428:
2417:
2400:
2390:
2360:
2359:
2305:
2275:
2252:
2219:
2218:
2189:
2152:
2122:
2121:
2120:Linear kernel:
2095:
2090:
2089:
2085:
2054:
2037:
2032:
2031:
2012:
2011:
1990:
1985:
1984:
1949:
1944:
1943:
1915:
1892:
1877:
1876:
1852:
1839:
1826:
1786:
1767:
1748:
1729:
1715:
1714:
1690:
1677:
1664:
1624:
1605:
1586:
1567:
1553:
1552:
1519:
1502:
1489:
1449:
1441:
1440:
1394:
1386:
1385:
1359:
1332:
1331:
1300:
1281:
1276:
1275:
1274:is p.d., given
1252:
1227:
1212:
1211:
1182:
1163:
1158:
1157:
1156:is p.d., given
1136:
1126:
1100:
1099:
1048:
1021:
1011:
1003:
1002:
998:
961:
956:
955:
922:
909:
886:
876:
829:
828:
796:
777:
772:
771:
740:
721:
716:
715:
679:
651:
650:
599:
598:
567:
554:
530:
525:
524:
484:
479:
478:
446:
427:
408:
407:
376:
357:
352:
351:
340:
305:
292:
276:
266:
219:
218:
190:
189:
142:
141:
115:
114:
111:
79:moment problems
71:operator theory
39:operator theory
35:
32:Integral kernel
28:
23:
22:
15:
12:
11:
5:
9190:
9188:
9180:
9179:
9177:Hilbert spaces
9174:
9164:
9163:
9159:
9158:
9145:
9136:
9127:
9118:
9109:
9100:
9090:
9081:
9072:
9063:
9054:
9045:
9032:
9025:
9006:
9004:
9001:
9000:
8999:
8994:
8989:
8984:
8979:
8974:
8967:
8964:
8962:environments.
8939:
8936:
8928:
8925:
8890:
8887:
8886:
8885:
8872:
8867:
8861:
8857:
8853:
8850:
8844:
8840:
8835:
8830:
8827:
8824:
8820:
8814:
8811:
8806:
8803:
8800:
8797:
8794:
8774:
8751:
8746:
8741:
8737:
8733:
8730:
8727:
8722:
8718:
8695:
8673:
8647:
8644:
8640:
8634:
8630:
8626:
8623:
8620:
8617:
8614:
8611:
8608:
8605:
8602:
8599:
8596:
8593:
8590:
8587:
8584:
8581:
8578:
8575:
8572:
8569:
8549:
8529:
8509:
8489:
8480:, is added to
8467:
8463:
8442:
8439:
8436:
8433:
8422:
8421:
8409:
8406:
8403:
8400:
8380:
8377:
8374:
8371:
8368:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8344:
8341:
8338:
8335:
8332:
8329:
8326:
8306:
8303:
8300:
8297:
8277:
8274:
8271:
8268:
8248:
8228:
8208:
8205:
8202:
8199:
8179:
8176:
8173:
8170:
8148:
8143:
8140:
8137:
8134:
8114:
8094:
8091:
8086:
8082:
8078:
8075:
8072:
8069:
8049:
8044:
8040:
8036:
8033:
8011:
8007:
7984:
7980:
7957:
7953:
7932:
7929:
7924:
7920:
7916:
7913:
7910:
7905:
7901:
7897:
7894:
7891:
7886:
7882:
7878:
7873:
7869:
7865:
7843:
7821:
7801:
7781:
7778:
7775:
7772:
7756:
7753:
7733:
7730:
7728:
7725:
7711:
7708:
7705:
7702:
7699:
7696:
7693:
7690:
7687:
7684:
7681:
7678:
7675:
7672:
7669:
7666:
7663:
7660:
7657:
7654:
7651:
7646:
7643:
7640:
7637:
7634:
7631:
7628:
7608:
7588:
7585:
7582:
7562:
7559:
7556:
7553:
7550:
7547:
7544:
7541:
7512:
7508:
7502:
7498:
7494:
7491:
7488:
7466:
7462:
7440:
7436:
7433:
7413:
7387:
7384:
7381:
7376:
7371:
7351:
7329:
7307:
7302:
7297:
7294:
7291:
7288:
7285:
7282:
7279:
7276:
7273:
7249:
7246:
7241:
7237:
7231:
7226:
7223:
7220:
7216:
7194:
7190:
7185:
7181:
7177:
7174:
7171:
7166:
7162:
7141:
7136:
7131:
7126:
7122:
7118:
7115:
7112:
7107:
7103:
7099:
7095:
7091:
7088:
7079:holds for any
7075:
7074:
7065:
7063:
7052:
7049:
7046:
7041:
7037:
7033:
7028:
7024:
7020:
7017:
7012:
7008:
7002:
6998:
6992:
6987:
6984:
6981:
6978:
6975:
6971:
6943:
6920:
6916:
6911:
6906:
6901:
6896:
6893:
6875:
6874:
6863:
6860:
6857:
6854:
6851:
6848:
6845:
6842:
6839:
6836:
6833:
6830:
6827:
6824:
6821:
6818:
6815:
6812:
6809:
6806:
6803:
6793:
6782:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6733:
6721:
6718:
6715:
6695:
6692:
6689:
6686:
6683:
6680:
6677:
6674:
6654:
6651:
6648:
6645:
6642:
6639:
6636:
6633:
6608:
6603:
6598:
6576:
6552:
6529:
6520:and distances
6509:
6493:
6490:
6477:
6474:
6471:
6468:
6465:
6462:
6442:
6422:
6402:
6399:
6396:
6393:
6390:
6387:
6384:
6379:
6375:
6369:
6365:
6361:
6356:
6352:
6348:
6345:
6340:
6336:
6332:
6329:
6326:
6323:
6320:
6317:
6314:
6311:
6308:
6305:
6285:
6282:
6279:
6257:
6253:
6249:
6246:
6243:
6240:
6237:
6217:
6214:
6211:
6191:
6171:
6166:
6162:
6158:
6155:
6152:
6149:
6146:
6143:
6140:
6137:
6134:
6131:
6128:
6125:
6122:
6119:
6116:
6113:
6110:
6090:
6070:
6067:
6064:
6061:
6058:
6036:
6032:
6028:
6025:
6022:
6019:
5999:
5975:
5955:
5934:
5917:
5900:
5892:
5891:
5879:
5876:
5873:
5853:
5850:
5847:
5827:
5824:
5821:
5818:
5815:
5812:
5807:
5803:
5799:
5796:
5793:
5783:
5771:
5768:
5765:
5762:
5759:
5756:
5753:
5750:
5747:
5744:
5739:
5735:
5710:
5706:
5703:
5700:
5697:
5694:
5691:
5660:
5633:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5607:
5603:
5599:
5596:
5587:is defined by
5575:
5571:
5568:
5565:
5560:
5556:
5535:
5532:
5529:
5509:
5488:
5484:
5481:
5478:
5475:
5472:
5467:
5463:
5459:
5456:
5453:
5450:
5429:
5425:
5422:
5419:
5416:
5396:
5376:
5354:
5351:
5314:
5310:
5306:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5279:
5276:
5273:
5270:
5267:
5264:
5261:
5258:
5255:
5235:
5215:
5195:
5192:
5187:
5183:
5162:
5159:
5156:
5136:
5133:
5130:
5127:
5124:
5121:
5101:
5078:
5075:
5072:
5069:
5066:
5061:
5057:
5053:
5050:
5030:
5027:
5024:
5021:
5018:
5015:
4988:
4984:
4979:
4976:
4973:
4968:
4964:
4960:
4957:
4954:
4949:
4945:
4936:
4932:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4890:
4887:
4884:
4881:
4878:
4858:
4838:
4835:
4832:
4829:
4826:
4823:
4803:
4800:
4797:
4794:
4791:
4788:
4768:
4765:
4762:
4759:
4756:
4753:
4733:
4730:
4727:
4724:
4721:
4718:
4698:
4694:
4689:
4685:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4634:
4629:
4625:
4619:
4614:
4610:
4606:
4603:
4600:
4597:
4594:
4568:
4564:
4539:
4534:
4530:
4526:
4506:
4486:
4473:
4472:
4463:
4461:
4450:
4445:
4440:
4436:
4432:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4404:
4400:
4394:
4389:
4385:
4380:
4371:
4367:
4363:
4358:
4355:
4352:
4348:
4343:
4339:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4288:
4283:
4279:
4273:
4268:
4264:
4245:
4244:
4235:
4233:
4222:
4219:
4215:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4176:
4171:
4167:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4045:
4042:
4041:
4040:
4029:
4026:
4023:
4020:
4015:
4011:
4008:
4004:
4001:
3998:
3995:
3992:
3989:
3985:
3981:
3978:
3974:
3971:
3967:
3964:
3961:
3958:
3938:
3933:
3928:
3921:
3917:
3913:
3907:
3900:
3896:
3889:
3882:
3878:
3874:
3867:
3863:
3858:
3854:
3850:
3843:
3839:
3835:
3829:
3822:
3818:
3811:
3805:
3801:
3797:
3790:
3786:
3781:
3761:
3757:
3752:
3748:
3744:
3740:
3735:
3731:
3726:
3720:
3716:
3712:
3707:
3704:
3700:
3695:
3688:
3684:
3680:
3676:
3671:
3667:
3659:
3655:
3650:
3646:
3642:
3638:
3633:
3629:
3625:
3617:
3613:
3609:
3602:
3598:
3593:
3573:
3568:
3564:
3560:
3557:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3529:
3525:
3520:
3515:
3512:
3508:
3505:
3499:
3495:
3492:
3487:
3484:
3480:
3459:
3436:
3431:
3426:
3413:
3402:
3399:
3394:
3389:
3384:
3378:
3374:
3368:
3364:
3358:
3353:
3350:
3347:
3343:
3338:
3333:
3328:
3323:
3317:
3313:
3307:
3303:
3297:
3292:
3289:
3286:
3282:
3278:
3273:
3269:
3263:
3259:
3253:
3248:
3245:
3242:
3238:
3233:
3228:
3223:
3219:
3213:
3209:
3205:
3200:
3196:
3192:
3187:
3183:
3177:
3173:
3167:
3162:
3159:
3156:
3153:
3150:
3146:
3124:
3120:
3117:
3114:
3111:
3108:
3103:
3099:
3095:
3092:
3089:
3086:
3062:
3051:
3050:
3049:
3037:
3034:
3031:
3028:
3024:
3020:
3017:
3014:
3011:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2945:
2927:
2923:
2902:
2897:
2893:
2889:
2886:
2883:
2880:
2877:
2870:
2867:
2862:
2859:
2855:
2847:
2844:
2839:
2836:
2831:
2827:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2773:
2768:
2763:
2758:
2753:
2748:
2744:
2733:Sobolev spaces
2729:
2717:
2714:
2711:
2708:
2704:
2700:
2697:
2694:
2691:
2687:
2681:
2677:
2674:
2671:
2667:
2663:
2660:
2656:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2623:
2611:
2608:
2605:
2602:
2597:
2592:
2587:
2583:
2579:
2575:
2570:
2565:
2561:
2557:
2553:
2549:
2546:
2543:
2539:
2535:
2532:
2528:
2524:
2520:
2516:
2513:
2502:
2490:
2487:
2484:
2481:
2476:
2471:
2466:
2462:
2458:
2454:
2449:
2439:
2435:
2431:
2424:
2420:
2415:
2411:
2407:
2403:
2397:
2393:
2389:
2386:
2382:
2378:
2374:
2370:
2367:
2352:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2314:
2309:
2304:
2300:
2296:
2292:
2287:
2282:
2278:
2274:
2271:
2267:
2261:
2256:
2251:
2248:
2245:
2241:
2237:
2233:
2229:
2226:
2212:
2198:
2193:
2188:
2184:
2180:
2176:
2171:
2167:
2161:
2156:
2151:
2148:
2144:
2140:
2136:
2132:
2129:
2104:
2099:
2084:
2081:
2080:
2079:
2065:
2059:
2053:
2048:
2042:
2019:
1997:
1993:
1970:
1965:
1960:
1954:
1940:
1926:
1920:
1914:
1911:
1908:
1903:
1897:
1891:
1886:
1864:
1859:
1855:
1851:
1846:
1842:
1838:
1833:
1829:
1823:
1818:
1815:
1812:
1808:
1804:
1801:
1798:
1793:
1789:
1785:
1782:
1779:
1774:
1770:
1766:
1763:
1760:
1755:
1751:
1747:
1744:
1741:
1736:
1732:
1728:
1725:
1722:
1702:
1697:
1693:
1689:
1684:
1680:
1676:
1671:
1667:
1661:
1656:
1653:
1650:
1646:
1642:
1639:
1636:
1631:
1627:
1623:
1620:
1617:
1612:
1608:
1604:
1601:
1598:
1593:
1589:
1585:
1582:
1579:
1574:
1570:
1566:
1563:
1560:
1539:
1535:
1530:
1524:
1518:
1513:
1507:
1501:
1496:
1492:
1482:
1477:
1472:
1469:
1466:
1462:
1456:
1452:
1448:
1426:
1421:
1418:
1415:
1411:
1405:
1399:
1393:
1382:
1381:
1380:
1366:
1362:
1356:
1353:
1350:
1346:
1342:
1339:
1328:
1316:
1312:
1307:
1303:
1299:
1296:
1293:
1288:
1284:
1259:
1255:
1249:
1245:
1241:
1234:
1230:
1224:
1220:
1208:
1197:
1194:
1189:
1185:
1181:
1178:
1175:
1170:
1166:
1143:
1139:
1133:
1129:
1123:
1118:
1115:
1112:
1108:
1084:
1080:
1075:
1070:
1065:
1060:
1055:
1051:
1041:
1035:
1031:
1028:
1024:
1018:
1014:
1010:
997:
994:
975:
969:
966:
952:
951:
940:
937:
934:
929:
925:
921:
916:
912:
908:
905:
900:
894:
891:
883:
879:
873:
868:
865:
862:
858:
852:
847:
844:
841:
837:
812:
808:
803:
799:
795:
792:
789:
784:
780:
757:
752:
747:
743:
739:
736:
733:
728:
724:
701:
697:
694:
691:
688:
685:
682:
676:
673:
670:
667:
664:
661:
658:
633:
629:
624:
619:
614:
609:
606:
579:
574:
570:
566:
561:
557:
553:
550:
547:
542:
539:
534:
512:
509:
506:
503:
499:
496:
491:
487:
462:
458:
453:
449:
445:
442:
439:
434:
430:
426:
422:
418:
415:
393:
388:
383:
379:
375:
372:
369:
364:
360:
350:holds for all
346:
345:
336:
334:
323:
320:
317:
312:
308:
304:
299:
295:
291:
288:
283:
279:
273:
269:
263:
258:
255:
252:
248:
242:
237:
234:
231:
227:
199:
176:
172:
167:
162:
157:
152:
149:
124:
110:
107:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
9189:
9178:
9175:
9173:
9170:
9169:
9167:
9155:
9149:
9146:
9140:
9137:
9131:
9128:
9122:
9119:
9113:
9110:
9104:
9101:
9094:
9091:
9085:
9082:
9076:
9073:
9067:
9064:
9058:
9055:
9049:
9046:
9042:
9036:
9033:
9028:
9022:
9018:
9011:
9008:
9002:
8998:
8997:Kernel method
8995:
8993:
8990:
8988:
8985:
8983:
8980:
8978:
8975:
8973:
8970:
8969:
8965:
8963:
8959:
8957:
8953:
8949:
8945:
8937:
8934:
8926:
8924:
8922:
8918:
8914:
8910:
8906:
8902:
8896:
8888:
8870:
8865:
8859:
8855:
8851:
8848:
8842:
8838:
8833:
8828:
8825:
8822:
8818:
8812:
8809:
8804:
8798:
8792:
8772:
8744:
8739:
8735:
8731:
8728:
8725:
8720:
8716:
8671:
8663:
8662:
8661:
8645:
8642:
8638:
8632:
8628:
8624:
8615:
8609:
8606:
8600:
8594:
8588:
8585:
8579:
8576:
8573:
8567:
8547:
8527:
8507:
8487:
8465:
8461:
8437:
8431:
8404:
8398:
8372:
8366:
8363:
8357:
8351:
8345:
8342:
8336:
8333:
8330:
8324:
8301:
8295:
8272:
8266:
8246:
8226:
8203:
8197:
8174:
8168:
8141:
8138:
8135:
8132:
8112:
8084:
8080:
8073:
8067:
8042:
8038:
8031:
8009:
8005:
7982:
7978:
7955:
7951:
7922:
7918:
7911:
7908:
7903:
7899:
7892:
7884:
7880:
7876:
7871:
7867:
7819:
7799:
7776:
7770:
7762:
7761:
7760:
7754:
7752:
7749:
7745:
7739:
7738:Kernel method
7731:
7726:
7724:
7706:
7703:
7700:
7694:
7691:
7685:
7682:
7679:
7673:
7670:
7667:
7661:
7658:
7655:
7649:
7644:
7638:
7635:
7632:
7626:
7606:
7586:
7583:
7580:
7560:
7557:
7551:
7548:
7545:
7539:
7531:
7526:
7510:
7500:
7496:
7489:
7486:
7464:
7460:
7434:
7431:
7411:
7402:
7400:
7399:isometrically
7382:
7379:
7349:
7295:
7292:
7289:
7283:
7280:
7277:
7261:
7247:
7244:
7239:
7235:
7229:
7224:
7221:
7218:
7214:
7188:
7183:
7179:
7175:
7172:
7169:
7164:
7160:
7139:
7129:
7124:
7120:
7116:
7113:
7110:
7105:
7101:
7097:
7089:
7086:
7073:
7066:
7064:
7050:
7047:
7039:
7035:
7031:
7026:
7022:
7015:
7010:
7006:
7000:
6996:
6990:
6985:
6982:
6979:
6976:
6973:
6969:
6961:
6960:
6957:
6904:
6894:
6891:
6883:
6878:
6861:
6855:
6852:
6849:
6843:
6840:
6834:
6831:
6828:
6822:
6819:
6813:
6810:
6807:
6801:
6794:
6780:
6774:
6771:
6768:
6762:
6759:
6753:
6750:
6747:
6741:
6734:
6719:
6716:
6713:
6693:
6690:
6684:
6681:
6678:
6672:
6652:
6649:
6643:
6640:
6637:
6631:
6624:
6623:
6622:
6601:
6574:
6566:
6550:
6541:
6527:
6507:
6499:
6491:
6489:
6472:
6469:
6466:
6460:
6440:
6420:
6397:
6394:
6391:
6385:
6382:
6377:
6367:
6363:
6359:
6354:
6350:
6343:
6338:
6327:
6318:
6312:
6283:
6280:
6277:
6255:
6251:
6247:
6241:
6215:
6212:
6209:
6189:
6169:
6164:
6153:
6144:
6138:
6126:
6120:
6117:
6114:
6108:
6088:
6068:
6062:
6059:
6034:
6026:
6023:
6020:
5997:
5987:
5973:
5953:
5943:
5933:
5929:
5915:
5899:
5895:
5877:
5874:
5871:
5851:
5848:
5845:
5822:
5816:
5813:
5805:
5801:
5797:
5794:
5784:
5769:
5766:
5763:
5757:
5754:
5751:
5745:
5737:
5733:
5725:
5724:
5723:
5701:
5698:
5695:
5692:
5689:
5681:
5676:
5672:
5658:
5650:
5645:
5628:
5622:
5619:
5613:
5605:
5601:
5594:
5566:
5563:
5558:
5554:
5533:
5530:
5527:
5507:
5479:
5476:
5473:
5470:
5465:
5457:
5454:
5451:
5420:
5417:
5414:
5394:
5374:
5365:
5360:
5352:
5350:
5347:
5345:
5341:
5337:
5333:
5328:
5312:
5301:
5298:
5295:
5289:
5286:
5283:
5277:
5271:
5265:
5262:
5259:
5256:
5253:
5233:
5213:
5193:
5190:
5185:
5181:
5160:
5157:
5154:
5131:
5128:
5125:
5119:
5099:
5090:
5073:
5070:
5067:
5059:
5055:
5051:
5048:
5028:
5025:
5019:
5013:
5004:
4986:
4982:
4974:
4966:
4962:
4955:
4947:
4943:
4934:
4930:
4926:
4920:
4917:
4914:
4908:
4885:
4882:
4879:
4856:
4836:
4833:
4827:
4821:
4798:
4795:
4792:
4786:
4766:
4763:
4757:
4751:
4731:
4728:
4722:
4716:
4696:
4687:
4673:
4667:
4661:
4655:
4649:
4646:
4643:
4637:
4632:
4627:
4623:
4617:
4612:
4608:
4604:
4598:
4592:
4584:
4566:
4562:
4553:
4532:
4528:
4504:
4484:
4471:
4464:
4462:
4448:
4443:
4438:
4434:
4422:
4416:
4410:
4402:
4398:
4392:
4387:
4383:
4378:
4369:
4365:
4361:
4356:
4353:
4350:
4341:
4327:
4321:
4315:
4309:
4303:
4300:
4297:
4291:
4286:
4281:
4277:
4271:
4266:
4262:
4254:
4253:
4250:
4243:
4236:
4234:
4220:
4217:
4203:
4197:
4191:
4188:
4185:
4179:
4174:
4169:
4165:
4161:
4158:
4152:
4146:
4143:
4137:
4131:
4124:
4123:
4120:
4118:
4114:
4095:
4092:
4089:
4083:
4080:
4074:
4071:
4068:
4062:
4051:
4043:
4027:
4024:
4021:
4018:
4009:
4006:
4002:
3999:
3993:
3990:
3987:
3983:
3979:
3972:
3969:
3965:
3962:
3956:
3936:
3931:
3926:
3919:
3915:
3911:
3905:
3898:
3894:
3887:
3880:
3876:
3872:
3865:
3861:
3856:
3852:
3848:
3841:
3837:
3833:
3827:
3820:
3816:
3809:
3803:
3799:
3795:
3788:
3784:
3779:
3759:
3755:
3750:
3746:
3742:
3738:
3733:
3729:
3724:
3718:
3714:
3710:
3705:
3702:
3698:
3693:
3686:
3682:
3678:
3674:
3669:
3665:
3657:
3648:
3644:
3640:
3636:
3631:
3627:
3615:
3611:
3607:
3600:
3596:
3591:
3571:
3566:
3558:
3555:
3548:
3545:
3539:
3533:
3527:
3523:
3518:
3513:
3510:
3506:
3503:
3497:
3493:
3490:
3485:
3482:
3478:
3457:
3434:
3429:
3414:
3400:
3397:
3392:
3387:
3376:
3372:
3366:
3362:
3356:
3351:
3348:
3345:
3341:
3331:
3326:
3321:
3315:
3311:
3305:
3301:
3295:
3290:
3287:
3284:
3280:
3276:
3271:
3267:
3261:
3257:
3251:
3246:
3243:
3240:
3236:
3231:
3226:
3221:
3211:
3207:
3203:
3198:
3194:
3185:
3181:
3175:
3171:
3165:
3160:
3157:
3154:
3151:
3148:
3144:
3115:
3112:
3109:
3106:
3101:
3093:
3090:
3087:
3076:
3075:Hilbert space
3060:
3052:
3035:
3032:
3029:
3026:
3018:
3015:
3012:
3009:
3005:
2996:
2993:
2990:
2984:
2978:
2975:
2972:
2966:
2963:
2960:
2954:
2946:
2943:
2925:
2921:
2895:
2887:
2884:
2881:
2868:
2865:
2860:
2857:
2853:
2845:
2842:
2837:
2834:
2829:
2821:
2818:
2815:
2809:
2803:
2800:
2797:
2791:
2766:
2751:
2746:
2742:
2734:
2730:
2715:
2712:
2709:
2706:
2698:
2695:
2692:
2689:
2685:
2675:
2672:
2669:
2661:
2658:
2654:
2650:
2644:
2641:
2638:
2632:
2625:Abel kernel:
2624:
2609:
2606:
2603:
2600:
2595:
2585:
2577:
2568:
2555:
2544:
2541:
2537:
2533:
2522:
2511:
2503:
2488:
2485:
2482:
2479:
2474:
2464:
2456:
2447:
2437:
2433:
2429:
2422:
2409:
2395:
2391:
2387:
2376:
2365:
2357:
2353:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2312:
2302:
2294:
2285:
2280:
2272:
2269:
2259:
2246:
2235:
2224:
2216:
2213:
2196:
2186:
2178:
2169:
2159:
2149:
2138:
2127:
2119:
2118:
2102:
2087:
2086:
2082:
2063:
2051:
2046:
2017:
1995:
1991:
1963:
1958:
1941:
1924:
1912:
1909:
1906:
1901:
1889:
1857:
1853:
1849:
1844:
1840:
1831:
1827:
1821:
1816:
1813:
1810:
1806:
1802:
1791:
1787:
1783:
1780:
1777:
1772:
1768:
1761:
1753:
1749:
1745:
1742:
1739:
1734:
1730:
1720:
1695:
1691:
1687:
1682:
1678:
1669:
1665:
1659:
1654:
1651:
1648:
1644:
1640:
1629:
1625:
1621:
1618:
1615:
1610:
1606:
1599:
1591:
1587:
1583:
1580:
1577:
1572:
1568:
1558:
1528:
1516:
1511:
1499:
1494:
1490:
1480:
1475:
1470:
1467:
1464:
1454:
1450:
1424:
1419:
1416:
1413:
1403:
1383:
1364:
1360:
1348:
1340:
1337:
1329:
1310:
1305:
1301:
1297:
1294:
1291:
1286:
1282:
1257:
1253:
1247:
1243:
1239:
1232:
1228:
1222:
1218:
1209:
1195:
1192:
1187:
1183:
1179:
1176:
1173:
1168:
1164:
1141:
1137:
1131:
1127:
1121:
1116:
1113:
1110:
1106:
1097:
1096:
1068:
1058:
1053:
1049:
1039:
1029:
1026:
1016:
1012:
1000:
999:
995:
993:
991:
973:
964:
938:
935:
927:
923:
919:
914:
910:
903:
898:
889:
881:
877:
871:
866:
863:
860:
856:
850:
845:
842:
839:
835:
827:
826:
825:
806:
801:
797:
793:
790:
787:
782:
778:
750:
745:
741:
737:
734:
731:
726:
722:
692:
689:
686:
680:
674:
668:
665:
662:
656:
648:
617:
607:
604:
595:
593:
572:
568:
564:
559:
555:
548:
545:
540:
537:
507:
497:
494:
489:
485:
475:
456:
451:
447:
443:
440:
437:
432:
428:
424:
416:
413:
386:
381:
377:
373:
370:
367:
362:
358:
344:
337:
335:
321:
318:
310:
306:
302:
297:
293:
286:
281:
277:
271:
267:
261:
256:
253:
250:
246:
240:
235:
232:
229:
225:
217:
216:
213:
160:
150:
147:
140:
108:
106:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
9148:
9139:
9130:
9121:
9112:
9103:
9093:
9084:
9075:
9066:
9057:
9048:
9035:
9016:
9010:
8960:
8941:
8898:
8423:
7758:
7741:
7530:pseudometric
7527:
7403:
7263:
7078:
7067:
6881:
6880:
6876:
6563:, we mean a
6542:
6495:
5988:
5945:
5935:
5931:
5901:
5897:
5893:
5679:
5678:
5674:
5648:
5647:
5366:
5362:
5348:
5329:
5091:
5005:
4476:
4465:
4248:
4237:
4053:
1210:The product
988:denotes the
953:
646:
645:is called a
596:
476:
349:
338:
112:
55:James Mercer
42:
36:
8921:collocation
5147:defined on
4744:except for
4583:eigenvalues
592:eigenvalues
9166:Categories
9003:References
7479:such that
7206:such that
6882:Definition
5838:, for all
5722:such that
5680:Definition
5649:Definition
5520:. For any
5387:be a set,
4709:satisfies
4113:S. Bochner
4048:See also:
2356:RBF kernel
1330:The limit
109:Definition
8852:−
8819:∑
8745:∈
8729:…
8639:δ
8629:σ
8607:⋅
8462:σ
8432:ϵ
8364:⋅
8142:∈
7832:of a set
7668:−
7584:≠
7435:∈
7296:∈
7215:∑
7189:∈
7173:…
7130:∈
7114:…
7090:∈
7048:≤
7016:ψ
6970:∑
6915:→
6905:×
6892:ψ
6820:≤
6650:≥
6602:×
6322:Φ
6307:Φ
6281:∈
6236:Φ
6148:Φ
6133:Φ
6066:→
6057:Φ
6027:⋅
6021:⋅
5875:∈
5849:∈
5767:∈
5761:∀
5752:∈
5746:⋅
5705:→
5699:×
5598:↦
5570:→
5531:∈
5483:→
5477:×
5458:⋅
5452:⋅
5424:→
5332:E. Cartan
5296:⋅
5257:∈
5191:∈
5158:×
5052:∈
5026:≥
4983:λ
4963:ψ
4944:ψ
4931:∑
4834:≥
4624:∫
4609:∫
4563:λ
4529:ψ
4399:ψ
4384:∫
4366:λ
4357:∑
4278:∫
4263:∫
4198:φ
4166:∫
4162:λ
4159:−
4147:φ
4093:−
4022:α
4007:θ
4000:θ
3994:ψ
3991:α
3988:−
3970:θ
3963:θ
3912:θ
3906:−
3895:θ
3877:∑
3857:ψ
3834:θ
3828:−
3817:θ
3800:∑
3780:ψ
3743:θ
3739:−
3730:θ
3715:∑
3699:ψ
3679:θ
3666:θ
3641:θ
3637:−
3628:θ
3612:∑
3597:χ
3592:ψ
3556:θ
3540:θ
3528:−
3511:θ
3504:θ
3479:ψ
3458:χ
3398:≥
3342:∑
3281:∑
3237:∑
3145:∑
3119:→
3113:×
3094:⋅
3088:⋅
3030:α
3019:∈
2994:−
2985:α
2979:
2926:ν
2892:‖
2885:−
2879:‖
2861:−
2838:−
2826:‖
2819:−
2813:‖
2710:α
2699:∈
2673:−
2662:α
2659:−
2604:α
2586:∈
2564:‖
2556:−
2548:‖
2545:α
2542:−
2483:σ
2465:∈
2434:σ
2419:‖
2410:−
2402:‖
2396:−
2336:≥
2324:≥
2303:∈
2187:∈
2117:include:
2052:×
1964:⊂
1913:×
1910:⋯
1907:×
1807:∑
1781:…
1743:…
1645:∏
1619:…
1581:…
1534:→
1517:×
1355:∞
1352:→
1311:∈
1295:…
1240:…
1193:≥
1184:λ
1177:…
1165:λ
1128:λ
1107:∑
1079:→
1069:×
1030:∈
968:¯
965:ξ
936:≥
893:¯
890:ξ
878:ξ
857:∑
836:∑
807:∈
798:ξ
791:…
779:ξ
751:∈
735:…
700:¯
628:→
618:×
505:∀
457:∈
441:…
417:∈
387:∈
371:…
319:≥
247:∑
226:∑
171:→
161:×
8966:See also
6270:for all
5651:: Space
5340:M. Krein
5041:for all
4010:′
3973:′
3920:′
3842:′
3751:′
3687:′
3649:′
3559:′
3514:′
3383:‖
3337:‖
2913:, where
9098:69–164.
8944:kriging
6296:. Then
5937:Theorem
5903:Theorem
5336:H. Weyl
4044:History
2940:is the
9023:
6665:, and
6565:metric
6228:, and
5441:, and
4680:
4554:, and
4477:where
4334:
4210:
1487:
1484:
1046:
1043:
954:where
5782:, and
3073:is a
49:or a
9021:ISBN
8954:and
8915:and
8905:PDEs
8288:and
8190:and
7573:for
7152:and
6433:and
5864:and
5367:Let
4729:>
4025:>
3033:>
2976:sinc
2713:>
2607:>
2486:>
1942:Let
1713:and
113:Let
89:for
7970:at
7070:1.4
6956:if
6587:on
4869:on
4468:1.3
4240:1.2
3053:If
2358:):
2030:to
2010:of
1384:If
1345:lim
649:if
474:.
341:1.1
212:if
37:In
9168::
8923:.
8911:,
8660:.
7525:.
7260:.
6540:.
5089:.
4028:0.
2784::
2217::
824:,
594:.
406:,
101:,
97:,
93:,
85:,
81:,
77:,
73:,
69:,
65:,
9029:.
8871:)
8866:h
8860:i
8856:x
8849:x
8843:(
8839:K
8834:n
8829:1
8826:=
8823:i
8813:n
8810:1
8805:=
8802:)
8799:x
8796:(
8793:f
8773:K
8750:X
8740:n
8736:x
8732:,
8726:,
8721:1
8717:x
8694:X
8672:f
8646:y
8643:x
8633:2
8625:+
8622:]
8619:)
8616:y
8613:(
8610:Z
8604:)
8601:x
8598:(
8595:Z
8592:[
8589:E
8586:=
8583:)
8580:y
8577:,
8574:x
8571:(
8568:K
8548:f
8528:Z
8508:x
8488:x
8466:2
8441:)
8438:x
8435:(
8408:)
8405:x
8402:(
8399:Z
8379:]
8376:)
8373:y
8370:(
8367:Z
8361:)
8358:x
8355:(
8352:Z
8349:[
8346:E
8343:=
8340:)
8337:y
8334:,
8331:x
8328:(
8325:K
8305:)
8302:y
8299:(
8296:Z
8276:)
8273:x
8270:(
8267:Z
8247:y
8227:x
8207:)
8204:y
8201:(
8198:Z
8178:)
8175:x
8172:(
8169:Z
8147:X
8139:y
8136:,
8133:x
8113:f
8093:]
8090:)
8085:i
8081:x
8077:(
8074:Z
8071:[
8068:E
8048:)
8043:i
8039:x
8035:(
8032:Z
8010:i
8006:x
7983:i
7979:x
7956:i
7952:f
7931:)
7928:)
7923:i
7919:x
7915:(
7912:f
7909:,
7904:i
7900:x
7896:(
7893:=
7890:)
7885:i
7881:f
7877:,
7872:i
7868:x
7864:(
7842:X
7820:x
7800:f
7780:)
7777:x
7774:(
7771:f
7710:)
7707:y
7704:,
7701:y
7698:(
7695:K
7692:+
7689:)
7686:y
7683:,
7680:x
7677:(
7674:K
7671:2
7665:)
7662:x
7659:,
7656:x
7653:(
7650:K
7645:=
7642:)
7639:y
7636:,
7633:x
7630:(
7627:d
7607:K
7587:y
7581:x
7561:0
7558:=
7555:)
7552:y
7549:,
7546:x
7543:(
7540:d
7511:n
7507:)
7501:n
7497:K
7493:(
7490:=
7487:K
7465:n
7461:K
7439:N
7432:n
7412:K
7386:)
7383:d
7380:,
7375:X
7370:(
7350:d
7328:X
7306:}
7301:X
7293:x
7290::
7287:)
7284:x
7281:,
7278:x
7275:(
7272:{
7248:0
7245:=
7240:i
7236:c
7230:n
7225:1
7222:=
7219:i
7193:R
7184:n
7180:c
7176:,
7170:,
7165:1
7161:c
7140:,
7135:X
7125:n
7121:x
7117:,
7111:,
7106:1
7102:x
7098:,
7094:N
7087:n
7072:)
7068:(
7051:0
7045:)
7040:j
7036:x
7032:,
7027:i
7023:x
7019:(
7011:j
7007:c
7001:i
6997:c
6991:n
6986:1
6983:=
6980:j
6977:,
6974:i
6942:X
6919:R
6910:X
6900:X
6895::
6862:.
6859:)
6856:z
6853:,
6850:y
6847:(
6844:d
6841:+
6838:)
6835:y
6832:,
6829:x
6826:(
6823:d
6817:)
6814:z
6811:,
6808:x
6805:(
6802:d
6781:,
6778:)
6775:x
6772:,
6769:y
6766:(
6763:d
6760:=
6757:)
6754:y
6751:,
6748:x
6745:(
6742:d
6732:,
6720:y
6717:=
6714:x
6694:0
6691:=
6688:)
6685:y
6682:,
6679:x
6676:(
6673:d
6653:0
6647:)
6644:y
6641:,
6638:x
6635:(
6632:d
6607:X
6597:X
6575:d
6551:X
6528:d
6508:K
6476:)
6473:y
6470:,
6467:x
6464:(
6461:K
6441:y
6421:x
6401:)
6398:y
6395:,
6392:x
6389:(
6386:K
6383:=
6378:H
6374:)
6368:y
6364:K
6360:,
6355:x
6351:K
6347:(
6344:=
6339:F
6335:)
6331:)
6328:y
6325:(
6319:,
6316:)
6313:x
6310:(
6304:(
6284:X
6278:x
6256:x
6252:K
6248:=
6245:)
6242:x
6239:(
6216:H
6213:=
6210:F
6190:K
6170:.
6165:F
6161:)
6157:)
6154:y
6151:(
6145:,
6142:)
6139:x
6136:(
6130:(
6127:=
6124:)
6121:y
6118:,
6115:x
6112:(
6109:K
6089:F
6069:F
6063:X
6060::
6035:F
6031:)
6024:,
6018:(
5998:F
5974:K
5954:K
5916:K
5890:.
5878:X
5872:x
5852:H
5846:f
5826:)
5823:x
5820:(
5817:f
5814:=
5811:)
5806:x
5802:K
5798:,
5795:f
5792:(
5770:X
5764:x
5758:,
5755:H
5749:)
5743:(
5738:x
5734:K
5709:R
5702:X
5696:X
5693::
5690:K
5659:H
5632:)
5629:x
5626:(
5623:f
5620:=
5617:)
5614:f
5611:(
5606:x
5602:e
5595:f
5574:R
5567:H
5564::
5559:x
5555:e
5534:X
5528:x
5508:H
5487:R
5480:H
5474:H
5471::
5466:H
5462:)
5455:,
5449:(
5428:R
5421:X
5418::
5415:f
5395:H
5375:X
5313:H
5309:)
5305:)
5302:y
5299:,
5293:(
5290:K
5287:,
5284:f
5281:(
5278:=
5275:)
5272:y
5269:(
5266:f
5263:,
5260:H
5254:f
5234:H
5214:K
5194:E
5186:i
5182:x
5161:E
5155:E
5135:)
5132:y
5129:,
5126:x
5123:(
5120:K
5100:E
5077:]
5074:b
5071:,
5068:a
5065:[
5060:1
5056:L
5049:x
5029:0
5023:)
5020:x
5017:(
5014:J
4987:n
4978:)
4975:t
4972:(
4967:n
4959:)
4956:s
4953:(
4948:n
4935:n
4927:=
4924:)
4921:t
4918:,
4915:s
4912:(
4909:K
4889:]
4886:b
4883:,
4880:a
4877:[
4857:x
4837:0
4831:)
4828:x
4825:(
4822:J
4802:)
4799:t
4796:,
4793:s
4790:(
4787:K
4767:0
4764:=
4761:)
4758:t
4755:(
4752:x
4732:0
4726:)
4723:x
4720:(
4717:J
4697:t
4693:d
4688:s
4684:d
4677:)
4674:t
4671:(
4668:x
4665:)
4662:s
4659:(
4656:x
4653:)
4650:t
4647:,
4644:s
4641:(
4638:K
4633:b
4628:a
4618:b
4613:a
4605:=
4602:)
4599:x
4596:(
4593:J
4567:n
4538:}
4533:n
4525:{
4505:x
4485:K
4470:)
4466:(
4449:,
4444:2
4439:]
4435:s
4431:d
4426:)
4423:s
4420:(
4417:x
4414:)
4411:s
4408:(
4403:n
4393:b
4388:a
4379:[
4370:n
4362:1
4354:=
4351:t
4347:d
4342:s
4338:d
4331:)
4328:t
4325:(
4322:x
4319:)
4316:s
4313:(
4310:x
4307:)
4304:t
4301:,
4298:s
4295:(
4292:K
4287:b
4282:a
4272:b
4267:a
4242:)
4238:(
4221:.
4218:t
4214:d
4207:)
4204:t
4201:(
4195:)
4192:t
4189:,
4186:s
4183:(
4180:K
4175:b
4170:a
4156:)
4153:s
4150:(
4144:=
4141:)
4138:s
4135:(
4132:f
4099:)
4096:y
4090:x
4087:(
4084:f
4081:=
4078:)
4075:y
4072:,
4069:x
4066:(
4063:K
4019:,
4014:)
4003:,
3997:(
3984:e
3980:=
3977:)
3966:,
3960:(
3957:K
3937:,
3932:2
3927:|
3916:i
3899:i
3888:|
3881:i
3873:=
3866:2
3862:H
3853:,
3849:|
3838:i
3821:i
3810:|
3804:i
3796:=
3789:1
3785:H
3760:,
3756:|
3747:i
3734:i
3725:|
3719:i
3711:=
3706:V
3703:T
3694:,
3683:i
3675:+
3670:i
3658:2
3654:)
3645:i
3632:i
3624:(
3616:i
3608:=
3601:2
3572:,
3567:2
3563:)
3552:(
3549:H
3546:+
3543:)
3537:(
3534:H
3524:)
3519:2
3507:+
3498:(
3494:H
3491:=
3486:D
3483:J
3435:d
3430:+
3425:R
3401:0
3393:2
3388:H
3377:i
3373:x
3367:i
3363:c
3357:n
3352:1
3349:=
3346:i
3332:=
3327:H
3322:)
3316:j
3312:x
3306:j
3302:c
3296:n
3291:1
3288:=
3285:j
3277:,
3272:i
3268:x
3262:i
3258:c
3252:n
3247:1
3244:=
3241:i
3232:(
3227:=
3222:H
3218:)
3212:j
3208:x
3204:,
3199:i
3195:x
3191:(
3186:j
3182:c
3176:i
3172:c
3166:n
3161:1
3158:=
3155:j
3152:,
3149:i
3123:R
3116:H
3110:H
3107::
3102:H
3098:)
3091:,
3085:(
3061:H
3048:.
3036:0
3027:,
3023:R
3016:y
3013:,
3010:x
3006:,
3003:)
3000:)
2997:y
2991:x
2988:(
2982:(
2973:=
2970:)
2967:y
2964:,
2961:x
2958:(
2955:K
2944:.
2922:B
2901:)
2896:2
2888:y
2882:x
2876:(
2869:2
2866:d
2858:k
2854:B
2846:2
2843:d
2835:k
2830:2
2822:y
2816:x
2810:=
2807:)
2804:y
2801:,
2798:x
2795:(
2792:K
2772:)
2767:d
2762:R
2757:(
2752:k
2747:2
2743:W
2728:.
2716:0
2707:,
2703:R
2696:y
2693:,
2690:x
2686:,
2680:|
2676:y
2670:x
2666:|
2655:e
2651:=
2648:)
2645:y
2642:,
2639:x
2636:(
2633:K
2622:.
2610:0
2601:,
2596:d
2591:R
2582:y
2578:,
2574:x
2569:,
2560:y
2552:x
2538:e
2534:=
2531:)
2527:y
2523:,
2519:x
2515:(
2512:K
2501:.
2489:0
2480:,
2475:d
2470:R
2461:y
2457:,
2453:x
2448:,
2438:2
2430:2
2423:2
2414:y
2406:x
2392:e
2388:=
2385:)
2381:y
2377:,
2373:x
2369:(
2366:K
2351:.
2339:1
2333:n
2330:,
2327:0
2321:r
2318:,
2313:d
2308:R
2299:y
2295:,
2291:x
2286:,
2281:n
2277:)
2273:r
2270:+
2266:y
2260:T
2255:x
2250:(
2247:=
2244:)
2240:y
2236:,
2232:x
2228:(
2225:K
2211:.
2197:d
2192:R
2183:y
2179:,
2175:x
2170:,
2166:y
2160:T
2155:x
2150:=
2147:)
2143:y
2139:,
2135:x
2131:(
2128:K
2103:d
2098:R
2064:0
2058:X
2047:0
2041:X
2018:K
1996:0
1992:K
1969:X
1959:0
1953:X
1939:.
1925:n
1919:X
1902:1
1896:X
1890:=
1885:X
1863:)
1858:i
1854:y
1850:,
1845:i
1841:x
1837:(
1832:i
1828:K
1822:n
1817:1
1814:=
1811:i
1803:=
1800:)
1797:)
1792:n
1788:y
1784:,
1778:,
1773:1
1769:y
1765:(
1762:,
1759:)
1754:n
1750:x
1746:,
1740:,
1735:1
1731:x
1727:(
1724:(
1721:K
1701:)
1696:i
1692:y
1688:,
1683:i
1679:x
1675:(
1670:i
1666:K
1660:n
1655:1
1652:=
1649:i
1641:=
1638:)
1635:)
1630:n
1626:y
1622:,
1616:,
1611:1
1607:y
1603:(
1600:,
1597:)
1592:n
1588:x
1584:,
1578:,
1573:1
1569:x
1565:(
1562:(
1559:K
1538:R
1529:i
1523:X
1512:i
1506:X
1500::
1495:i
1491:K
1481:,
1476:n
1471:1
1468:=
1465:i
1461:)
1455:i
1451:K
1447:(
1425:n
1420:1
1417:=
1414:i
1410:)
1404:i
1398:X
1392:(
1365:n
1361:K
1349:n
1341:=
1338:K
1315:N
1306:n
1302:a
1298:,
1292:,
1287:1
1283:a
1258:n
1254:a
1248:n
1244:K
1233:1
1229:a
1223:1
1219:K
1196:0
1188:n
1180:,
1174:,
1169:1
1142:i
1138:K
1132:i
1122:n
1117:1
1114:=
1111:i
1083:R
1074:X
1064:X
1059::
1054:i
1050:K
1040:,
1034:N
1027:i
1023:)
1017:i
1013:K
1009:(
974:j
939:0
933:)
928:j
924:x
920:,
915:i
911:x
907:(
904:K
899:j
882:i
872:n
867:1
864:=
861:j
851:n
846:1
843:=
840:i
811:C
802:n
794:,
788:,
783:1
756:X
746:n
742:x
738:,
732:,
727:1
723:x
696:)
693:x
690:,
687:y
684:(
681:K
675:=
672:)
669:y
666:,
663:x
660:(
657:K
632:C
623:X
613:X
608::
605:K
578:)
573:j
569:x
565:,
560:i
556:x
552:(
549:K
546:=
541:j
538:i
533:K
511:)
508:i
502:(
498:0
495:=
490:i
486:c
461:R
452:n
448:c
444:,
438:,
433:1
429:c
425:,
421:N
414:n
392:X
382:n
378:x
374:,
368:,
363:1
359:x
343:)
339:(
322:0
316:)
311:j
307:x
303:,
298:i
294:x
290:(
287:K
282:j
278:c
272:i
268:c
262:n
257:1
254:=
251:j
241:n
236:1
233:=
230:i
198:X
175:R
166:X
156:X
151::
148:K
123:X
34:.
20:)
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