2156:
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50:
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1608:
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2151:{\displaystyle f(x)={\begin{cases}-0.1522x^{3}+0.9937x,&{\text{if }}x\in ,\\-0.01258x^{3}-0.4189x^{2}+1.4126x-0.1396,&{\text{if }}x\in ,\\0.1403x^{3}-1.3359x^{2}+3.2467x-1.3623,&{\text{if }}x\in ,\\0.1579x^{3}-1.4945x^{2}+3.7225x-1.8381,&{\text{if }}x\in ,\\0.05375x^{3}-0.2450x^{2}-1.2756x+4.8259,&{\text{if }}x\in ,\\-0.1871x^{3}+3.3673x^{2}-19.3370x+34.9282,&{\text{if }}x\in .\end{cases}}}
459:
222:
3037:) using various digital filtering techniques (for example, convolution with a frequency-limited impulse signal). In this application there is a specific requirement that the harmonic content of the original signal be preserved without creating aliased harmonic content of the original signal above the original
2183:
Depending on the underlying discretisation of fields, different interpolants may be required. In contrast to other interpolation methods, which estimate functions on target points, mimetic interpolation evaluates the integral of fields on target lines, areas or volumes, depending on the type of field
2814:
is a powerful non-linear interpolation tool. Many popular interpolation tools are actually equivalent to particular
Gaussian processes. Gaussian processes can be used not only for fitting an interpolant that passes exactly through the given data points but also for regression; that is, for fitting a
164:
of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting
3066:
problems, the constraint that the interpolant has to go exactly through the data points is relaxed. It is only required to approach the data points as closely as possible (within some other constraints). This requires parameterizing the potential interpolants and having some way of measuring the
1594:
More generally, the shape of the resulting curve, especially for very high or low values of the independent variable, may be contrary to commonsense; that is, to what is known about the experimental system which has generated the data points. These disadvantages can be reduced by using spline
2207:
difference at the endpoints of the integration path. Mimetic interpolation ensures that the error of estimating the line integral of an electric field is the same as the error obtained by interpolating the potential at the end points of the integration path, regardless of the length of the
2168:
Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation, while the interpolant is smoother and easier to evaluate than the high-degree polynomials used in polynomial interpolation. However, the global nature of the basis functions leads to
1350:
2222:
are also considered mimetic, even if it is the field values that are conserved (not the integral of the field). Apart from linear interpolation, area weighted interpolation can be considered one of the first mimetic interpolation methods to have been developed.
701:
1621:
Linear interpolation uses a linear function for each of intervals . Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline.
3077:
studies how to find the best approximation to a given function by another function from some predetermined class, and how good this approximation is. This clearly yields a bound on how well the interpolant can approximate the unknown function.
1521:
1587:) ≈ −1.003. However, these maxima and minima may exceed the theoretical range of the function; for example, a function that is always positive may have an interpolant with negative values, and whose inverse therefore contains false
935:
818:
2682:
2781:
3033:
In the domain of digital signal processing, the term interpolation refers to the process of converting a sampled digital signal (such as a sampled audio signal) to that of a higher sampling rate (
1563:
Polynomial interpolation can estimate local maxima and minima that are outside the range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at
1633:
cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points. The natural cubic spline interpolating the points in the table above is given by
2352:
566:
1164:
1549:. Furthermore, the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation overcomes most of the problems of linear interpolation.
2398:
2275:
2456:
3041:
of the signal (that is, above fs/2 of the original signal sample rate). An early and fairly elementary discussion on this subject can be found in
Rabiner and Crochiere's book
2500:
1108:
1062:
984:
2594:
1359:
and spline interpolation (described below), is proportional to higher powers of the distance between the data points. These methods also produce smoother interpolants.
2542:
at these points). In general, an interpolant need not be a good approximation, but there are well known and often reasonable conditions where it will. For example, if
1402:
2985:
434:
1016:
3137:
283:
216:
3288:
3104:
2959:
2801:
2540:
2520:
251:
475:
The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as
2859:
3238:
2199:. As a result, mimetic interpolation conserves line, area and volume integrals. Conservation of line integrals might be desirable when interpolating the
3374:
826:
709:
2169:
ill-conditioning. This is completely mitigated by using splines of compact support, such as are implemented in Boost.Math and discussed in Kress.
3395:
Stern, Ari; Tong, Yiying; Desbrun, Mathieu; Marsden, Jerrold E. (2015), Chang, Dong Eui; Holm, Darryl D.; Patrick, George; Ratiu, Tudor (eds.),
3167:
1556:) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see
3426:
3264:
180:
deduced only from the red points. The interpolated curves have polynomial formulas much simpler than that of the original epitrochoid curve.
2865:
Sometimes, we know not only the value of the function that we want to interpolate, at some points, but also its derivative. This leads to
1552:
However, polynomial interpolation also has some disadvantages. Calculating the interpolating polynomial is computationally expensive (see
1545:−1 going through all the data points. The interpolation error is proportional to the distance between the data points to the power
3197:
2881:
150:
2603:
3546:
2827:
Other forms of interpolation can be constructed by picking a different class of interpolants. For instance, rational interpolation is
1355:
In words, the error is proportional to the square of the distance between the data points. The error in some other methods, including
1128:
The following error estimate shows that linear interpolation is not very precise. Denote the function which we want to interpolate by
3313:
93:
71:
3670:
3681:
3712:
3659:
1553:
38:
3143:, then the problem is treated as "interpolation of operators". The classical results about interpolation of operators are the
2687:
3612:
470:
463:
31:
3698:
3648:
3202:
3161:
3707:
1112:
Linear interpolation is quick and easy, but it is not very precise. Another disadvantage is that the interpolant is not
505:
One of the simplest methods is linear interpolation (sometimes known as lerp). Consider the above example of estimating
2840:
165:
gain in simplicity may outweigh the loss from interpolation error and give better performance in calculation process.
696:{\displaystyle y=y_{a}+\left(y_{b}-y_{a}\right){\frac {x-x_{a}}{x_{b}-x_{a}}}{\text{ at the point }}\left(x,y\right)}
3223:
2993:
2924:
2188:
480:
3192:
2280:
1345:{\displaystyle |f(x)-g(x)|\leq C(x_{b}-x_{a})^{2}\quad {\text{where}}\quad C={\frac {1}{8}}\max _{r\in }|g''(r)|.}
3605:
R.E. Crochiere and L.R. Rabiner. (1983). Multirate
Digital Signal Processing. Englewood Cliffs, NJ: Prentice–Hall
1396:
Consider again the problem given above. The following sixth degree polynomial goes through all the seven points:
3481:
3456:
3144:
2877:
2862:
can be used if the number of data points is infinite or if the function to be interpolated has compact support.
2844:
1376:
1356:
64:
58:
3017:
3564:"Conservative interpolation of edge and face data on n dimensional structured grids using differential forms"
3330:
3177:
2941:
in three dimensions. They can be applied to gridded or scattered data. Mimetic interpolation generalizes to
2938:
2219:
1381:
Polynomial interpolation is a generalization of linear interpolation. Note that the linear interpolant is a
2892:
3727:
2930:
2872:
When each data point is itself a function, it can be useful to see the interpolation problem as a partial
2215:
1390:
161:
75:
3005:
2929:
Multivariate interpolation is the interpolation of functions of more than one variable. Methods include
3737:
3148:
2934:
2866:
2357:
2234:
2178:
1596:
142:
2403:
1557:
3397:"Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms"
2815:
curve through noisy data. In the geostatistics community
Gaussian process regression is also known as
157:; that is, estimate the value of that function for an intermediate value of the independent variable.
3575:
3468:
3342:
3074:
2597:
2211:
1626:
1616:
500:
442:
of interpolation, differing in such properties as: accuracy, cost, number of data points needed, and
177:
3403:, Fields Institute Communications, vol. 73, New York, NY: Springer New York, pp. 437–475,
2461:
1663:
1367:
3182:
3172:
2836:
1067:
1021:
943:
2545:
3486:
3432:
3404:
3366:
3303:
3282:
2204:
2196:
111:
2192:
1516:{\displaystyle f(x)=-0.0001521x^{6}-0.003130x^{5}+0.07321x^{4}-0.3577x^{3}+0.2255x^{2}+0.9038x.}
1607:
491:
3608:
3542:
3422:
3358:
3309:
3270:
3260:
3107:
3038:
2832:
3228:
3583:
3517:
3476:
3414:
3350:
2964:
2811:
1588:
413:
3457:"First- and Second-Order Conservative Remapping Schemes for Grids in Spherical Coordinates"
989:
3702:
3685:
3674:
3663:
3652:
3113:
1382:
443:
410:
Interpolation provides a means of estimating the function at intermediate points, such as
259:
192:
146:
3579:
3472:
3396:
3346:
3732:
3667:
3089:
2944:
2848:
2786:
2525:
2505:
2200:
236:
3721:
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3490:
3370:
3233:
3068:
3063:
3055:
3656:
3436:
3187:
3140:
2917:
dots correspond to the interpolated point and neighbouring samples, respectively.
168:
127:
458:
221:
3695:
3645:
3418:
149:, which represent the values of a function for a limited number of values of the
447:
173:
134:
123:
107:
3689:
3588:
3563:
3034:
1386:
1113:
119:
3362:
3274:
3242:. Vol. 14 (11th ed.). Cambridge University Press. pp. 706–710.
1158:
is twice continuously differentiable. Then the linear interpolation error is
930:{\displaystyle {\frac {y-y_{a}}{x-x_{a}}}={\frac {y_{b}-y_{a}}{x_{b}-x_{a}}}}
813:{\displaystyle {\frac {y-y_{a}}{y_{b}-y_{a}}}={\frac {x-x_{a}}{x_{b}-x_{a}}}}
17:
3354:
2873:
1630:
439:
3254:
2231:
Interpolation is a common way to approximate functions. Given a function
2852:
2816:
479:
interpolation (see below) is almost as easy, but in higher-dimensional
138:
940:
This previous equation states that the slope of the new line between
509:(2.5). Since 2.5 is midway between 2 and 3, it is reasonable to take
476:
3227:
3059:
is used to find data points outside the range of known data points.
3409:
1606:
1366:
490:
483:, this could be a favourable choice for its speed and simplicity.
457:
220:
2677:{\displaystyle \|f-s\|_{\infty }\leq C\|f^{(4)}\|_{\infty }h^{4}}
1541:
data points, there is exactly one polynomial of degree at most
43:
3331:"Mimetic Interpolation of Vector Fields on Arakawa C/D Grids"
524:
Generally, linear interpolation takes two data points, say (
3482:
10.1175/1520-0493(1999)127<2204:FASOCR>2.0.CO;2
2144:
2919:
Their heights above the ground correspond to their values.
2896:
Comparison of some 1- and 2-dimensional interpolations.
2876:
problem between each data point. This idea leads to the
2776:{\displaystyle h\max _{i=1,2,\dots ,n-1}|x_{i+1}-x_{i}|}
3713:
Interpolation via the
Chebyshev transform in Boost.Math
3164:– for interpolating within on a triangle or tetrahedron
495:
Plot of the data with linear interpolation superimposed
176:. The points in red are connected by blue interpolated
27:
Method for estimating new data within known data points
1371:
Plot of the data with polynomial interpolation applied
3506:"Optimal Error Bounds for Cubic Spline Interpolation"
3116:
3092:
2967:
2947:
2789:
2690:
2606:
2548:
2528:
2508:
2464:
2406:
2360:
2283:
2237:
1642:
1405:
1167:
1070:
1024:
992:
946:
829:
712:
569:
416:
262:
239:
195:
141:, one often has a number of data points, obtained by
189:
This table gives some values of an unknown function
3603:Crochiere, Ronald E.; Rabiner, Lawrence R. (1983).
3329:Pletzer, Alexander; Hayek, Wolfgang (2019-01-01).
3131:
3098:
2979:
2953:
2795:
2775:
2676:
2588:
2534:
2514:
2494:
2450:
2392:
2346:
2269:
2203:, for instance, since the line integral gives the
2184:(scalar, vector, pseudo-vector or pseudo-scalar).
2150:
1611:Plot of the data with spline interpolation applied
1515:
1344:
1102:
1056:
1010:
978:
929:
812:
695:
428:
277:
245:
210:
172:An interpolation of a finite set of points on an
3708:Barycentric rational interpolation in Boost.Math
3151:. There are also many other subsequent results.
2695:
1271:
228:
2187:A key feature of mimetic interpolation is that
2596:(four times continuously differentiable) then
1018:is the same as the slope of the line between
225:Plot of the data points as given in the table
8:
3562:Pletzer, Alexander; Fillmore, David (2015).
2655:
2635:
2620:
2607:
2347:{\displaystyle x_{1},x_{2},\dots ,x_{n}\in }
3504:Hall, Charles A.; Meyer, Weston W. (1976).
3067:error. In the simplest case this leads to
3287:: CS1 maint: location missing publisher (
1595:interpolation or restricting attention to
3587:
3521:
3480:
3408:
3115:
3091:
2966:
2946:
2788:
2768:
2762:
2743:
2734:
2698:
2689:
2668:
2658:
2642:
2623:
2605:
2559:
2547:
2527:
2507:
2463:
2439:
2417:
2405:
2386:
2385:
2359:
2320:
2301:
2288:
2282:
2263:
2262:
2236:
2112:
2086:
2070:
2027:
2001:
1985:
1945:
1919:
1903:
1863:
1837:
1821:
1781:
1755:
1739:
1696:
1676:
1658:
1641:
1495:
1479:
1463:
1447:
1431:
1404:
1385:. We now replace this interpolant with a
1334:
1312:
1301:
1288:
1274:
1260:
1248:
1241:
1231:
1218:
1200:
1168:
1166:
1091:
1078:
1069:
1045:
1032:
1023:
991:
967:
954:
945:
918:
905:
893:
880:
873:
861:
843:
830:
828:
801:
788:
776:
763:
751:
738:
726:
713:
711:
669:
660:
647:
635:
622:
611:
598:
580:
568:
415:
261:
238:
194:
122:, a method of constructing (finding) new
94:Learn how and when to remove this message
2891:
256:
167:
57:This article includes a list of general
3215:
2989:
2860:Whittaker–Shannon interpolation formula
3280:
3688:interpolation with visualisation and
462:Piecewise constant interpolation, or
7:
3696:Sol Tutorials - Interpolation Tricks
560:), and the interpolant is given by:
3627:Colin Bennett, Robert C. Sharpley,
3198:Radial basis function interpolation
3168:Brahmagupta's interpolation formula
3043:Multirate Digital Signal Processing
521:(3) = 0.1411, which yields 0.5252.
2659:
2624:
2393:{\displaystyle s:\to \mathbb {R} }
2270:{\displaystyle f:\to \mathbb {R} }
63:it lacks sufficient corresponding
25:
3401:Geometry, Mechanics, and Dynamics
3259:(Second ed.). Mineola, N.Y.
2451:{\displaystyle f(x_{i})=s(x_{i})}
1575:) ≈ 1.003 and a local minimum at
160:A closely related problem is the
3568:Journal of Computational Physics
3016:
3004:
2992:
2851:. Another possibility is to use
454:Piecewise constant interpolation
48:
3510:Journal of Approximation Theory
3377:from the original on 2022-06-07
1253:
1247:
3541:. Dover Books on Mathematics.
3126:
3120:
2769:
2735:
2649:
2643:
2583:
2580:
2568:
2565:
2495:{\displaystyle i=1,2,\dots ,n}
2445:
2432:
2423:
2410:
2382:
2379:
2367:
2341:
2329:
2259:
2256:
2244:
2135:
2123:
2050:
2038:
1968:
1956:
1886:
1874:
1804:
1792:
1719:
1707:
1652:
1646:
1415:
1409:
1335:
1331:
1325:
1313:
1307:
1281:
1238:
1211:
1201:
1197:
1191:
1182:
1176:
1169:
1097:
1071:
1051:
1025:
1005:
993:
973:
947:
471:Nearest-neighbor interpolation
464:nearest-neighbor interpolation
272:
266:
205:
199:
32:Interpolation (disambiguation)
1:
3203:Simple rational approximation
1103:{\displaystyle (x_{b},y_{b})}
1057:{\displaystyle (x_{a},y_{a})}
979:{\displaystyle (x_{a},y_{a})}
3539:Geometric Integration Theory
3523:10.1016/0021-9045(76)90040-X
3419:10.1007/978-1-4939-2441-7_19
3029:In digital signal processing
2600:has an error bound given by
2589:{\displaystyle f\in C^{4}()}
3224:Sheppard, William Fleetwood
2841:trigonometric interpolation
397:
379:
361:
343:
325:
307:
295:
3754:
3629:Interpolation of Operators
3253:Steffensen, J. F. (2006).
2925:Multivariate interpolation
2922:
2878:displacement interpolation
2598:cubic spline interpolation
2189:vector calculus identities
2176:
2165:(2.5) = 0.5972.
1614:
1374:
498:
481:multivariate interpolation
468:
254:
231:
153:. It is often required to
36:
29:
3589:10.1016/j.jcp.2015.08.029
3537:Whitney, Hassler (1957).
2961:dimensional spaces where
2845:trigonometric polynomials
2191:are satisfied, including
2354:one can form a function
1554:computational complexity
1377:Polynomial interpolation
1363:Polynomial interpolation
1357:polynomial interpolation
671: at the point
126:based on the range of a
37:Not to be confused with
3355:10.1175/MWR-D-18-0146.1
3239:Encyclopædia Britannica
3178:Imputation (statistics)
3162:Barycentric coordinates
2939:trilinear interpolation
2937:in two dimensions, and
2220:trilinear interpolation
78:more precise citations.
3461:Monthly Weather Review
3455:Jones, Philip (1999).
3335:Monthly Weather Review
3302:Kress, Rainer (1998).
3183:Lagrange interpolation
3133:
3100:
2981:
2980:{\displaystyle n>3}
2955:
2931:bilinear interpolation
2920:
2807:Via Gaussian processes
2797:
2777:
2678:
2590:
2536:
2516:
2496:
2452:
2394:
2348:
2271:
2227:Function approximation
2152:
1612:
1537:Generally, if we have
1517:
1372:
1346:
1104:
1058:
1012:
980:
931:
814:
697:
496:
466:
430:
429:{\displaystyle x=2.5.}
279:
247:
226:
212:
181:
130:of known data points.
3631:, Academic Press 1988
3229:"Interpolation"
3193:Newton–Cotes formulas
3173:Fractal interpolation
3149:Marcinkiewicz theorem
3134:
3101:
2982:
2956:
2935:bicubic interpolation
2895:
2882:transportation theory
2867:Hermite interpolation
2798:
2778:
2679:
2591:
2537:
2517:
2497:
2453:
2395:
2349:
2277:with a set of points
2272:
2179:Mimetic interpolation
2173:Mimetic interpolation
2153:
1610:
1597:Chebyshev polynomials
1518:
1370:
1347:
1105:
1059:
1013:
1011:{\displaystyle (x,y)}
981:
932:
815:
698:
513:(2.5) midway between
494:
469:Further information:
461:
431:
280:
248:
224:
213:
171:
3145:Riesz–Thorin theorem
3132:{\displaystyle f(x)}
3114:
3110:, and the function
3090:
3075:Approximation theory
2965:
2945:
2888:In higher dimensions
2843:is interpolation by
2787:
2688:
2604:
2546:
2526:
2506:
2462:
2404:
2358:
2281:
2235:
2161:In this case we get
1640:
1627:natural cubic spline
1617:Spline interpolation
1603:Spline interpolation
1530:= 2.5, we find that
1403:
1165:
1068:
1022:
990:
944:
827:
710:
567:
501:Linear interpolation
487:Linear interpolation
414:
278:{\displaystyle f(x)}
260:
237:
211:{\displaystyle f(x)}
193:
151:independent variable
30:For other uses, see
3580:2015JCoPh.302...21P
3473:1999MWRv..127.2204J
3347:2019MWRv..147....3P
3106:as a variable in a
2208:integration path.
1589:vertical asymptotes
1132:, and suppose that
3701:2021-01-31 at the
3684:2016-09-18 at the
3673:2016-08-20 at the
3662:2016-09-18 at the
3651:2016-09-18 at the
3305:Numerical Analysis
3129:
3096:
2977:
2951:
2921:
2833:rational functions
2793:
2773:
2733:
2674:
2586:
2532:
2512:
2492:
2448:
2390:
2344:
2267:
2205:electric potential
2197:divergence theorem
2148:
2143:
1625:For instance, the
1613:
1558:Runge's phenomenon
1534:(2.5) = ~0.59678.
1513:
1373:
1342:
1311:
1100:
1054:
1008:
976:
927:
810:
693:
497:
467:
426:
275:
243:
227:
208:
182:
112:numerical analysis
3644:Online tools for
3607:. Prentice-Hall.
3428:978-1-4939-2440-0
3266:978-0-486-15483-1
3108:topological space
3099:{\displaystyle x}
2954:{\displaystyle n}
2796:{\displaystyle C}
2694:
2535:{\displaystyle f}
2515:{\displaystyle s}
2115:
2030:
1948:
1866:
1784:
1699:
1270:
1268:
1251:
925:
868:
808:
758:
672:
667:
517:(2) = 0.9093 and
446:of the resulting
438:We describe some
408:
407:
246:{\displaystyle x}
104:
103:
96:
16:(Redirected from
3745:
3632:
3625:
3619:
3618:
3600:
3594:
3593:
3591:
3559:
3553:
3552:
3534:
3528:
3527:
3525:
3501:
3495:
3494:
3484:
3467:(9): 2204–2210.
3452:
3446:
3445:
3444:
3443:
3412:
3392:
3386:
3385:
3383:
3382:
3326:
3320:
3319:
3299:
3293:
3292:
3286:
3278:
3250:
3244:
3243:
3231:
3220:
3138:
3136:
3135:
3130:
3105:
3103:
3102:
3097:
3049:Related concepts
3020:
3008:
2999:Nearest neighbor
2996:
2986:
2984:
2983:
2978:
2960:
2958:
2957:
2952:
2916:
2912:
2908:
2904:
2900:
2880:problem used in
2837:Padé approximant
2812:Gaussian process
2802:
2800:
2799:
2794:
2782:
2780:
2779:
2774:
2772:
2767:
2766:
2754:
2753:
2738:
2732:
2683:
2681:
2680:
2675:
2673:
2672:
2663:
2662:
2653:
2652:
2628:
2627:
2595:
2593:
2592:
2587:
2564:
2563:
2541:
2539:
2538:
2533:
2521:
2519:
2518:
2513:
2501:
2499:
2498:
2493:
2457:
2455:
2454:
2449:
2444:
2443:
2422:
2421:
2399:
2397:
2396:
2391:
2389:
2353:
2351:
2350:
2345:
2325:
2324:
2306:
2305:
2293:
2292:
2276:
2274:
2273:
2268:
2266:
2157:
2155:
2154:
2149:
2147:
2146:
2116:
2113:
2091:
2090:
2075:
2074:
2031:
2028:
2006:
2005:
1990:
1989:
1949:
1946:
1924:
1923:
1908:
1907:
1867:
1864:
1842:
1841:
1826:
1825:
1785:
1782:
1760:
1759:
1744:
1743:
1700:
1697:
1681:
1680:
1522:
1520:
1519:
1514:
1500:
1499:
1484:
1483:
1468:
1467:
1452:
1451:
1436:
1435:
1351:
1349:
1348:
1343:
1338:
1324:
1316:
1310:
1306:
1305:
1293:
1292:
1269:
1261:
1252:
1249:
1246:
1245:
1236:
1235:
1223:
1222:
1204:
1172:
1109:
1107:
1106:
1101:
1096:
1095:
1083:
1082:
1063:
1061:
1060:
1055:
1050:
1049:
1037:
1036:
1017:
1015:
1014:
1009:
985:
983:
982:
977:
972:
971:
959:
958:
936:
934:
933:
928:
926:
924:
923:
922:
910:
909:
899:
898:
897:
885:
884:
874:
869:
867:
866:
865:
849:
848:
847:
831:
819:
817:
816:
811:
809:
807:
806:
805:
793:
792:
782:
781:
780:
764:
759:
757:
756:
755:
743:
742:
732:
731:
730:
714:
702:
700:
699:
694:
692:
688:
673:
670:
668:
666:
665:
664:
652:
651:
641:
640:
639:
623:
621:
617:
616:
615:
603:
602:
585:
584:
435:
433:
432:
427:
284:
282:
281:
276:
252:
250:
249:
244:
229:
217:
215:
214:
209:
99:
92:
88:
85:
79:
74:this article by
65:inline citations
52:
51:
44:
21:
3753:
3752:
3748:
3747:
3746:
3744:
3743:
3742:
3718:
3717:
3703:Wayback Machine
3686:Wayback Machine
3675:Wayback Machine
3664:Wayback Machine
3653:Wayback Machine
3641:
3636:
3635:
3626:
3622:
3615:
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3597:
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3449:
3441:
3439:
3429:
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3380:
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3327:
3323:
3316:
3301:
3300:
3296:
3279:
3267:
3252:
3251:
3247:
3222:
3221:
3217:
3212:
3207:
3157:
3112:
3111:
3088:
3087:
3086:If we consider
3084:
3071:approximation.
3051:
3031:
3024:
3021:
3012:
3009:
3000:
2997:
2963:
2962:
2943:
2942:
2927:
2918:
2914:
2910:
2906:
2902:
2898:
2897:
2890:
2825:
2809:
2803:is a constant.
2785:
2784:
2758:
2739:
2686:
2685:
2664:
2654:
2638:
2619:
2602:
2601:
2555:
2544:
2543:
2524:
2523:
2504:
2503:
2502:(that is, that
2460:
2459:
2435:
2413:
2402:
2401:
2356:
2355:
2316:
2297:
2284:
2279:
2278:
2233:
2232:
2229:
2193:Stokes' theorem
2181:
2175:
2142:
2141:
2110:
2082:
2066:
2057:
2056:
2025:
1997:
1981:
1975:
1974:
1943:
1915:
1899:
1893:
1892:
1861:
1833:
1817:
1811:
1810:
1779:
1751:
1735:
1726:
1725:
1694:
1672:
1659:
1638:
1637:
1619:
1605:
1491:
1475:
1459:
1443:
1427:
1401:
1400:
1383:linear function
1379:
1365:
1317:
1297:
1284:
1237:
1227:
1214:
1163:
1162:
1153:
1144:
1124:
1087:
1074:
1066:
1065:
1041:
1028:
1020:
1019:
988:
987:
963:
950:
942:
941:
914:
901:
900:
889:
876:
875:
857:
850:
839:
832:
825:
824:
797:
784:
783:
772:
765:
747:
734:
733:
722:
715:
708:
707:
678:
674:
656:
643:
642:
631:
624:
607:
594:
593:
589:
576:
565:
564:
559:
550:
541:
532:
503:
489:
473:
456:
412:
411:
258:
257:
235:
234:
191:
190:
187:
147:experimentation
100:
89:
83:
80:
70:Please help to
69:
53:
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
3751:
3749:
3741:
3740:
3735:
3730:
3720:
3719:
3716:
3715:
3710:
3705:
3693:
3640:
3639:External links
3637:
3634:
3633:
3620:
3613:
3595:
3554:
3548:978-0486445830
3547:
3529:
3516:(2): 105–122.
3496:
3447:
3427:
3387:
3321:
3314:
3294:
3265:
3245:
3234:Chisholm, Hugh
3214:
3213:
3211:
3208:
3206:
3205:
3200:
3195:
3190:
3185:
3180:
3175:
3170:
3165:
3158:
3156:
3153:
3128:
3125:
3122:
3119:
3095:
3083:
3082:Generalization
3080:
3050:
3047:
3030:
3027:
3026:
3025:
3022:
3015:
3013:
3010:
3003:
3001:
2998:
2991:
2976:
2973:
2970:
2950:
2923:Main article:
2889:
2886:
2849:Fourier series
2824:
2821:
2808:
2805:
2792:
2771:
2765:
2761:
2757:
2752:
2749:
2746:
2742:
2737:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2697:
2693:
2671:
2667:
2661:
2657:
2651:
2648:
2645:
2641:
2637:
2634:
2631:
2626:
2622:
2618:
2615:
2612:
2609:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2562:
2558:
2554:
2551:
2531:
2511:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2447:
2442:
2438:
2434:
2431:
2428:
2425:
2420:
2416:
2412:
2409:
2388:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2343:
2340:
2337:
2334:
2331:
2328:
2323:
2319:
2315:
2312:
2309:
2304:
2300:
2296:
2291:
2287:
2265:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2228:
2225:
2201:electric field
2177:Main article:
2174:
2171:
2159:
2158:
2145:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2111:
2109:
2106:
2103:
2100:
2097:
2094:
2089:
2085:
2081:
2078:
2073:
2069:
2065:
2062:
2059:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2026:
2024:
2021:
2018:
2015:
2012:
2009:
2004:
2000:
1996:
1993:
1988:
1984:
1980:
1977:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1944:
1942:
1939:
1936:
1933:
1930:
1927:
1922:
1918:
1914:
1911:
1906:
1902:
1898:
1895:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1862:
1860:
1857:
1854:
1851:
1848:
1845:
1840:
1836:
1832:
1829:
1824:
1820:
1816:
1813:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1780:
1778:
1775:
1772:
1769:
1766:
1763:
1758:
1754:
1750:
1747:
1742:
1738:
1734:
1731:
1728:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1695:
1693:
1690:
1687:
1684:
1679:
1675:
1671:
1668:
1665:
1664:
1662:
1657:
1654:
1651:
1648:
1645:
1615:Main article:
1604:
1601:
1524:
1523:
1512:
1509:
1506:
1503:
1498:
1494:
1490:
1487:
1482:
1478:
1474:
1471:
1466:
1462:
1458:
1455:
1450:
1446:
1442:
1439:
1434:
1430:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1375:Main article:
1364:
1361:
1353:
1352:
1341:
1337:
1333:
1330:
1327:
1323:
1320:
1315:
1309:
1304:
1300:
1296:
1291:
1287:
1283:
1280:
1277:
1273:
1267:
1264:
1259:
1256:
1244:
1240:
1234:
1230:
1226:
1221:
1217:
1213:
1210:
1207:
1203:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1171:
1149:
1140:
1120:
1114:differentiable
1099:
1094:
1090:
1086:
1081:
1077:
1073:
1053:
1048:
1044:
1040:
1035:
1031:
1027:
1007:
1004:
1001:
998:
995:
975:
970:
966:
962:
957:
953:
949:
938:
937:
921:
917:
913:
908:
904:
896:
892:
888:
883:
879:
872:
864:
860:
856:
853:
846:
842:
838:
835:
821:
820:
804:
800:
796:
791:
787:
779:
775:
771:
768:
762:
754:
750:
746:
741:
737:
729:
725:
721:
718:
704:
703:
691:
687:
684:
681:
677:
663:
659:
655:
650:
646:
638:
634:
630:
627:
620:
614:
610:
606:
601:
597:
592:
588:
583:
579:
575:
572:
555:
546:
537:
528:
499:Main article:
488:
485:
455:
452:
425:
422:
419:
406:
405:
402:
399:
396:
394:
391:
388:
387:
384:
381:
378:
376:
373:
370:
369:
366:
363:
360:
358:
355:
352:
351:
348:
345:
342:
340:
337:
334:
333:
330:
327:
324:
322:
319:
316:
315:
312:
309:
306:
304:
301:
298:
297:
294:
292:
289:
286:
285:
274:
271:
268:
265:
255:
253:
242:
232:
207:
204:
201:
198:
186:
183:
102:
101:
56:
54:
47:
39:Interpellation
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3750:
3739:
3736:
3734:
3731:
3729:
3728:Interpolation
3726:
3725:
3723:
3714:
3711:
3709:
3706:
3704:
3700:
3697:
3694:
3691:
3687:
3683:
3680:
3676:
3672:
3669:
3665:
3661:
3658:
3654:
3650:
3647:
3643:
3642:
3638:
3630:
3624:
3621:
3616:
3610:
3606:
3599:
3596:
3590:
3585:
3581:
3577:
3573:
3569:
3565:
3558:
3555:
3550:
3544:
3540:
3533:
3530:
3524:
3519:
3515:
3511:
3507:
3500:
3497:
3492:
3488:
3483:
3478:
3474:
3470:
3466:
3462:
3458:
3451:
3448:
3438:
3434:
3430:
3424:
3420:
3416:
3411:
3406:
3402:
3398:
3391:
3388:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3336:
3332:
3325:
3322:
3317:
3315:9781461205999
3311:
3307:
3306:
3298:
3295:
3290:
3284:
3276:
3272:
3268:
3262:
3258:
3257:
3256:Interpolation
3249:
3246:
3241:
3240:
3235:
3230:
3225:
3219:
3216:
3209:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3163:
3160:
3159:
3154:
3152:
3150:
3146:
3142:
3139:mapping to a
3123:
3117:
3109:
3093:
3081:
3079:
3076:
3072:
3070:
3069:least squares
3065:
3064:curve fitting
3060:
3058:
3057:
3056:extrapolation
3048:
3046:
3044:
3040:
3039:Nyquist limit
3036:
3028:
3019:
3014:
3007:
3002:
2995:
2990:
2988:
2974:
2971:
2968:
2948:
2940:
2936:
2932:
2926:
2894:
2887:
2885:
2883:
2879:
2875:
2870:
2868:
2863:
2861:
2856:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2829:interpolation
2822:
2820:
2818:
2813:
2806:
2804:
2790:
2763:
2759:
2755:
2750:
2747:
2744:
2740:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2691:
2669:
2665:
2646:
2639:
2632:
2629:
2616:
2613:
2610:
2599:
2577:
2574:
2571:
2560:
2556:
2552:
2549:
2529:
2522:interpolates
2509:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2440:
2436:
2429:
2426:
2418:
2414:
2407:
2376:
2373:
2370:
2364:
2361:
2338:
2335:
2332:
2326:
2321:
2317:
2313:
2310:
2307:
2302:
2298:
2294:
2289:
2285:
2253:
2250:
2247:
2241:
2238:
2226:
2224:
2221:
2217:
2213:
2209:
2206:
2202:
2198:
2194:
2190:
2185:
2180:
2172:
2170:
2166:
2164:
2138:
2132:
2129:
2126:
2120:
2117:
2107:
2104:
2101:
2098:
2095:
2092:
2087:
2083:
2079:
2076:
2071:
2067:
2063:
2060:
2053:
2047:
2044:
2041:
2035:
2032:
2022:
2019:
2016:
2013:
2010:
2007:
2002:
1998:
1994:
1991:
1986:
1982:
1978:
1971:
1965:
1962:
1959:
1953:
1950:
1940:
1937:
1934:
1931:
1928:
1925:
1920:
1916:
1912:
1909:
1904:
1900:
1896:
1889:
1883:
1880:
1877:
1871:
1868:
1858:
1855:
1852:
1849:
1846:
1843:
1838:
1834:
1830:
1827:
1822:
1818:
1814:
1807:
1801:
1798:
1795:
1789:
1786:
1776:
1773:
1770:
1767:
1764:
1761:
1756:
1752:
1748:
1745:
1740:
1736:
1732:
1729:
1722:
1716:
1713:
1710:
1704:
1701:
1691:
1688:
1685:
1682:
1677:
1673:
1669:
1666:
1660:
1655:
1649:
1643:
1636:
1635:
1634:
1632:
1628:
1623:
1618:
1609:
1602:
1600:
1598:
1592:
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1561:
1559:
1555:
1550:
1548:
1544:
1540:
1535:
1533:
1529:
1526:Substituting
1510:
1507:
1504:
1501:
1496:
1492:
1488:
1485:
1480:
1476:
1472:
1469:
1464:
1460:
1456:
1453:
1448:
1444:
1440:
1437:
1432:
1428:
1424:
1421:
1418:
1412:
1406:
1399:
1398:
1397:
1394:
1392:
1388:
1384:
1378:
1369:
1362:
1360:
1358:
1339:
1328:
1321:
1318:
1302:
1298:
1294:
1289:
1285:
1278:
1275:
1265:
1262:
1257:
1254:
1242:
1232:
1228:
1224:
1219:
1215:
1208:
1205:
1194:
1188:
1185:
1179:
1173:
1161:
1160:
1159:
1157:
1152:
1148:
1143:
1139:
1136:lies between
1135:
1131:
1126:
1123:
1119:
1116:at the point
1115:
1110:
1092:
1088:
1084:
1079:
1075:
1046:
1042:
1038:
1033:
1029:
1002:
999:
996:
968:
964:
960:
955:
951:
919:
915:
911:
906:
902:
894:
890:
886:
881:
877:
870:
862:
858:
854:
851:
844:
840:
836:
833:
823:
822:
802:
798:
794:
789:
785:
777:
773:
769:
766:
760:
752:
748:
744:
739:
735:
727:
723:
719:
716:
706:
705:
689:
685:
682:
679:
675:
661:
657:
653:
648:
644:
636:
632:
628:
625:
618:
612:
608:
604:
599:
595:
590:
586:
581:
577:
573:
570:
563:
562:
561:
558:
554:
549:
545:
540:
536:
531:
527:
522:
520:
516:
512:
508:
502:
493:
486:
484:
482:
478:
472:
465:
460:
453:
451:
449:
445:
441:
436:
423:
420:
417:
403:
400:
395:
392:
390:
389:
385:
382:
377:
374:
372:
371:
367:
364:
359:
356:
354:
353:
349:
346:
341:
338:
336:
335:
331:
328:
323:
320:
318:
317:
313:
310:
305:
302:
300:
299:
293:
290:
288:
287:
269:
263:
240:
233:
230:
223:
219:
202:
196:
184:
179:
178:spline curves
175:
170:
166:
163:
162:approximation
158:
156:
152:
148:
144:
140:
136:
131:
129:
125:
121:
118:is a type of
117:
116:interpolation
113:
109:
98:
95:
87:
77:
73:
67:
66:
60:
55:
46:
45:
40:
33:
19:
18:Interpolating
3738:Video signal
3692:source code.
3668:cubic spline
3628:
3623:
3604:
3598:
3571:
3567:
3557:
3538:
3532:
3513:
3509:
3499:
3464:
3460:
3450:
3440:, retrieved
3400:
3390:
3379:. Retrieved
3338:
3334:
3324:
3308:. Springer.
3304:
3297:
3255:
3248:
3237:
3218:
3188:Missing data
3141:Banach space
3085:
3073:
3061:
3054:
3052:
3042:
3032:
2928:
2871:
2864:
2857:
2828:
2826:
2810:
2230:
2210:
2186:
2182:
2167:
2162:
2160:
1624:
1620:
1593:
1584:
1580:
1576:
1572:
1568:
1564:
1562:
1551:
1546:
1542:
1538:
1536:
1531:
1527:
1525:
1395:
1380:
1354:
1155:
1150:
1146:
1141:
1137:
1133:
1129:
1127:
1121:
1117:
1111:
939:
556:
552:
547:
543:
538:
534:
529:
525:
523:
518:
514:
510:
506:
504:
474:
437:
409:
188:
159:
154:
132:
128:discrete set
115:
108:mathematical
105:
90:
84:October 2016
81:
62:
3341:(1): 3–16.
2823:Other forms
448:interpolant
174:epitrochoid
155:interpolate
135:engineering
124:data points
76:introducing
3722:Categories
3690:JavaScript
3679:polynomial
3614:0136051626
3442:2022-06-15
3381:2022-06-07
3210:References
3035:Upsampling
2869:problems.
2400:such that
1389:of higher
1387:polynomial
450:function.
444:smoothness
120:estimation
59:references
3657:quadratic
3574:: 21–40.
3491:122744293
3410:0707.4470
3371:125214770
3363:1520-0493
3283:cite book
3275:867770894
3053:The term
2874:advection
2756:−
2727:−
2718:…
2660:∞
2656:‖
2636:‖
2630:≤
2625:∞
2621:‖
2614:−
2608:‖
2553:∈
2484:…
2383:→
2327:∈
2311:…
2260:→
2121:∈
2093:−
2061:−
2036:∈
2008:−
1992:−
1954:∈
1935:−
1910:−
1872:∈
1853:−
1828:−
1790:∈
1771:−
1746:−
1730:−
1705:∈
1667:−
1631:piecewise
1579:≈ 4.708,
1567:≈ 1.566,
1470:−
1438:−
1425:0.0001521
1422:−
1279:∈
1225:−
1206:≤
1186:−
1154:and that
912:−
887:−
855:−
837:−
795:−
770:−
745:−
720:−
654:−
629:−
605:−
110:field of
3699:Archived
3682:Archived
3671:Archived
3660:Archived
3649:Archived
3437:15194760
3375:Archived
3226:(1911).
3155:See also
3147:and the
3011:Bilinear
2853:wavelets
2216:bilinear
2195:and the
2114:if
2029:if
1947:if
1865:if
1783:if
1698:if
1441:0.003130
1322:″
398:−0
380:−0
362:−0
143:sampling
3576:Bibcode
3469:Bibcode
3343:Bibcode
3236:(ed.).
3023:Bicubic
2817:Kriging
2105:34.9282
2096:19.3370
1979:0.05375
1733:0.01258
1457:0.07321
542:) and (
440:methods
185:Example
139:science
106:In the
72:improve
3677:, and
3646:linear
3611:
3545:
3489:
3435:
3425:
3369:
3361:
3312:
3273:
3263:
2907:yellow
2847:using
2839:, and
2835:using
2684:where
2212:Linear
2080:3.3673
2064:0.1871
2020:4.8259
2011:1.2756
1995:0.2450
1938:1.8381
1929:3.7225
1913:1.4945
1897:0.1579
1856:1.3623
1847:3.2467
1831:1.3359
1815:0.1403
1774:0.1396
1765:1.4126
1749:0.4189
1686:0.9937
1670:0.1522
1505:0.9038
1489:0.2255
1473:0.3577
1391:degree
477:linear
61:, but
3733:Video
3487:S2CID
3433:S2CID
3405:arXiv
3367:S2CID
3232:. In
2911:green
2899:Black
1250:where
404:2794
386:9589
368:7568
350:1411
332:9093
314:8415
3609:ISBN
3543:ISBN
3423:ISBN
3359:ISSN
3310:ISBN
3289:link
3271:OCLC
3261:ISBN
2972:>
2933:and
2915:blue
2901:and
2858:The
2783:and
2458:for
2218:and
1145:and
1064:and
986:and
424:2.5.
137:and
3584:doi
3572:302
3518:doi
3477:doi
3465:127
3415:doi
3351:doi
3339:147
3062:In
2903:red
2831:by
2696:max
1629:is
1560:).
1272:max
145:or
133:In
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2427:=
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2017:+
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1987:3
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1969:]
1966:4
1963:,
1960:3
1957:[
1951:x
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871:=
863:a
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761:=
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676:(
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574:=
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91:(
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20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
