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can be seen as functions with finite energy (L^1 and L^2)(is a function of the time which has a compact support and is in L^00) and a compact spectrum so it’s possible to represent their information (which has the power of the continuum) with discrete series. Moreover we can cut a part of this series and a part of the spectrum of the signal to obtain an MP3. (Using spectral analysis techniques). Gibbs phenomenon can be seen when we compress pictures made of color spots with sharp outlines (the ending titles of a DiVX) so to compress animated cartoons is better not to use a method related with
Fourier series (for example we can use GIF).
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sense, and the fact that the series converges for every periodic distribution in this sense. (This is arguably the sense that is used in most practical contexts nowadays, as
Fourier series usually come hand-in-hand with other distribution concepts such as delta functions. It also gives a concrete
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I think that convergence of
Fourier series is not useless for engineers. In fact Fourier series and Fourier transform is used in signals compression (for example JPEG or MP3) in this contexts it's very important to know in which sense the information is reconstructed. Musical signals, for example,
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The original definition of a trigonometric series does not require f to be periodic. This is reflected in the convergence criteria. For example, a function of bounded variation or a function with existing left and right derivatives does not need to be periodic. It would be good to add a paragraph
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in harmonic analysis. The
Fourier transform for jpegs and mp3s can be viewed strictly in the discrete context, in which case the convergence is moot (since the Fourier series is a finite sum.) The remaining comments do not pertain to the convergence of Fourier series.
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article, because it's about
Fourier series whereas the article is about Fourier transforms. If it belongs anywhere then it would be in this article, so I'm posting it here in case someone wants to integrate it into this article. By the way, feel free to
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is a contradiction. Any nonzero function which is compactly supported has necessarely unbounded spectrum and any function with compact spectrum has necessarely unbounded support. This is a very elementary version of the
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As Y. Katznelson shows in his textbook: when f is absolutely continuous, n times its n-th
Fourier coefficient is not only bounded (as currently says this article with other words) but tends to 0 with n -:
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Dirichlet made an important historical contribution, as well as Jordan, Wilbraham, Riemann, Cantor, etc.. A more detailed history is available in the French version of the
Fourier Series article, see
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which cites the book "Introduction To
Classical Real Analysis". It's not stated explicitly in the book. Also, the claim can't be right. For instance, consider
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I do not have access to
Jackson's book, so could someone else double-check the formula for the speed of uniform convergence? If f is a polynomial of degree
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allows the
Fourier transform to be extended to a unitary operator on the Hilbert space of all square-integrable functions, i.e., all functions satisfying
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The proposition that the family of absolutely converging Fourier series is a Banach Algebra is wrong, It is a vectorial space which is not complete.
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I think one should add in the section on summability that the uniform convergence by Fejer's theorem only holds for continuous, periodic functions.
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Should we say something about when Fourier series converge uniformly? (I don't know the details off the top of my head, else I would do it.) --
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way to understand "almost everywhere" convergence, since finite errors on sets of measure zero obviously don't affect weak convergence.)
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855:{\displaystyle \lim _{N\rightarrow \infty }\int _{-\pi }^{\pi }\left|f(x)-\sum _{n=-N}^{N}{\widehat {f}}(n)\,e^{inx}\right|^{2}\,dx=0.}
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has a modulus of continuity which is identically zero. But clearly the approximation error will not be zero for any finite
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supposed to take, change the coefficient 1000 above to 1,000,000 or 1,000,000,000 - it's bound to be wrong eventually. --
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which is not that of uniform convergence like for the space of continuous functions. Look in literature about the
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Under "Summability" the first summation says "summation of A sub n" but I believe that should be "A sub k."
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For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:
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It is known as a Banach algebra with ordinary pointwise multiplication of functions, but its
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I'm not confident enough to make this change myself. Someone who knows more please do this:
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that explains where this periodicity requirment comes from and when it is necessary.
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http://fr.wikipedia.org/search/?title=S%C3%A9rie_de_Fourier&oldid=30788202
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Where does Dirichlet's line of proof fit into all this? The text of it is at
689:{\displaystyle \sum _{n=-\infty }^{\infty }{\widehat {f}}(n)\,e^{inx}=f(x).}
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https://en.wikipedia.org/Convergence_of_Fourier_series#Uniform_convergence
1007:(1966). On the convergence and growth of partial sums of Fourier series.
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Deleted content from Fourier inversion theorem - suitable for here?
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is a square-integrable periodic function on the interval
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is square-integrable, then for "almost every" value of
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Hi, I'm about to delete the above material from the
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This was not proved until 1966 in (Carleson, 1966).
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About the case of an absolutely continuous function
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1141:Pretty sure uniform convergence bound is wrong
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109:and see a list of open tasks.
1497:B-Class mathematics articles
1110:13:38, 14 October 2014 (UTC)
205:02:35, 4 January 2008 (UTC)
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1483:18:18, 22 April 2024 (UTC)
1461:17:43, 22 April 2024 (UTC)
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869:? That would say that if
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237:and search for Dirichlet.
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1241:{\displaystyle \alpha =1}
1025:Fourier inversion theorem
877:between 0 and 2π we have
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1215:{\displaystyle \alpha }
1081:{\displaystyle f^{(p)}}
865:What about convergence
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295:— Steven G. Johnson
191:Uniform convergence
180:Mistake in formula?
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112:Mathematics
103:mathematics
59:Mathematics
1491:Categories
1413:Discussion
1014:, 135–157.
1009:Acta Math.
197:Walt Pohl
1445:own norm
1433:contribs
1421:unsigned
1123:unsigned
428:In case
262:unsigned
1400:Svennik
139:on the
30:B-class
1475:UKe-CH
1453:UKe-CH
239:Loisel
220:Ayacop
173:Loisel
36:scale.
1055:then
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1094:N=1
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131:Mid
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245:)
226:)
218:--
203:)
1477:(
1455:(
1427:(
1402:(
1386:K
1366:K
1343:.
1332:N
1327:N
1315:K
1308:|
1304:)
1301:x
1298:(
1295:)
1292:f
1287:N
1283:S
1279:(
1273:)
1270:x
1267:(
1264:f
1260:|
1236:1
1233:=
1190:)
1187:x
1181:(
1169:=
1166:)
1163:x
1160:(
1157:f
1129:(
1104:(
1098:N
1090:N
1074:)
1071:p
1068:(
1064:f
1053:p
1036:(
982:.
977:x
974:n
971:i
967:e
961:)
958:n
955:(
946:f
938:N
933:N
927:=
924:n
908:N
900:=
897:)
894:x
891:(
888:f
875:x
871:f
847:=
844:x
841:d
834:2
829:|
823:x
820:n
817:i
813:e
807:)
804:n
801:(
792:f
784:N
779:N
773:=
770:n
759:)
756:x
753:(
750:f
746:|
715:N
684:.
681:)
678:x
675:(
672:f
669:=
664:x
661:n
658:i
654:e
648:)
645:n
642:(
633:f
614:=
611:n
580:.
577:x
574:d
567:x
564:n
561:i
554:e
548:)
545:x
542:(
539:f
512:2
508:1
503:=
500:)
497:n
494:(
485:f
455:]
449:,
440:[
430:f
423:L
406:.
397:x
394:d
387:2
382:|
378:)
375:x
372:(
369:f
365:|
297:(
268:(
241:(
222:(
199:(
143:.
42::
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