Fourier series

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31: 22.20 Methods of Computation

… ► $\mathrm{am}(x,k)$ can be computed from its definition (22.16.1) or from its Fourier series (22.16.9). …

32: 3.11 Approximation Techniques

… ►In fact, (3.11.11) is the Fourier-series expansion of $f(\cos \theta )$; compare (3.11.6) and §1.8(i). … ►to the maximum error of the minimax polynomial $p_n(x)$ is bounded by $1+L_n$, where $L_n$ is the $n$th Lebesgue constant for Fourier series; see §1.8(i). …

33: Bibliography C

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Introduction to the Theory of Fourier’s Series and Integrals. 3rd edition, Macmillan, London.

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Notes:

Reprinted by Dover, New York, 1950.

External Links:

ZentralBlatt

Referenced by:

§6.16(i).

Encodings:

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An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (90), pp. 297–301.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§3.11(v).

Encodings:

BibTeX

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34: Bibliography S

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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

BibTeX

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35: 18.2 General Orthogonal Polynomials

… ►For such a system, functions $f\in L_w^2((a,b))$ and sequences $\{{\lambda }_n\}$ ($n=0,1,2,\mathrm{\dots }$) satisfying can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2): …

36: Errata

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Originally it was stated that these Fourier series converge “…conditionally when $\nu$ is real and .” It has been corrected to read “If then they converge, but, if $\theta \ne \frac{1}{2}\pi$, they do not converge absolutely.”

Reported by Hans Volkmer on 2021-06-04

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37: 28.11 Expansions in Series of Mathieu Functions

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28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28

38: 28.15 Expansions for Small $q$

§28.15 Expansions for Small $q$

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§28.15(i) Eigenvalues ${\lambda }_{\nu }\left(q\right)$

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§28.15(ii) Solutions ${\mathrm{me}}_{\nu }(z,q)$

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39: 18.17 Integrals

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§18.17(v) Fourier Transforms

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Jacobi

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Ultraspherical

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Legendre

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Hermite

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40: Bibliography R

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Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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External Links:

ISSN 0098-3500, Document

Referenced by:

1st item.

Encodings:

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Fourier analysis and signal processing by use of the Möbius inversion formula. IEEE Trans. Acoustics, Speech, Signal Processing 38, pp. 458–470.

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External Links:

ZentralBlatt

Referenced by:

§27.17.

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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.

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External Links:

ISBN 978-0-12-585002-5, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

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Elliptic hypergeometric series on root systems. Adv. Math. 181 (2), pp. 417–447.

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External Links:

ISSN 0001-8708, Document, MathReview, ZentralBlatt

Referenced by:

§20.11(iii).

Encodings:

BibTeX

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Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:

McGraw-Hill Series in Higher Mathematics

External Links:

MathReview (F. Smithies), ZentralBlatt

Referenced by:

§1.18(x), §2.6(i).

Encodings:

BibTeX

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