Fourier coefficients

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21: Bibliography T

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Angular momentum coupling coefficients. Comput. Phys. Comm. 1 (5), pp. 337–342.

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Single-precision Fortran

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11st item.

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Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt

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§22.11.

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Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts, Vol. 43, Cambridge University Press, Cambridge.

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ISBN 0-521-45718-1, MathReview (Stefan Kühnlein), ZentralBlatt

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§5.16.

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Introduction to the Theory of Fourier Integrals. Third edition, Chelsea Publishing Co., New York.

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Original edition was published in 1937 by Oxford University Press. Expanded bibliography and some improvements were made for the second and third editions.

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ISBN 0-8284-0324-4, MathReview, ZentralBlatt

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§1.14(i), §1.14(ii), §1.14(iv), §1.14(v), §1.14(vii), §1.8(iv), §10.22(v), §10.32(iii), §5.13.

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Fourier Series. Prentice-Hall Inc., Englewood Cliffs, N.J..

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Translated from the Russian by Richard A. Silverman. Second English edition published by Dover Publications Inc., New York, 1976.

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MathReview (R. P. Boas, Jr.), ZentralBlatt

Referenced by:

§1.8(i), §1.8(ii).

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22: 15.17 Mathematical Applications

… ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

23: 27.10 Periodic Number-Theoretic Functions

… ►Every function periodic (mod $k$) can be expressed as a finite Fourier series of the form … ►is a periodic function of $n(modk)$ and has the finite Fourier-series expansion …where ►

27.10.8 $$a_k(m)=\sum _{d|\left(m,k\right)}g(d)f\left(\frac{k}{d}\right)\frac{d}{k}.$$

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$\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer, $m$: positive integer, $f(n)$: function, $g(m)$: periodic function and $a_k(m)$: coefficients

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http://dlmf.nist.gov/27.10.E8

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Annotations for §27.10 and Ch.27

… ►The finite Fourier expansion of a primitive Dirichlet character $\chi (modk)$ has the form …

24: Bibliography V

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Computational Frameworks for the Fast Fourier Transform. Frontiers in Applied Mathematics, Vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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ISBN 0-89871-285-8, MathReview (S. Hitotumatu), ZentralBlatt

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§3.11(v).

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On the coefficients of the modular invariant $J(\tau )$ . Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 56, pp. 389–400.

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MathReview (J. Lehner), ZentralBlatt

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§23.17(ii).

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

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Fourier series representation of Ferrers function $𝖯$ .

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FullText

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(14.13.1).

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25: 2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►where the coefficients $b_s$ are defined by the expansion … ►Then …where the coefficients $a_s$ are given by (2.3.7). … ►The coefficients $b_s$ are defined as in §2.3(iii). …

26: 1.14 Integral Transforms

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§1.14(i) Fourier Transform

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Inversion

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Convolution

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Uniqueness

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Fourier Transform

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27: 2.10 Sums and Sequences

… ►where …This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►

§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

►Let $f(z)$ be analytic on the annulus , with Laurent expansion … ►We need a “comparison function” $g(z)$ with the properties: …

28: 3.11 Approximation Techniques

… ►with coefficients given in Table 3.11.1. … ►

Example. The Discrete Fourier Transform

… ►is called a discrete Fourier transform pair. ►

The Fast Fourier Transform

►The direct computation of the discrete Fourier transform (3.11.38), that is, of …

29: Bibliography S

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

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§2.3(iv).

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Prolate spheroidal wave functions, Fourier analysis and uncertainty. I. Bell System Tech. J. 40, pp. 43–63.

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MathReview (I. Marx), ZentralBlatt

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§30.15(v).

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A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.

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ISBN 0-8493-8273-4, MathReview (J. Horváth), ZentralBlatt

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§18.17(v).

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

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

Referenced by:

§17.3(i).

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30: Bibliography F

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A computer program for the calculation of angular-momentum coupling coefficients. Comput. Phys. Comm. 70 (1), pp. 147–153.

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Notes:

Fortran for exact calculation using integers

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ISSN 0010-4655, Document, Code, MathReview Entry

Referenced by:

§34.13, §34.13, 5th item.

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

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