Fourier coefficients

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31: Bibliography C

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An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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External Links:

ISSN 0010-4655, Document, MathReview

Referenced by:

§10.74(vii).

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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).

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Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.

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Notes:

Erratum: see same journal 125(1996), no. 2, p. 391.

External Links:

ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

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A method for numerical calculation of Fourier integrals. Numer. Math. 8 (5), pp. 422–436.

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External Links:

ISSN 0029-599X, Document, MathReview (M. P. S. Madan), ZentralBlatt

Referenced by:

§3.5(vii).

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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).

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32: 29.6 Fourier Series

§29.6 Fourier Series

… ►with ${\alpha }_p$, ${\beta }_p$, and ${\gamma }_p$ as in (29.3.11) and (29.3.12), and … ►An alternative version of the Fourier series expansion (29.6.1) is given by … ►with ${\alpha }_p$, ${\beta }_p$, and ${\gamma }_p$ as in (29.3.13) and (29.3.14), and ►

29.6.20 $$\sum _{p=0}^{\mathrm{\infty }}{A}_{2p+1}^2=1,$$

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Symbols:

$p$: nonnegative integer and $A_{2p}$: coefficients

Referenced by:

§29.6(ii)

Permalink:

http://dlmf.nist.gov/29.6.E20

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See also:

Annotations for §29.6(ii), §29.6 and Ch.29

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33: Errata

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A sentence indicating that explicit representations for the coefficients $c_k$ are given in Volkmer (2023) was inserted just below (16.11.5).

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Equations: (3.3.3_1), (3.3.3_2), (5.15.9) (suggested by Calvin Khor on 2021-09-04), (8.15.2), Pochhammer symbol representation in (10.17.1) for $a_k(\nu )$ coefficient, Pochhammer symbol representation in (11.9.4) for $a_k(\mu ,\nu )$ coefficient, and (12.14.4_5).

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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$ℱ\left(f\right)\left(x\right)$	$ℱf\left(x\right)$
Fourier Cosine	${ℱ}_c\left(f\right)\left(x\right)$	${ℱ}_{c}f\left(x\right)$
Fourier Sine	${ℱ}_s\left(f\right)\left(x\right)$	${ℱ}_{s}f\left(x\right)$
Laplace	$ℒ\left(f\right)\left(s\right)$	$ℒf\left(s\right)$	$ℒ(f(t);s)$
Mellin	$ℳ\left(f\right)\left(s\right)$	$ℳf\left(s\right)$	$ℳ(f;s)$
Hilbert	$\mathscr{H}\left(f\right)\left(s\right)$	$\mathscr{H}f\left(s\right)$	$\mathscr{H}(f;s)$
Stieltjes	$𝒮\left(f\right)\left(s\right)$	$𝒮f\left(s\right)$	$𝒮(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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Several changes have been made to

1. make consistent use of the Fourier transform notations $ℱ\left(f\right)$, $ℱ\left(\varphi \right)$ and $ℱ\left(u\right)$ where $f$ is a function of one real variable, $\varphi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. introduce the partial differential operator $𝐃$ in (1.16.30);

3. clarify the definition (1.16.32) of the partial differential operator $P(𝐃)$; and

4. clarify the use of $P(𝐃)$ and $P(𝐱)$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

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An entire new Subsection 1.16(viii) Fourier Transforms of Special Distributions, was contributed by Roderick Wong.

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34: Bibliography Z

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Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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External Links:

ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

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Notes:

English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905

External Links:

ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

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The computation of the smaller coefficients of $J(\tau )$ . Bull. Amer. Math. Soc. 45 (12), pp. 917–919.

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External Links:

FullText, MathReview (H. Rademacher), ZentralBlatt

Referenced by:

§23.17(ii).

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35: 30.15 Signal Analysis

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30.15.2 $${\mathrm{\Lambda }}_n=\frac{2\gamma }{\pi }{\left(K_n^0(\gamma )A_n^0({\gamma }^2)\right)}^2;$$

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Defines:

$\mathrm{\Lambda }$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ , $K_n^m(\gamma )$: connection coefficient and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.15.E2

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See also:

Annotations for §30.15(i), §30.15 and Ch.30

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§30.15(iii) Fourier Transform

… ►Equations (30.15.4) and (30.15.6) show that the functions ${\varphi }_n$ are $\sigma$-bandlimited, that is, their Fourier transform vanishes outside the interval $[-\sigma ,\sigma ]$. …

36: 18.2 General Orthogonal Polynomials

… ►Then, with the coefficients (18.2.11_4) associated with the monic OP’s $p_n$, the orthonormal recurrence relation for $q_n$ takes the form … ►The monic and orthonormal OP’s, and their determination via recursion, are more fully discussed in §§3.5(v) and 3.5(vi), where modified recursion coefficients are listed for the classical OP’s in their monic and orthonormal forms. … ►The recurrence coefficients ${\alpha }_n$ and ${\beta }_n$ in (18.2.11_5) can be expressed in terms of the determinants (18.2.27) and (18.2.28) by …Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii). … ►for certain coefficients $a_{n,j}$ with $s,t$ independent of $n$. …

37: 18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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… ►For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). …

38: Bibliography O

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Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

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Fourier Expansions. A Collection of Formulas. Academic Press, New York-London.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.8(v), §22.11, §4.27.

Encodings:

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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt

Referenced by:

§36.12(iii).

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii), §2.7(ii).

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39: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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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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Tables of the integrals and differential coefficients of $\mathrm{Gi}(+x)$ and $\mathrm{Hi}(-x)$ . Quart. J. Mech. Appl. Math. 7 (3), pp. 379–384.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

(9.12.30), (9.12.31), §9.12(viii), §9.16, 2nd item.

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