Fourier transform

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11: 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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12: 24.19 Methods of Computation

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A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

13: 13.10 Integrals

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§13.10(iv) Fourier Transforms

… ►For additional Fourier transforms see Erdélyi et al. (1954a, §§1.14, 2.14, 3.3) and Oberhettinger (1990, §§1.22, 2.22). …

14: 1.8 Fourier Series

… ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. …

15: 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).

Encodings:

BibTeX

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16: 13.23 Integrals

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§13.23(ii) Fourier Transforms

… ►For additional Fourier transforms see Erdélyi et al. (1954a, §§1.14, 2.14, 3.3) and Oberhettinger (1990, §§1.22, 2.22). …

17: 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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External Links:

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

Referenced by:

§3.11(v).

Encodings:

BibTeX

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18: 1.15 Summability Methods

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1.15.36 $$h(x,y)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-y\left|t\right|}{\mathrm{e}}^{-\mathrm{i}xt}F(t)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F(x)$: Fourier transform of $f(t)$ , $h(x,y)$: function and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

►where $F(t)$ is the Fourier transform of $f(x)$ (§1.14(i)). … ►

1.15.44 $${\sigma }_R(\theta )=\frac{1}{\sqrt{2\pi }}{\int }_{-R}^R\left(1-\frac{\left|t\right|}{R}\right){\mathrm{e}}^{-\mathrm{i}\theta t}F(t)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F(x)$: Fourier transform of $f(t)$ , ${\sigma }_R(\theta )$: function and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.15.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

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19: Bibliography W

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Asymptotic expansions of Fourier transforms of functions with logarithmic singularities. J. Math. Anal. Appl. 64 (1), pp. 173–180.

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External Links:

Document, MathReview (B. L. Gupta), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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20: 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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External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

§18.17(v).

Encodings:

BibTeX

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