Fourier transform

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1: 1.14 Integral Transforms

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§1.14(i) Fourier Transform

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Inversion

… ►In this subsection we let $F_c(x)={ℱ}_{c}f\left(x\right)$, $F_s(x)={ℱ}_{s}f\left(x\right)$, $G_c(x)={ℱ}_{c}g\left(x\right)$, and $G_s(x)={ℱ}_{s}g\left(x\right)$. … ►

Fourier Transform

… ►where $ℱf\left(x\right)$ is given by (1.14.1). …

2: 1.16 Distributions

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§1.16(vii) Fourier Transforms of Tempered Distributions

… ►Then its Fourier transform is … ►The Fourier transform $ℱ\left(u\right)$ of a tempered distribution is again a tempered distribution, and … ►

§1.16(viii) Fourier Transforms of Special Distributions

… ►Since $\sqrt{2\pi }ℱ\left(\delta \right)=1$, we have …

3: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …

4: 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. …

5: Wolter Groenevelt

… ►Groenevelt has research interests in special functions, (matrix valued) orthogonal polynomials, moment problems, generalized Fourier transforms in relations with mathematical objects such as Lie algebras, quantum groups and affine Hecke algebras. …

6: 15.14 Integrals

… ►Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …

7: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
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Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform"	both $1/(2\pi )$ and the phrase “Fourier transform”.
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8: 30.15 Signal Analysis

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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 $[-\sigma ,\sigma ]$. …

9: Errata

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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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10: 3.11 Approximation Techniques

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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 …The method of the fast Fourier transform (FFT) exploits the structure of the matrix $𝛀$ with elements ${\omega }_n^{jk}$, $j,k=0,1,\mathrm{\dots },n-1$. …