Fourier transform of

SearchAdvancedHelp

(0.001 seconds)

1—10 of 42 matching pages

1: 1.16 Distributions

… ►

§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 …

2: 1.14 Integral Transforms

… ►

§1.14(i) Fourier Transform

… ►

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). …

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: 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). …

6: Guide to Searching the DLMF

… ►

Table 1: Query Examples

► ►►►►►

Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
…
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”.
…

►

…

7: 37.11 Spherical Harmonics

… ►See Braaksma and Meulenbeld (1968). ►

Fourier Transform Involving Spherical Harmonics

►The Fourier transform of a function $f$ on ${ℝ}^d$ is a function $ℱ\left(f\right)$ on ${ℝ}^d$ which is defined by the Fourier integral (1.16.29). … ►

37.11.30 $$ℱ\left(f\right)\left(\rho \xi \right)={\mathrm{i}}^{n}Y(\xi ){(2\pi \rho )}^{1-\frac{1}{2}d}{\int }_0^{\mathrm{\infty }}{f}_0(r)J_{\frac{1}{2}d+n-1}\left(\rho r\right)r^{\frac{1}{2}d}dr,$$ $\rho >0$, $\xi \in {𝕊}^{d-1}$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $ℱ\left(f\right)\left(s\right)$: Fourier transform, $dx$: differential of $x$ , $\in$: element of, $\mathrm{i}$: imaginary unit, $\int$: integral, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ and ${𝕊}^{d-1}$: unit sphere of dimension $d-1$

Proved:

Stein and Weiss (1971, Ch. IV, Theorem 3.10)(proved)

Referenced by:

§37.11(iv)

Permalink:

http://dlmf.nist.gov/37.11.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(iv), §37.11(iv), §37.11, Part 37.III and Ch.37

… ►

37.11.31 $$ℱ\left(f\right)\left(\rho \cos \theta ,\rho \sin \theta )\right)={\mathrm{i}}^n{e}_n(\theta ){\int }_0^{\mathrm{\infty }}{f}_0(r)J_n\left(\rho r\right)rdr,$$ $\rho >0$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $ℱ\left(f\right)\left(s\right)$: Fourier transform, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer

Proved:

Stein and Weiss (1971, Ch. IV, Theorem 1.6)(proved)

Referenced by:

(37.4.22)

Permalink:

http://dlmf.nist.gov/37.11.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11(iv), §37.11(iv), §37.11, Part 37.III and Ch.37

…

8: 37.21 Physical Applications

… ►Important in this method is the Fourier transform (37.4.22) ($\alpha =0$) of the Zernike polynomials. …

9: 30.15 Signal Analysis

… ►

§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 $[-\sigma ,\sigma ]$. …

10: 3.11 Approximation Techniques

… ►

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 ${\omega }_n^{jk}$, $j,k=0,1,\mathrm{\dots },n-1$. …