Fourier

(0.001 seconds)

1—10 of 76 matching pages

1: 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. …

2: 28.4 Fourier Series

§28.4 Fourier Series

… ►

§28.4(ii) Recurrence Relations

… ►

§28.4(iii) Normalization

… ►

§28.4(v) Change of Sign of $q$

… ►

§28.4(vi) Behavior for Small $q$

…

3: 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. …

4: 1.8 Fourier Series

§1.8 Fourier Series

… ► … ►

Uniqueness of Fourier Series

… ►

§1.8(ii) Convergence

… ►

5: 28.14 Fourier Series

§28.14 Fourier Series

►The Fourier series … ►

28.14.4

{qc_{2m+2}-\left(a-(\nu+2m)^{2}\right)c_{2m}+qc_{2m-2}=0},

a=\lambda_{\nu}\left(q\right),c_{2m}=c_{2m}^{\nu}(q)

ⓘ

Symbols:: $\lambda_{\NVar{\nu+2n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter, $a$ : parameter and $c_{2m}(q)$ : coefficients
Referenced by:: item (d), item (d)
Permalink:: http://dlmf.nist.gov/28.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

… ►

28.14.5

\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;

ⓘ

Symbols:: $m$ : integer, $q=h^{2}$ : parameter, $\nu$ : complex parameter and $c_{2m}(q)$ : coefficients
Referenced by:: item (d)
Permalink:: http://dlmf.nist.gov/28.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.14 and Ch.28

…

6: 1.14 Integral Transforms

… ►

§1.14(i) Fourier Transform

… ►

Inversion

… ►

Convolution

… ►

Uniqueness

… ►

Fourier Transform

…

7: 29.20 Methods of Computation

… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. …The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. …

8: 1.16 Distributions

… ►

§1.16(vii) Fourier Transforms of Tempered Distributions

… ►Then its Fourier transform is … ►The Fourier transform

\mathscr{F}\left(u\right)

of a tempered distribution is again a tempered distribution, and … ►

§1.16(viii) Fourier Transforms of Special Distributions

… ►The second to last equality follows from the Fourier integral formula (1.17.8). …

9: 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 or series`	at least one of the words “Fourier” or “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”.
…

► …

10: 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). …