# Fourier

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##### 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. …
##### 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. …
##### 5: 28.14 Fourier Series
###### §28.14 Fourier Series
The Fourier series …
28.14.5 $\sum_{m=-\infty}^{\infty}\left(c_{2m}^{\nu}(q)\right)^{2}=1;$
##### 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). …
##### 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). …