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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. …
2: 1.8 Fourier Series
§1.8 Fourier Series
Uniqueness of Fourier Series
§1.8(ii) Convergence
3: 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
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: 28.14 Fourier Series
§28.14 Fourier Series
The Fourier series …
28.14.4 q c 2 m + 2 - ( a - ( ν + 2 m ) 2 ) c 2 m + q c 2 m - 2 = 0 , a = λ ν ( q ) , c 2 m = c 2 m ν ( q ) ,
28.14.5 m = - ( c 2 m ν ( q ) ) 2 = 1 ;
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 ( u ) 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 π ) 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). …