discrete Fourier transform

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1: 3.11 Approximation Techniques

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Example. The Discrete Fourier Transform

… ►is called a discrete Fourier transform pair. … ►The direct computation of the discrete Fourier transform (3.11.38), that is, of …

2: 24.19 Methods of Computation

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A method related to “Stickelberger codes” is applied in Buhler et al. (2001); in particular, it allows for an efficient search for the irregular pairs $(2n,p)$. Discrete Fourier transforms are used in the computations. See also Crandall (1996, pp. 120–124).

3: 18.3 Definitions

… ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). …

4: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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MathReview (L. Berg)

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§7.14(iii).

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A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)

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§18.16(iii).

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Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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ISSN 0096-3518, FullText, ZentralBlatt

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§10.74(vii).

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Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.

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Notes:

Reprinted in 1993.

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ISBN 0-471-08703-3; 0-471-58986-1, MathReview (A. E. Heins), ZentralBlatt

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§1.14(v), Ch.3.

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Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

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ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt

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§24.16(i).

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5: Bibliography W

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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ZentralBlatt

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§15.8(v), §15.8(v).

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Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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Document, ZentralBlatt

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§16.6.

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On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

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ISSN 1044-677X, FullText

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§28.4(vii).

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Asymptotic expansions of Fourier transforms of functions with logarithmic singularities. J. Math. Anal. Appl. 64 (1), pp. 173–180.

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Document, MathReview (B. L. Gupta), ZentralBlatt

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§2.3(iv).

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On the connection formulas of the third Painlevé transcendent. Discrete Contin. Dyn. Syst. 23 (1-2), pp. 541–560.

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ISSN 1078-0947, Document, MathReview (Galina V. Filipuk), ZentralBlatt

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§32.13(ii).

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6: Bibliography K

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A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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Document, MathReview (G. Gasper), ZentralBlatt

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§18.10(i).

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

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§15.14, §15.14.

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

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§18.14(i).

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Fourier Analysis. 2nd edition, Cambridge University Press, Cambridge.

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ISBN 0-521-38991-7, MathReview, ZentralBlatt

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§1.15(iii), §20.13.

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

External Links:

MathReview, ZentralBlatt

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§3.5(viii).

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7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ► The analogous orthonormality is … ►

§1.18(v) Point Spectra and Eigenfunction Expansions

… ► … ►For $f(x)$ even in $x$ this yields the Fourier cosine transform pair (1.14.9) & (1.14.11), and for $f(x)$ odd the Fourier sine transform pair (1.14.10) & (1.14.12). … ► …

8: Bibliography L

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Single-precision Fortran, maximum accuracy 8S.

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ISSN 0010-4655, Document, Code, ZentralBlatt

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4th item.

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Discrete-variable representations and their utilization. In Advances in Chemical Physics, pp. 263–310.

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ISBN 9780470141731, Document, Link

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§18.38(i).

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An Introduction to Fourier Analysis and Generalised Functions. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York.

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ISBN 0521091284, MathReview (G. Temple), ZentralBlatt

Referenced by:

§1.16(ix), §1.16(v).

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

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§10.74(vii).

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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

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FullText, MathReview (J. Bryant), ZentralBlatt

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§2.3(iv).

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9: Bibliography Z

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A proof of Andrews’ $q$-Dyson conjecture. Discrete Math. 54 (2), pp. 201–224.

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ISSN 0012-365X, Document, MathReview (Ira Gessel), ZentralBlatt

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§17.14.

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Distribution Theory and Transform Analysis, An Introduction and Generalized Functions with Applications. Dover, New York.

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Notes:

Reprint of 1965 McGraw-Hill Publication

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ISBN 0-486-65479-6, MathReview Entry

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§1.16(ix).

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Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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§35.7(ii).

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A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

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Notes:

English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905

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ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt

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§10.74(vii).

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10: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

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§18.39(i).

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Singularity analysis of generating functions. SIAM J. Discrete Math. 3 (2), pp. 216–240.

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ISSN 0895-4801, Document, MathReview (E. Rodney Canfield), ZentralBlatt

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§2.10(iv).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

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§32.7(ii).

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Discrete Painlevé equations and their appearance in quantum gravity. Comm. Math. Phys. 142 (2), pp. 313–344.

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ISSN 0010-3616, Link, MathReview (Nikolai A. Kostov), ZentralBlatt

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§32.15.

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Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

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Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.15.

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