arithmetic Fourier transform

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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: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►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. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

3: 22.20 Methods of Computation

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§22.20(ii) Arithmetic-Geometric Mean

… ►Then as $n\to \mathrm{\infty }$ sequences $\{a_n\}$, $\{b_n\}$ converge to a common limit $M=M(a_0,b_0)$, the arithmetic-geometric mean of $a_0,b_0$. … ►The rate of convergence is similar to that for the arithmetic-geometric mean. … ►using the arithmetic-geometric mean. … ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute $\mathrm{am}(x,k)$. …

4: Bibliography K

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Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

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ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt

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2nd item.

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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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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

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MathReview, ZentralBlatt

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§3.5(viii).

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5: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

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§22.19(i), §22.20(vi).

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A Course in Arithmetic. Graduate Texts in Mathematics, Vol. 7, Springer-Verlag, New York.

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Notes:

Translated from the French.

External Links:

MathReview, ZentralBlatt

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§20.12(i), §20.7(viii), §20.9(i), §20.9(ii), §22.18(iv), §23.15(i), §23.18, §23.19, §23.20(v).

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

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§10.74(vii).

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Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt

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§24.10(ii).

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A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.

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ISBN 0-8493-8273-4, MathReview (J. Horváth), ZentralBlatt

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§18.17(v).

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6: Bibliography P

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A Kummer-type transformation for a ${}_2{}^{}F_2^{}$ hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt

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§16.6.

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Complex Interval Arithmetic and its Applications. Mathematical Research, Vol. 105, Wiley-VCH Verlag Berlin GmbH, Berlin.

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ISBN 3-527-40134-2, MathReview (G. Alefeld), ZentralBlatt

Referenced by:

§3.1(ii).

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Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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ISSN 1065-2469, Document, MathReview, ZentralBlatt

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§10.37.

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Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt

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§1.14(vii).

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Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:

Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.

External Links:

ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).

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7: 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

Referenced by:

§10.74(vii).

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The higher arithmetic. American Scientist 97, pp. 364–368.

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FullText

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§3.1(iv).

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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.

External Links:

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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8: Bibliography C

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An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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ISSN 0010-4655, Document, MathReview

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§10.74(vii).

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Macdonald’s evaluation conjectures and difference Fourier transform. Invent. Math. 122 (1), pp. 119–145.

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Notes:

Erratum: see same journal 125(1996), no. 2, p. 391.

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ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§17.14.

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Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.1(iv).

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The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§19.8(i).

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Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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ISSN 0002-9920, MathReview (Willard Parker)

Referenced by:

§19.8(i).

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