Hilbert transform
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1: 1.14 Integral Transforms
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§1.14(v) Hilbert Transform
►The Hilbert transform of a real-valued function is defined in the following equivalent ways: … ►Inversion
… ►Inequalities
… ►Fourier Transform
…2: 3.5 Quadrature
3: Bibliography F
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The Plasma Dispersion Function: The Hilbert Transform of the Gaussian.
Academic Press, London-New York.
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4: Errata
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Section 1.14
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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.
Transform | New | Abbreviated | Old |
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Notation | Notation | Notation | |
Fourier | |||
Fourier Cosine | |||
Fourier Sine | |||
Laplace | |||
Mellin | |||
Hilbert | |||
Stieltjes |
Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.
5: 1.15 Summability Methods
6: Bibliography S
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Linear transformations in Hilbert space.
American Mathematical Society Colloquium Publications, Vol. 15, American Mathematical Society, Providence, RI.
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7: 18.38 Mathematical Applications
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►It has elegant structures, including -soliton solutions, Lax pairs, and Bäcklund transformations.
While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations.
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Riemann–Hilbert Problems
… ►Radon Transform
…8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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§1.18(i) Hilbert spaces
… ► … ►§1.18(iii) Linear Operators on a Hilbert Space
… ►In the following let be a Hilbert space. … ►Self-Adjoint and Symmetric Operators
…9: Bibliography W
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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach.
J. Math. Pures Appl. (9) 85 (5), pp. 698–718.
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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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Some transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
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The Airy transform.
Amer. Math. Monthly 86 (4), pp. 271–277.
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The Laplace Transform.
Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.
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10: Bibliography C
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A quadrature formula for the Hankel transform.
Numer. Algorithms 9 (2), pp. 343–354.
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An algorithm for the Fourier-Bessel transform.
Comput. Phys. Comm. 23 (4), pp. 343–353.
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Quadratic transformations of Appell functions.
SIAM J. Math. Anal. 7 (2), pp. 291–304.
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Optimized fast Hankel transform filters.
Geophysical Prospecting 38 (5), pp. 545–568.
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Methods of mathematical physics. Vol. I.
Interscience Publishers, Inc., New York, N.Y..
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