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 $f(t)$ is defined in the following equivalent ways: … ►

Inversion

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Inequalities

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Fourier Transform

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2: 3.5 Quadrature

… ►A special case is the rule for Hilbert transforms (§1.14(v)): ►

3.5.46 $$\mathscr{H}f\left(x\right)=\frac{1}{\pi }{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{f(t)}{t-x}dt,$$ $x\in ℝ$,

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Symbols:

$\mathscr{H}\left(f\right)\left(x\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\in$: element of, ${⨍}_a^b$: Cauchy principal value and $ℝ$: real line

Permalink:

http://dlmf.nist.gov/3.5.E46

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Hilberttransform was changed to $\mathscr{H}f\left(x\right)$ from $\mathscr{H}(f;x)$.

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

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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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Notes:

Erratum: Math. Comp. v. 26 (1972), no. 119, p. 814.

External Links:

MathReview (R. Balescu)

Referenced by:

§7.21.

Encodings:

BibTeX

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4: Errata

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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
	Notation	Notation	Notation
Fourier	$ℱ\left(f\right)\left(x\right)$	$ℱf\left(x\right)$
Fourier Cosine	${ℱ}_c\left(f\right)\left(x\right)$	${ℱ}_{c}f\left(x\right)$
Fourier Sine	${ℱ}_s\left(f\right)\left(x\right)$	${ℱ}_{s}f\left(x\right)$
Laplace	$ℒ\left(f\right)\left(s\right)$	$ℒf\left(s\right)$	$ℒ(f(t);s)$
Mellin	$ℳ\left(f\right)\left(s\right)$	$ℳf\left(s\right)$	$ℳ(f;s)$
Hilbert	$\mathscr{H}\left(f\right)\left(s\right)$	$\mathscr{H}f\left(s\right)$	$\mathscr{H}(f;s)$
Stieltjes	$𝒮\left(f\right)\left(s\right)$	$𝒮f\left(s\right)$	$𝒮(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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5: 1.15 Summability Methods

… ►Moreover, ${lim}_{y\to 0+}\mathrm{\Im }\mathrm{\Phi }(x+\mathrm{i}y)$ is the Hilbert transform of $f(x)$ (§1.14(v)). …

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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Notes:

Reprint of the 1932 original

External Links:

ISBN 0-8218-1015-4, Document, Link, MathReview (P. R. Halmos)

Referenced by:

§1.18(x).

Encodings:

BibTeX

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7: 18.38 Mathematical Applications

… ►It has elegant structures, including $N$-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. … ►

Riemann–Hilbert Problems

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Radon Transform

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8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(i) Hilbert spaces

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§1.18(iii) Linear Operators on a Hilbert Space

… ►In the following let $V$ be a Hilbert space. … ►

Self-Adjoint and Symmetric Operators

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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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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

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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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External Links:

ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

Encodings:

BibTeX

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

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (A. G. Ramm), ZentralBlatt

Referenced by:

(9.10.13), §9.10(ix), §9.10(v).

Encodings:

BibTeX

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The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.

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External Links:

FullText, MathReview (J. D. Tamarkin), ZentralBlatt

Referenced by:

§1.14(vi), §1.15(viii).

Encodings:

BibTeX

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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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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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

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External Links:

ISSN 0010-4655, Document, MathReview

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Quadratic transformations of Appell functions. SIAM J. Math. Anal. 7 (2), pp. 291–304.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.16(ii), §19.22(i).

Encodings:

BibTeX

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Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

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External Links:

Document

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y..

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External Links:

MathReview (J. B. Diaz), ZentralBlatt

Referenced by:

(14.30.8_5).

Encodings:

BibTeX

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