Hilbert

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1: Alexander A. Its

… ► Matveev), published by Springer in 1994, and Painlevé Transcendents: The Riemann-Hilbert Approach (with A. …

2: 25.17 Physical Applications

… ►This relates to a suggestion of Hilbert and Pólya that the zeros are eigenvalues of some operator, and the Riemann hypothesis is true if that operator is Hermitian. …

3: Brian D. Sleeman

… ►He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D. …

4: 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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5: 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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6: 31.16 Mathematical Applications

… ►Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space: … ►

7: 18.38 Mathematical Applications

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Riemann–Hilbert Problems

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8: 31.15 Stieltjes Polynomials

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31.15.12

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Defines:

$\rho (z)$: weight function (locally)

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(iii), §31.15 and Ch.31

►The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space $L_{\rho }^2(Q)$. …

9: Bibliography F

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Painlevé Transcendents: The Riemann-Hilbert Approach. Mathematical Surveys and Monographs, Vol. 128, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-3651-X, MathReview Entry, ZentralBlatt

Referenced by:

§32.11(ii), §32.11(iii), §32.11(iv), §32.12(ii), §32.12(iii), §32.4(i), Profile Alexander A. Its.

Encodings:

BibTeX

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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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10: 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 Hilbert transform 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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