Sobolev

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1: 37.19 Other Orthogonal Polynomials of $d$ Variables

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§37.19(iv) Sobolev OPs on the Ball and on the Simplex

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37.19.7 $$⟨f,g⟩={\int }_{{𝔹}^d}{\mathrm{\nabla }}^{s}f(𝐱){\mathrm{\nabla }}^{s}g(𝐱)d𝐱+\sum _{k=0}^{\lfloor \frac{s}{2}\rfloor -1}{\int }_{{𝕊}^{d-1}}{\mathrm{\Delta }}^{k}f(𝝃){\mathrm{\Delta }}^{k}g(𝝃)d\sigma (𝝃),$$

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Defines:

$⟨f,g⟩$: Sobolev inner product on the ball (locally), $𝝃$: $d$-dimensional vector restricted to ${𝕊}^{d-1}$ (locally), $\mathrm{\nabla }$: gradient (locally) and $\mathrm{\Delta }$: Laplacian (locally)

Symbols:

$dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $k$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${𝕊}^{d-1}$: unit sphere of dimension $d-1$ , ${𝔹}^d$: unit ball, $f$: $d$-variable integrable function on ${𝔹}^d$ and $g$: $d$-variable integrable function on ${𝔹}^d$

Permalink:

http://dlmf.nist.gov/37.19.E7

Encodings:

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See also:

Annotations for §37.19(iv), §37.19 and Ch.37

►which is useful for studying approximation order in the Sobolev space; see Li and Xu (2014). … …

2: 18.36 Miscellaneous Polynomials

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§18.36(ii) Sobolev Orthogonal Polynomials

►Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. …

3: Bibliography I

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On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:

ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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4: Bibliography E

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The Sobolev orthogonality and spectral analysis of the Laguerre polynomials $\{L_n^{-k}\}$ for positive integers $k$ . J. Comput. Appl. Math. 171 (1-2), pp. 199–234.

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External Links:

ISSN 0377-0427, Document, Link, MathReview (Khélifa Trimèche)

Referenced by:

§18.36(v).

Encodings:

BibTeX

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5: Bibliography K

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Uniform asymptotic approximations for the Meixner-Sobolev polynomials. Anal. Appl. (Singap.) 10 (3), pp. 345–361.

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External Links:

ISSN 0219-5305, Document, FullText, MathReview (Diego E. Dominici), ZentralBlatt

Referenced by:

§18.24, §18.24.

Encodings:

BibTeX

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On differential equations for Sobolev-type Laguerre polynomials. Trans. Amer. Math. Soc. 350 (1), pp. 347–393.

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External Links:

ISSN 0002-9947, FullText, MathReview (Hans W. Volkmer), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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6: Bibliography M

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Orthogonal polynomials on Sobolev spaces: Old and new directions. J. Comput. Appl. Math. 48 (1-2), pp. 113–131.

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External Links:

ISSN 0377-0427, Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§18.36(ii), §18.36(ii), Erratum (V1.1.9) for Section 18.36(ii).

Encodings:

BibTeX

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On Sobolev orthogonal polynomials. Expo. Math. 33 (3), pp. 308–352.

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External Links:

ISSN 0723-0869, Document, Link, MathReview (Brian Simanek), ZentralBlatt

Referenced by:

§18.36(ii), §18.36(ii), Erratum (V1.1.9) for Section 18.36(ii).

Encodings:

BibTeX

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7: Bibliography G

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Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation, Oxford University Press, New York.

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Notes:

OPQ, a package of Matlab routines for generating classical and Sobolev orthogonal polynomials, accompanies this book.

External Links:

OPQ, ISBN 0-19-850672-4, MathReview Entry, ZentralBlatt

Referenced by:

(18.2.27), (18.2.28), (18.2.29), (18.2.30), §18.2(ix), §18.40(ii), §18.40(ii), §18.40(i), 3rd item, §3.5(v), Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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