Divergent Jacobi polynomial series

Christopher Meaney

doi:10.1090/S0002-9939-1983-0684639-4

Divergent Jacobi polynomial series

Christopher Meaney

Research output: Contribution to journal › Article › peer-review

23 Citations (Scopus)

Abstract

Fix real numbers α ≥ β ≥ - 1/2, with α > - 1/2, and equip [-1, 1] with the measure dμ(x) = (1 - x)^α(1 + x)^β dx. For p = 4(α + l)/(2α + 3) there exists f ∈ L^p(μ) such that f(x) = 0 a.e. on [-1, 0] and the appropriate Jacobi polynomial series for f diverges a.e. on [-1.1] This implies failure of localization for spherical harmonic expansions of elements of L^2d/(d+1)(X), where X is a sphere or projective space of dimension d > 1.

Original language	English
Pages (from-to)	459-462
Number of pages	4
Journal	Proceedings of the American Mathematical Society
Volume	87
Issue number	3
DOIs	https://doi.org/10.1090/S0002-9939-1983-0684639-4
Publication status	Published - 1983
Externally published	Yes

Keywords

Compact two-point homogeneous space
Jacobi polynomial
Laplace-Beltrami operator
Localization
Spherical harmonic
Zonal

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10.1090/S0002-9939-1983-0684639-4

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title = "Divergent Jacobi polynomial series",

abstract = "Fix real numbers α ≥ β ≥ - 1/2, with α > - 1/2, and equip [-1, 1] with the measure dμ(x) = (1 - x)α(1 + x)β dx. For p = 4(α + l)/(2α + 3) there exists f ∈ Lp(μ) such that f(x) = 0 a.e. on [-1, 0] and the appropriate Jacobi polynomial series for f diverges a.e. on [-1.1] This implies failure of localization for spherical harmonic expansions of elements of L2d/(d+1)(X), where X is a sphere or projective space of dimension d > 1.",

keywords = "Compact two-point homogeneous space, Jacobi polynomial, Laplace-Beltrami operator, Localization, Spherical harmonic, Zonal",

author = "Christopher Meaney",

year = "1983",

doi = "10.1090/S0002-9939-1983-0684639-4",

language = "English",

volume = "87",

pages = "459--462",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "3",

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TY - JOUR

T1 - Divergent Jacobi polynomial series

AU - Meaney, Christopher

PY - 1983

Y1 - 1983

N2 - Fix real numbers α ≥ β ≥ - 1/2, with α > - 1/2, and equip [-1, 1] with the measure dμ(x) = (1 - x)α(1 + x)β dx. For p = 4(α + l)/(2α + 3) there exists f ∈ Lp(μ) such that f(x) = 0 a.e. on [-1, 0] and the appropriate Jacobi polynomial series for f diverges a.e. on [-1.1] This implies failure of localization for spherical harmonic expansions of elements of L2d/(d+1)(X), where X is a sphere or projective space of dimension d > 1.

AB - Fix real numbers α ≥ β ≥ - 1/2, with α > - 1/2, and equip [-1, 1] with the measure dμ(x) = (1 - x)α(1 + x)β dx. For p = 4(α + l)/(2α + 3) there exists f ∈ Lp(μ) such that f(x) = 0 a.e. on [-1, 0] and the appropriate Jacobi polynomial series for f diverges a.e. on [-1.1] This implies failure of localization for spherical harmonic expansions of elements of L2d/(d+1)(X), where X is a sphere or projective space of dimension d > 1.

KW - Compact two-point homogeneous space

KW - Jacobi polynomial

KW - Laplace-Beltrami operator

KW - Localization

KW - Spherical harmonic

KW - Zonal

UR - https://www.scopus.com/pages/publications/84966199061

U2 - 10.1090/S0002-9939-1983-0684639-4

DO - 10.1090/S0002-9939-1983-0684639-4

M3 - Article

AN - SCOPUS:84966199061

SN - 0002-9939

VL - 87

SP - 459

EP - 462

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -

Divergent Jacobi polynomial series

Abstract

Keywords

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