Completeness of the Grid Translates of Functions

Swagato K. Ray; Rudra P. Sarkar; Yitzhak Weit

doi:10.1007/s00041-013-9294-1

Completeness of the Grid Translates of Functions

Swagato K. Ray, Rudra P. Sarkar, Yitzhak Weit

Department of Mathematics

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

Abstract

If f ∈ L^p(ℝ^d) is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point x₀ ∈ ℝ^d and if the inverse image of the neighborhoods of f(x ₀) shrinks regularly to x ₀, then span (Formula Presented) is a dense subset of L^p(ℝ^d), 1 ≤ p< ∞ where f ^m(x)=f(x)^m and {e _i} is the natural basis of R^d. The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.

Original language	English
Pages (from-to)	186-198
Number of pages	13
Journal	Journal of Fourier Analysis and Applications
Volume	20
Issue number	1
DOIs	https://doi.org/10.1007/s00041-013-9294-1
State	Published - Feb 2014

Keywords

Euclidean space
Grid-translate
Homogeneous group
Riemannian symmetric space

ASJC Scopus subject areas

Analysis
General Mathematics
Applied Mathematics

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10.1007/s00041-013-9294-1

Cite this

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title = "Completeness of the Grid Translates of Functions",

abstract = "If f ∈ Lp(ℝd) is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point x0 ∈ ℝd and if the inverse image of the neighborhoods of f(x 0) shrinks regularly to x 0, then span (Formula Presented) is a dense subset of Lp(ℝd), 1 ≤ p< ∞ where f m(x)=f(x)m and \{e i\} is the natural basis of Rd. The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.",

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author = "Ray, \{Swagato K.\} and Sarkar, \{Rudra P.\} and Yitzhak Weit",

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journal = "Journal of Fourier Analysis and Applications",

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AU - Ray, Swagato K.

AU - Sarkar, Rudra P.

AU - Weit, Yitzhak

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Y1 - 2014/2

N2 - If f ∈ Lp(ℝd) is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point x0 ∈ ℝd and if the inverse image of the neighborhoods of f(x 0) shrinks regularly to x 0, then span (Formula Presented) is a dense subset of Lp(ℝd), 1 ≤ p< ∞ where f m(x)=f(x)m and {e i} is the natural basis of Rd. The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.

AB - If f ∈ Lp(ℝd) is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point x0 ∈ ℝd and if the inverse image of the neighborhoods of f(x 0) shrinks regularly to x 0, then span (Formula Presented) is a dense subset of Lp(ℝd), 1 ≤ p< ∞ where f m(x)=f(x)m and {e i} is the natural basis of Rd. The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.

KW - Euclidean space

KW - Grid-translate

KW - Homogeneous group

KW - Riemannian symmetric space

UR - https://www.scopus.com/pages/publications/84897024759

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EP - 198

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 1

ER -

Completeness of the Grid Translates of Functions

Abstract

Keywords

ASJC Scopus subject areas

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