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Proceedings of the American Mathematical Society

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Compactness in $ L\sp 2$ and the Fourier transform

Author: Robert L. Pego
Journal: Proc. Amer. Math. Soc. 95 (1985), 252-254
MSC: Primary 42A38; Secondary 43A15
MathSciNet review: 801333
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Abstract: The Riesz-Tamarkin compactness theorem in $ {L^p}({{\mathbf{R}}^n})$ employs notions of $ {L^p}$-equicontinuity and uniform $ {L^p}$-decay at $ \infty $. When $ 1 \leqslant p \leqslant 2$, we show that these notions correspond under the Fourier transform, and establish new necessary and sufficient criteria for compactness in $ {L^2}({{\mathbf{R}}^n})$.

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Keywords: Compactness, Fourier transform, $ {L^p}$, $ {L^2}$, $ {L^p}$-equicontinuity
Article copyright: © Copyright 1985 American Mathematical Society