Compactness in 𝐿² and the Fourier transform

Pego, Robert L.

doi:10.1090/S0002-9939-1985-0801333-9

Compactness in $L^ 2$ and the Fourier transform
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by Robert L. Pego PDF

Proc. Amer. Math. Soc. 95 (1985), 252-254 Request permission

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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