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\(l-1\) summability of multiple Fourier integrals and positivity. (English) Zbl 0881.42007

Summary: Let \(f\in L^1(\mathbb{R}^d)\), and let \(\widehat f\) be its Fourier integral. We study summability of the \(l-1\) partial integral \(S^{(1)}_{R,d}(f;x)= \int_{|v|_1\leq R}e^{iv\cdot x}\widehat f(v)dv\), \(x\in\mathbb{R}^d\); note that the integral ranges over the \(l_1\)-ball in \(\mathbb{R}^d\) centred at the origin with radius \(R>0\). As a central result, we prove that for \(\delta\geq 2d-1\) the \(l-1\) Riesz \((R,\delta)\) means of the inverse Fourier integral are positive, the lower bound being best possible. Moreover, we give an \(l-1\) analogue of Schoenberg’s modification of Bochner’s theorem on positive definite functions on \(\mathbb{R}^d\) as well as an extension of Pólya’s sufficiency condition.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A82 Positive definite functions in one variable harmonic analysis
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