Berens, Hubert; Xu, Yuan \(l-1\) summability of multiple Fourier integrals and positivity. (English) Zbl 0881.42007 Math. Proc. Camb. Philos. Soc. 122, No. 1, 149-172 (1997). 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. Cited in 5 ReviewsCited in 29 Documents MSC: 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42A82 Positive definite functions in one variable harmonic analysis Keywords:multiple Fourier integrals; summability; partial integral; Bochner’s theorem on positive definite functions; Pólya’s sufficiency condition PDFBibTeX XMLCite \textit{H. Berens} and \textit{Y. Xu}, Math. Proc. Camb. Philos. Soc. 122, No. 1, 149--172 (1997; Zbl 0881.42007) Full Text: DOI