Almost everywhere convergence of Fourier integrals
Copyright policy )Almost everywhere convergence of Fourier integrals
In this paper, we prove that if \(f \in L^ p (\mathbb{R}^ n)\), for certain \(p\) and \(n\), satisfies \[ \lim_{\lambda \downarrow 1} \varlimsup_{R \to \infty} \int_{R < | \xi | \leq \lambda R} \bigl | \widehat f(\xi) \bigr | d \xi = 0, \] then, for almost all \(x \in \mathbb{R}^ n\), \(\int_{| \xi | \leq R} \widehat f (\xi) e^{ix \cdot \xi} d \xi\) converges to \(f(x)\) as \(R \to \infty\). This result not only generalizes the classical Fourier inversion formula, but also includes a result of \textit{D. S. Lubinsky} and \textit{F. Móricz} [Arch. Math. 61, No. 1, 82-87 (1993; Zbl 0793.42004)] as a special case.
- National Tsing Hua University Taiwan
- National Central University Taiwan
Bochner-Riesz means, Fourier inversion formula, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Fourier integrals, almost everywhere convergence
Bochner-Riesz means, Fourier inversion formula, Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type, Fourier integrals, almost everywhere convergence
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