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Spherical convergence of the Fourier integral of the indicator function of an \(N\)-dimensional domain. (English. Russian original) Zbl 0920.42004

Sb. Math. 189, No. 7, 1101-1113 (1997); translation from Mat. Sb. 189, No. 7, 145-157 (1998).
Let \({\mathcal D}= {\mathcal D}^N\) be a compact subdomain of the \(N\)-dimensional Euclidean space \(\mathbb{R}^N\). For \(f= \chi_{\mathcal D}\), the indicator function of \(D\), consider its Fourier transform \[ \widehat f(\omega): \int_{\mathbb{R}^N} e^{i\langle x,\omega\rangle}f(x)dx,\quad \omega\in \mathbb{R}^N. \] A spherical sum is defined by \[ f_\Omega(a):= (2\pi)^{-N} \int_{|\omega|\leq \Omega} e^{-i\langle\omega, a\rangle} \widehat f(\omega)d\omega,\quad \Omega>0. \] The author characterizes the rate of pointwise convergence of \(f_\Omega(a)\) \((a\not\in \partial{\mathcal D})\) as \(\Omega\to\infty\) by the so-called convergence exponent \(\sigma(a|\partial{\mathcal D})\), the latter being the least upper bound of those exponents \(\gamma>0\) for which \[ | f_\Omega(a)- f(a)|\leq O(\Omega^{-\gamma+ \varepsilon})\quad\text{for each }\varepsilon> 0\quad\text{as }\Omega\to\infty. \] Among others, the following is proved. If \(\partial{\mathcal D}\) is smooth and \(a\not\in K(\partial{\mathcal D})\), the focal surface of \(\partial{\mathcal D}\), then \(\sigma(a|\partial D)= 1\) for each dimension \(N\geq 1\).

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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