The rate of convergence of Fourier expansions in the plane: a geometric viewpoint
Let \(f= g\chi_B\), where \(g\) is a smooth function on \(\mathbb{R}^2\) and \(B\) is a bounded planar domain with smooth boundary, and let \[ S_\lambda f(x)= \int_{\lambda A}\widehat f(\xi)\exp(2\pi i\xi\cdot x)\,d\xi, \] where \(A\) is a bounded planar domain with smooth boundary and strictly starshaped with respect to the origin and \[ \widehat f(\xi):= \int_{\mathbb{R}^2} f(y)\exp(-2\pi i\xi\cdot y)\,dy. \] One of the main results in the paper is the following Theorem 1: There exist constants \(c= c(x,A,B,g)\) and \(\ell= \ell(x,A,B)\) such that \[ \begin{aligned}| S_\lambda f(x)|\leq c\lambda^{-1/2}\quad &\text{if }x\in\overline B,\\ | S_\lambda f(x)- g(x)|\leq c\lambda^{-1/2}\quad &\text{if }x\in B,\\ | S_\lambda f(x)- \ell g(x)|\leq c\lambda^{-1/2}\quad &\text{if }x\in\partial B.\end{aligned} \] We note that in the special case when \(A\) is a disc, it is well known that \(\ell(x,A,B)= 1/2\).
scientific article; zbMATH DE number 2075750