Decomposability of continuous functions from Nikol'skij classes into multiple Fourier integrals
The author considers convergence questions for the Riesz means at the critical index. Let \(\Omega \subseteq R^ N\) be an N-dimensional domain, \(\{E_{\lambda}\}\) be a spectral decomposition of an arbitrary self- adjoint extension of the Laplace operator in \(L^ 2(\Omega)\). The Riesz means of order \(s\geq 0\) are defined by the formula \(E_{\lambda}f(x)=\int^{\lambda}_{0}(1-t/\lambda)^ sdE_ t(x).\) \textit{S. A. Alimov}, \textit{V. A. Il'in} and \textit{E. M. Nikisin} [Usp. Mat. Nauk 31, No.6 (192), 28-83 (1976; Zbl 0345.42005)] gave exact conditions for the convergence of the Riesz means \(E^ s_{\lambda}f\to f\) for \(f\in H^ a_ p(\Omega)\) when \(ap>N\), where \(H^ a_ p\) denotes the Nikolski space (equivalently, the Besov space \(B_ p^{a,\infty})\). One cannot expect uniform convergence for \(f\in H^ a_ p(\Omega)\) when ap\(\leq N\) because this would imply the existence of an unbounded f whose Riesz means converge uniformly to it. \textit{S. A. Alimov} [Sib. Mat. Zh. 19, 721-734 (1978; Zbl 0397.47023)] considered the case of ap\(\leq N\) and showed that if one assumed additionally that \(f\in W_ p^{\ell}(\Omega)\), where \(\ell +s>(N-1)/2\) and \(p\ell =N\) then the Riesz means converge to f uniformly. He also showed that the condition \(\ell +s>(N-1)/2\) is essential by finding a continuous \(f\in W_ p^{\ell}(\Omega)\) when \(\ell +s(N-1)/2\) and \(p\ell =N\) whose Riesz means were unbounded at some point of \(\Omega\). This note is devoted to the necessity of the condition \(p\ell =N\). The author shows that if \(0<s<(N-1)/2\), \(p\geq 1\), \(a>0\), and \(ap<N\), where \(a+s\) is an integer, there exists a continuous function \(f\in H^ a_ p(R^ N)\) such that \(\limsup_{\lambda \to \infty}E^ s_{\lambda}f(0)=+\infty\).
scientific article; zbMATH DE number 4148858