Local theorems for the absolute convergence of multiple lacunary Fourier series
Let \(T^ m\) be the m-dimensional torus identified with \(Q=[-\pi,\pi]\) m and let its dual be identified with \({\mathbb{Z}}^ m.\) For \(\beta >0\), the set of all \(f\in L^ 1(T^ m)\) for which \(\sum_{n\in {\mathbb{Z}}^ m}|\hat f(n)|^{\beta}<\infty\) is denoted by \(A(\beta)\). Given \(E\subset {\mathbb{Z}}^ m,\) a function \(f\in L^ 1(T^ m)\) is said to be E-spectral if and only if \(\hat f(n)=0\) for all \(n\in {\mathbb{Z}}^ m-E\). The class of all the E-spectral functions in \(L^ p(T^ m)\) is denoted by \(L^ p_ E(T^ m)\). For arbitrarily fixed \(x_ 0=(x_{01},...,x_{0m})\) in Q and \(\delta\), \(0<\delta\leq \pi\), let \(I\equiv I(x_ 0,\delta)=\{x\in Q: | x_ j-x_{0j}|\leq \delta\) for each \(j=1,...,m\}\). For \(1<p\leq 2\) and \(\alpha >0\), \(\alpha =\ell +a\) with \(0<a\leq 1\) and \(\ell\) non-negative integer, we say that \(f\in Lip_ 2(\alpha,p,I)\) if \((a)\quad D^{\theta}f(x)\equiv (\partial^{\theta_ 1+...+\theta_ m}/\partial x_ 1^{\theta_ 1}...\partial x_ m^{\theta_ m}f(x)\) exists a.e. and belongs to \(L^ p(T^ m)\) whenever \(\theta_ 1+\theta_ 2+...+\theta_ m\leq\ell \), and (b) for \(\theta_ 1+...+\theta_ m=\ell\) we have \[ \| D^{\theta}f(x+2h)+D^{\theta}f(x-2h)- 2D^{\theta}f(x)\|_{p,I}=O(| h|^{\alpha}), \] where \(x,h\in Q\) and \(\|\cdot \|_{p,I}=\| (\cdot)\chi_ I\|_ p (\chi_ I\) characteristic function; \(| h|\) and \(\|\cdot \|_ p\) have usual meanings). For a certain sort of sets \(E\subset {\mathbb{Z}}^ m,\) more general than Hadamard sets, the following results are proved: \[ (1)\quad Lip_ 2(\alpha,p,I)\cap L^ p_ E(T^ m)\subset A(\beta)\text{ whenever }\beta >mp/(\alpha p+mp-m); \] \[ (2)\quad f\in Lip(\alpha,I)\cap L^ 1_ E(T^ m)\Rightarrow \hat f(n_ k)=O(| n_ k|^{-\alpha})\text{ for all }n_ k\in E, \] where Lip\((\alpha,I)\) is defined analogously. These are m-dimensional analogues of the extended version of the results due to \textit{M. E. Noble} [ibid. 128, 55-62; correction, 256 (1954; Zbl 0055.295)]. In view of the method of proof employed here, when there are no gaps (i.e., \(E={\mathbb{Z}}^ m)\) we need to take \(I=Q\), and then one interestingly gets Minakshisundaram- Szász's m-dimensional analogue of the classical Bernshtein's theorem without using spherical means and through a rectangular type method of summation.
scientific article; zbMATH DE number 3873942