On the convergence of series of Fourier coefficients of vector valued functions
Let \(X\) be a Banach space with norm \(\|\cdot\|\), and let \(x(t)\) be a \(X\) valued function on \(-\infty<t<\infty\) which is \(2\pi\) periodic and integrable in Bochner sense. Define the Fourier coefficients \(c_ n\) of \(x(t)\) as usual. The paper studies sufficient conditions for the unconditional convergence of \(\sum_{n=-\infty}^ \infty c_ n\) in \(X\) and the convergence of \(\sum_{n=-\infty}^ \infty \| c_ n\|\) when \(X\) is a Hilbert space. Those are analogues to the known criteria for the absolute convergence of ordinary Fourier series of complex valued functions. The following is a typical result: The class of \(x(t)\) with \(\| x(\cdot)\|\in L^ p(T)\) is denoted by \({\mathbf L}^ p(T)\). Write \(||| x(\cdot)|||_ p=\left[{1\over 2\pi}\int^ \pi_{-\pi} \| x(t)\|^ p dt\right]^{1/p}\). Define the \(r\)-th integrated modulus of continuity of \(x(t)\) by \(M_ p^{(r)}(\delta)=\sup_{| h|\leq\delta}||| \Delta_ h^{(r)}x(\cdot)|||_ p\), \(\delta>0\). Theorem. Let \(1\leq p\leq 2\), \(x(t)\in {\mathbf L}^ p(T)\), and \(X\) be weakly complete. If \(\sum_{n=1}^ \infty n^{-1/p'}M_ p^{(r)}(1/n)<\infty\), then \(\sum_{n=-\infty}^ \infty c_ n\) is unconditionally convergent. The paper also studies the case when \(X\) is the space of random variables.
scientific article; zbMATH DE number 7241