On Hardy transformation of Fourier coefficients of a function of several variables (Q1813584)

The following multi-variable extension of Hardy's theorem on Fourier coefficients of a function of one variable is proved: Let \(T_ n=[- \pi,\pi]^ n\), \(p\geq1\) and let \(f\in L^ p(T_ n)\) be a function, even and \(2\pi\)-periodic with respect to every variable. For \(q=(q_ 1,\ldots,q_ n)\), \(q_ i=0,1,2,\ldots,\) let \(a_ q\) be the corresponding Fourier coefficients of \(f\). Set \(A_ m=\sum_{0\leq q\leq m}2^{-\lambda(q)}a_ q\), \(m=(m_ 1,\ldots,m_ n)\), \(m_ i=0,1,2,\ldots,\) where \(\lambda(q)\) is the number of zero components of \(q\). Then the numbers \(A_ m/\prod^ n_{j=1}m_ j\) are also the Fourier coefficients of a function from \(L^ p(T_ n)\), even with respect to every variable.