About integral averages by the radius of the spherical partial Fourier sums (Q2901594)

For every \(p\geq 2,\) a precise order of growth of norms of \(p\)--integral averages by the radius of the spherical partial Fourier sums was found. In the present paper the results are formulated for the space of bounded functions on the \(m\)--measured torus, where \(m\geq 3\). Let \(p\geq 2,\) let \(S_R(f,x)\) be a spherical partial sum \(\sum\limits_{| k| \leq R}\widehat{f}(k)e^{ikx}\) of a correspondent Fourier-Lebesgue series with Fourier coefficients \(\widehat{f}(k),\) let \(H_{R,p}(f,x)\) be a \(L_p[0,R]\)--norm of the function \(S_r(f,x)/R^p,\) \(r\in (0,R),\) and \(H_{R,p}=\sup\limits_{| f| \leq 1}\| H_{R,p}(f)\|_{\infty}\). It is proved that there exist constants \(c_{1m}\) and \(c_{2m}\) such that \(c_{1m}\cdot R^{\frac{m-1}{2}-\frac{1}{p}}\leq H_{R,p}\leq c_{2m}\cdot R^{\frac{m-1}{2}-\frac{1}{p}}\) as \(R\rightarrow \infty,\) \(m\geq 3\). For \(p=1\) it is proved that \(R^{\frac{m-3}{2}}\leq c_{3m}\cdot H_{R,1}\) and \(H_{R,1}\leq c_{4m}\cdot R^{\frac{m-2}{2}}\) as \(R\rightarrow \infty,\) \(m\geq 3,\) for some constants \(c_{3m}\) and \(c_{4m}\).