Lebesgue constants of Dirichlet kernels of monotone type and convergence of multiple trigonometric series (Q920329)

A bounded integer set U in \(R_+^ m\) is said to be of monotone type, \(U\in M(R_+^ m)\), if \(k\in U\) and \(\ell \leq k\) with \(\ell\) an integer vector in \(R_+^ m\) implies \(\ell \in U\). For the Dirichlet kernel \(D_ U(x)=\sum_{k\in U}\exp (ikx)\) with \(U\in M(R^ m_+)\) the following estimate in the \(L^ 1(T^ m)\) norm is derived: \[ \| D_ U\| \leq 2\pi m!| U|^{(m-1)/(2m)}(\ln | U| +1). \] Further, it is shown that for \(p>2m/(m+1)\), the series \(\sum^{\infty}_{k=1}a_ k\exp (ikx)\) with \(\sum^{\infty}_{k=1}a^ p_ k(\Pi (k))^{p-2}<\infty\) is a Fourier series of some function \(f\in L^ p(T^ m)\) and converges in the sense of Pringsheim almost everywhere while for \(p\in (1,2m/(m+1))\) this needs not to be true.