Trigonometric series of classes $L^p(\mathbb T)$, $p\in\left]1;\infty\right[$ and their conservative means

I. N. Brui

Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 677-686

Résumé

Suppose that a lower triangular matrix $\mu\colon[\mu_m^{(n)}]$ defines a conservative summation method for series, i.e., $$ \sup_{n\in{\mathbb Z}_0}\sum_{m=0}^n|\mu_m^{(n)}-\mu_{m+1}^{(n)}|\infty,\qquad \forall m\in{\mathbb Z}_0 \quad \lim_{n\to\infty}\mu_m^{(n)}=\rho_m\in\mathbb R, $$ and the sequence $(\rho_m)$, $m\in{\mathbb Z}_0$, is bounded away from zero. Then the trigonometric series $\sum_{m=-\infty}^\infty\gamma_me^{imx}$ is the Fourier series of a function $f\in L^p(\mathbb T)$, where $p\in\left]1;\infty\right[$, if and only if the sequence of $p$-norms of its $\mu$-means is bounded: $$ \sup_{n\in{\mathbb Z}_0}\biggl\|\sum_{m=-n}^n\mu_{|m|}^{(n)}\gamma_me^{imx}\biggr\|_p\infty. $$ In the case of the Fejér method, we have the test due to W. and G. Young (1913). In the case of the Fourier method, we obtain the converse of the Riesz theorem (1927).