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

Abstract

Suppose that a lower triangular matrix μ:[μ ⁽ⁿ⁾ _m ] defines a conservative summation method for series, i.e.,

$$\mathop {\sup }\limits_{n \in \mathbb{Z}_0 } \sum\limits_{m = 0}^n {\left| {\mu _m^{(n)} - \mu _{m + 1}^{(n)} } \right|}< \infty , \forall m \in \mathbb{Z}_0 , \mathop {\lim }\limits_{n \to \infty } \mu _m^{(n)} = \rho _m \in \mathbb{R}$$

and the sequence (ρ _m ,m ∈ ℤ₀), is bounded away from zero. Then the trigonometric series$\sum\nolimits_{m = - \infty }^\infty {\gamma _m e^{imx} }$ is the Fourier series of a functionf ∈L ^p($\mathbb{T}$), wherep ε ]1; ∞[, if and only if the sequence ofp-norms of its μ-means is bounded:

$$_{n \in \mathbb{Z}_0 }^{\sup } \left\| {\sum\limits_{m = - n}^n {\mu _{\left| m \right|}^{(n)} \gamma _m e^{imx} } } \right\|_p< \infty$$

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).