Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

I N Brui

Institution:

(1) Byelorussian Law Institute, Baranovichi

Abstract:

Suppose that a lower triangular matrix μ:μ _m ⁽ⁿ⁾ ] 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). Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 677–686, November, 1997. Translated by N. K. Kulman

Keywords:

trigonometric series conservative summation method Young test Fejér method Riesz theorem

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