The Smoothness of Functions of Two Variables and Double Trigonometric Series

Omar Dzagnidze
Source: Real Anal. Exchange Volume 34, Number 2 (2008), 451-470.

Abstract

The notion of smoothness (according to Riemann) is introduced for functions of two variables and some of their properties are established. As an application we prove the uniform smoothness of an everywhere continuous sum of a double trigonometric series in the complex form which is obtained by twice term-by-term integration, over every variable rectangle $[0,x] \times [0,y] \subset [0,2\pi]$ of a double trigonometric series in the complex form absolutely converging at some point. An analogous consideration is given to a double trigonometric series in the real form, the absolute values of whose coefficients form a converging series.

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Primary Subjects: 26B05, 42B05
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1256835198
Zentralblatt MATH identifier: 05652547
Mathematical Reviews number (MathSciNet): MR2569198

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