## Real Analysis Exchange

- Real Anal. Exchange
- Volume 34, Number 2 (2008), 451-470.

### The Smoothness of Functions of Two Variables and Double Trigonometric Series

#### 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.

#### Article information

**Source**

Real Anal. Exchange Volume 34, Number 2 (2008), 451-470.

**Dates**

First available in Project Euclid: 29 October 2009

**Permanent link to this document**

http://projecteuclid.org/euclid.rae/1256835198

**Zentralblatt MATH identifier**

05652547

**Mathematical Reviews number (MathSciNet)**

MR2569198

**Subjects**

Primary: 26B05: Continuity and differentiation questions 42B05: Fourier series and coefficients

**Keywords**

smoothness unilateral and symmetrical differentiability of functions of two variables smoothness in an angular sense smoothness of the sum of a double trigonometric series

#### Citation

Dzagnidze, Omar. The Smoothness of Functions of Two Variables and Double Trigonometric Series. Real Analysis Exchange 34 (2008), no. 2, 451--470. http://projecteuclid.org/euclid.rae/1256835198.