Convergence of Walsh-Fourier series of functions with bounded osscillation

Abstract

The notion of function of the bounded variation was introduced by Jordan. Generalizing this notion Wiener has considered the $V_{p}$ class of functions. Young introduced the notion of functions of $\Phi$-variation. Waterman has studied the class of functions of the bounded $\Lambda$-variation, and Chanturia has defined the notion of the modulus of variation of a function. In 1990 Kita and Yoneda introduced the notion of the generalized Wiener's class $BV\left( p\left( n\right) \uparrow p\right)$. Generalizing the class $BV\left( p\left( n\right) \uparrow p\right)$ Akhobadze \cite{Akh1,Akh2} has considered the $BV\left( p\left( n\right) \uparrow p,\varphi \right) $ and$\,$ $B\Lambda \left( p\left( n\right) \uparrow p,\varphi \right)$ classes of functions. We study the pointwise convergence of Walsh-Fourier series and we also estimate the Walsh-Fourier coefficients of the functions on the class $BO\left( p\left( n\right) \uparrow \infty \right)$. These results were obtained in collaboration with Ushangi Goginava, professor of the University of Tbilisi, Georgia.