Spaces of functions of generalized bounded variation. II: Questions of uniform convergence of Fourier series (Q2773662)

Let \(X\) be a symmetric space of sequences. The space \(\text{BV}(X)\) of functions \(f\:[0,1]\to \mathbb R\) of generalized bounded variation comprises continuous functions such that NEWLINE\[NEWLINE \|f|_{\text{BV}(X)}\|=\sup \Big\{\Big\|\sum f(J_i)e^i|X\Big\|:\;\bigcup_i J_i\subset [0,1]\Big\} +\sup\limits_{t\in [0,1]}|f(t)|<\infty, NEWLINE\]NEWLINE where \(J_i=(a_i,b_i)\) are disjoint intervals, \(f(J_i)=f(b_i)-f(a_i)\), and \(\{e^i\}\) is the standard basis in the space of sequences. First, it is demonstrated how the known criterion of uniform convergence of the Fourier series such as the Salem-Bernstein-Oskolkov criterion and the Waterman criterion are obtained on the base of a new method proposed by the author. Next, the author establishes several new criteria, actually using all conventional spaces \(X\). Distinct criteria are compared. Sharp estimates for the coefficients of the Fourier series of \(f\in \text{BV}(X)\) are also presented.NEWLINENEWLINENEWLINE[See also Part I in Sib. Math. J. 40, No.~5, 837-850 (1995); translation from Sib. Mat. Zh. 40, No. 5, 997-1011 (1999; Zbl 0944.46025)].