On the convergence criteria of a Fourier series (Q2640084)

Let f be \(2\pi\)-periodic and L-integrable over (0,2\(\pi\)) and let its Fourier series at a point x be \[ (1/2)a_ 0+\sum^{\infty}_{n=1}(a_ n \cos nx+b_ n \sin nx). \] For fixed real numbers x and s, we write \[ \phi (t)=f(x+t)+f(x-t)-2s,\quad g_ 1(t)=(1/t)\int^{t}_{0}g(u)du\quad (g\in L(0,\pi)), \] \[ P(t)=\phi (t)-\phi_ 1(t),\quad Q(t)=P(t)+P_ 1(t), \] \[ S_ n(x)=(1/2)a_ 0+\sum^{n}_{m=1}(a_ m \cos mx+b_ m \sin mx). \] The author proved the following theorem: Theorem 1: let \(\phi_ 1(t)\to 0\) as \(t\to 0+\), and let \[ \lim_{n\to 0+}\int^{\delta}_{0}t^{-1}| P(t+n)-P(t)| dt=0, \] where \(\delta >0\). Then the sequence \((S_ n(x))\) converges to s. Furthermore the author has proved a new result for weaker conditions than those of \textit{N. K. Bary} [A treatise on trigonometric series, Vol. 1 (1964; Zbl 0129.280)] and himself [Indian J. Pure Appl. Math. 10, 1576- 1581 (1979; Zbl 0433.42005)] and investigated a new criterion for the convergence of a Fourier series at a point which includes the de la Vallée-Poussin criterion and the well known Lebesgue-Gergen criterion for the convergence of Fourier series at a point.