On the study of Fourier series by \(K^\lambda\)-summability method (Q2772873)

In this paper a new theorem has been established on \(K^\lambda\)-summability of Fourier series; the method \(K^\lambda\) was first introduced by Karamata. Let us define, for \(n= 0,1,2,3,\dots\), the number \({n\brack m}\), for \(0\leq m\leq n\), by NEWLINE\[NEWLINEx(x+ 1)(x+ 2)\cdots(x+ n-1):= \sum^n_{m=0} {n\brack m} x^m,NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\overline{|x+n}\over \overline{|x}}= \prod^{n- 1}_{\nu= 0} (x+\nu)= \sum^n_{m= 0} {n\brack m} x^n,\quad \overline{|x}:= 1\cdot 2\cdots(x- 1).NEWLINE\]NEWLINE The numbers \({n\brack m}\) are known as the absolute values of the Stirling numbers sequence of the first kind. Let \(\{S_n\}\) be the sequence of partial sums of the series \(\sum a_n\) and let us write \(S^\lambda= {\overline{|\lambda}\over \overline{|\lambda+ n}} \sum^n_{m=0} {n\brack m} \lambda^m S_m\), to denote the \(n\)th \(K^\lambda\)-mean of order \(\lambda> 0\). If \(S^\lambda_n\to S\), as \(n\to\infty\), where \(S\) is a fixed finite quantity, then the sequence \(\{S_n\}\) or the series \(\sum a_n\) is said to be summable by the Karamata method \(K^\lambda\) of order \(\lambda> 0\) to the sum \(S\) and we can write \(S^\lambda_n\to S\) (\(K^\lambda)\), as \(n\to\infty\). Let the \(2\pi\)-periodic function \(f(t)\in L(-\pi,\pi)\), the Fourier series of \(f(t)\) is given by NEWLINE\[NEWLINEf(t)\sim{1\over 2} a_0+ \sum^\infty_{n=1} (a_n\cos nt+ b_n\sin nt),NEWLINE\]NEWLINE we write \(\phi(t):= f(x- t)+ f(x+ t)- 2f(x)\). The following theorem generalizes several previously known theorems.NEWLINENEWLINENEWLINETheorem: If NEWLINE\[NEWLINE\int^t_0 |\phi(u)|du= o\Biggl({t^\delta\over \xi({1\over t})}\Biggr),\quad t\to +0,NEWLINE\]NEWLINE where \(0< \delta\leq 1\) and \(\xi(t)\) is positive nondecreasing with \(t\) such that \(\xi(n)\to\infty\) as \(n\to\infty\) and \(\log n= o(\xi(n))\) as \(n\to\infty\), then the Fourier series is summable \(K^\lambda\) \((\lambda> 0)\) to the sum \(f(x)\) at the point \(t=x\).