On the study of conjugate series of Fourier series by matrix summability method (Q2772875)

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 The conjugate series of the Fourier series is NEWLINE\[NEWLINE\sum^\infty_{n=1} (a_n\sin t- b_n\cos nt),\quad \psi(t)= \psi(x,t):= f(x+ t)- f(x- t).NEWLINE\]NEWLINE In this paper a new theorem on matrix summability of conjugate series of Fourier series has been established.NEWLINENEWLINENEWLINETheorem: Let \(\|T\|= (a_{n,k})\) be an infinite triangular matrix with \(a_{n,k}\geq 0\), \(A_{n,\tau}:= \sum^\tau_{k=0} a_{n,n-k}\), also \(A_{n,n}= 1\) for each \(n\geq 0\). Let \(\{a_{n,k}\}^n_{k=0}\) be a real nonnegative and nondecreasing sequence with respect to \(k\). If NEWLINE\[NEWLINE\int^t_0|\psi(u)|du= o\Biggl({1\over \chi(1/t)}\Biggr)\quad\text{as }t\to+0NEWLINE\]NEWLINE provided \(\chi(t)\) is a nonnegative, nondecreasing function of \(t\) such that NEWLINE\[NEWLINE\int^n_1 {A_{n,u}\over u\chi(u)} du= O(1)\quad\text{as }t\to\infty,NEWLINE\]NEWLINE then the conjugate series is \(T\)-summable to \(-{1\over 2\pi} \int^\pi_0 \psi(t)\text{ cot}\frac{t}{2} dt\) at every point where this integral exists.NEWLINENEWLINENEWLINEThis theorem generalizes previous results.