On the non-absolute summability of a Fourier series and the conjugate of a Fourier series by a Nörlund method

Source abstract

Let { S n } be the sequence of partial sums of the infinite seriesΣα n . Let { p n } be a sequence of constants real or complex and let us set The sequence { t n } of Nörlund means (5) or simply ( N, p n ) means of the sequence { S n } generated by the sequence of coefficients { p n } is defined by the following sequence -to-sequence transformation The series ∑α n or the sequence { S n } is said to be summable ( N, p n ) to the sum S , if and is said to be absolutely summable ( N, p n ) or summable | N, p n |, if the sequence { t n } is of bounded variation, that is, the series ∑| t n − t n −1 | is convergent (2).