On the absolute Riesz summability of Fourier series and its conjugate series

Abstract

The series E1 a. is said to be absolutely summable (R, X, r), or simply summable R, X, rI, r_O, if Arw(,)/ W is of bounded variation in (A, oo), where A is a finite positive number [6; 7](1). We may, for example, take A =X1. The above definition can also be put in the following equivalent form by defining X suitably at nonintegral points and by a change of variable. ALTERNATIVE DEFINITION. Let X =X(X) be a continuous, differentiable, and monotonic increasing function of X in (K, oo), K being a positive constant, and let it tend to infinity with co. Suppose that >1 is a given infinite series, and write Cr(c@) = jX(co) X(n) } an (r ? 0).