Notes on Fourier Analysis (X). On the summability of Fourier series.

Notes on Fourier Series

A function or a real variable f is said to be periodic with period P if f(x+ P ) = f(x) holds for all x. Hence, if we know the values of f on an interval of length P , we know its values everywhere. If f is a function defined on an interval [a, b), we can extend f to a function defined for all x which is periodic of period b− a. We simply define f(x) to be f(x+ n(b− a)), where n is the integer ...

Logarithmic Summability of Fourier Series

A set of regular summations logarithmic methods is introduced. This set includes Riesz and Nörlund logarithmic methods as limit cases. The application to logarithmic summability of Fourier series of continuous and integrable functions are given. The kernels of these logarithmic methods for trigonometric system are estimated.

On the Summability of the Differentiated Fourier Series

On the Summability of Fourier Series. Fifth Note.

and evaluating the generalized sum of (1) as the lim rm(x; f). The matrix ?2 will be subjected to the restrictions (i-iv) below. These restrictions are of such general nature, however, that they are satisfied by a great majority of the definitions of summability known in the literature. Our first condition on 21 is significant only in the case where 91 is of infinite reference. It is automatica...

On the Restricted Cesaro Summability of Double Fourier Series

and po.biu, v), («, v), has been considered by a number of writers. Gergen and Littauer [4, Theorems IV and V] have treated the problem of the boundedness, and convergence in the Pringsheim sense, of ...