Summation of Derived Fourier Series: An Application to Fourier Expansions on Compact Lie Groups

Abstract

Introduction. Recently' the author investigated the question of convergence and summability of multiple Fourier Series from a new view point. It was shown that there exists, for Fourier series of all dimensions, a general type of summation processes under which the convergence or non-convergence of the corresponding partial sums at a given point depends only on the behavior of the function in this given point, and that continuity (in a rather weakened sense) of the function at the point is sufficient for convergence. Multiple Fourier series are Fourier expansions on the torus. The torus is a special case of a compact Lie group; on the other hand there exist Fourier expansions on every group.2 In the present note we shall see that our previous result may be easily unheld for Fourier expansions on general compact Lie groups. In fact the construction of the underlying group space does not enter materially into the formulation of the convergence criterion; what matters essentially is only the dimension of the group, that is the total number of its real parameters. In part I we shall give an extension of our previous result concerning Fourier series on the torus to trigonometric series which, formally, are partial derivatives of such series. This extension will be required in Part II, but the result itself is of interest. In part II we shall treat in detail the expansions on (closed) semi-simple Lie groups which are unitary according to the definition of H. Weyl;3 these groups are the very opposite to the torus. We shall discuss only class functions. The result can be very easily extended