A generalization of the Salem test

Abstract

The Salem test is a criterion for uniform convergence of a Fourier series on the circle group. An improved test for convergence a point and uniform convergence on an arbitrary set is obtained, as well as a necessary and sufficient condition for the convergence of the Fourier series of a function a Lebesgue point. The result with which we are concerned is a test for the convergence of the Fourier series of a summable function. It provides conditions for convergence a point and for uniform convergence on a set. This result generalizes the Salem test for uniform convergence [3, 7]. Bary's opinion of the Salem test was that it appears at a first glance to be hardly suitable for application [3, pp. 305-310], but it has served as the starting point of many other investigations. For example, various notions of generalized bounded variation, such as Salem's ?-bounded variation [3] and Waterman's A-bounded variation [9, 10] arise directly from the ideas of the Salem test. Investigations into the everywhere convergence and uniform convergence of Fourier series under every change of variable [1, 2, 4, 5, 6, 8] also originated from these ideas. We shall consider a summable realor complex-valued function on the circle group T. For odd integers n, let Tn(X I t) _f f(X + tln) f (x + (t + 7r)/n) T~(xt) =f(x+t/)fx(t+7r)/)/n)+(+2)n f (x + (t + 27r)/n) f (x + (t + 37r)/n) + 3 + f (x + (t + (n 1)7r)/n) f (x + (t + n7r)/n) n and let Qn (x, t) be obtained from Tn (x , t) by substituting -t and r for t and 7r respectively. The Salem test can then be stated as follows. Salem Test. If f is continuous on T and Tn(x , 0) and Qn(x , 0) converge uniformly to 0, then the Fourier series of f converges uniformly. Received by the editors January 13, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 42A20; Secondary 42A16. (? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page