A class of continuous functions and convergence criteria for their Fourier series
Nash, John P.
Doctor of Philosophy
The present paper has a two-fold purpose. First, to define a class of functions wider than the class of functions of ecart fini and to obtain sufficient conditions for the existence of functions of this class. Second, to show that if continuous functions of this new class have moduli of continuity satisfying a slight restriction---less restrictive, in fact, than the Dini-Lipschitz condition---then their Fourier series converge uniformly. These functions do not satisfy the classical convergence criteria. An interesting example of a function of this kind, for which the Dini-Lipschitz criterion is not satisfied, is exhibited.