FUNCTIONS OF BOUNDED MEAN OSCILLATION AND RELATED TOPICS

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It is very well known that the functions of bounded mean oscillation are characterized by means of Poisson integral. In these investigations, we have succeeded in finding out that the same characterization is possible through a rather wide class of singular integrals : in particular, we can characterize the functions of bounded mean oscillation by means of Cesaro means of their Fourier series. Also, analogue of Bernstein inequality for the space BMO is obtained. See appended pamphlet, part 1.As to the approximation theory, results on exponential type operators are obtained. They are also included together with a preliminary version of Lecture notes on Approximation Theory of periodic functions. In our intension, some other topics such as approximation by positive linear operapors, operating functions on some subclasses of A, would be included ; we hope, in a near future, to publish a completed version.