A NEW CHARACTERIZATION OF BESOV-SPACES ON THE REAL LINE

We give a new characterization of functions f defined on the real line (-infinity, infinity) in order to belong to a Besov space B-alpha(p,r) for some 0 < alpha < 1 and 1 less than or equal to p, r less than or equal to infinity. These conditions are in terms of the Riesz mean of f in case 1 less than or equal to p less than or equal to infinity, and in terms of the Dirichlet integral of f in case 1 < p < infinity. An analogous characterization of periodic functions on the torus [-pi, pi) was initiated by Fournier and Self, via the partial sums of their Fourier series. The novelty in our treatment is that we use norms involving integrals, instead of norms involving sums of infinite series. Our approach is also appropriate to building up a complete characterization of Besov spaces on the torus. (C) 1995 Academic Press, Inc.