On complete characterization of coefficients of a.e. converging orthogonal series资料.pdfVIP

ON COMPLETE CHARACTERIZATION OF COEFFICIENTS OF A.E. CONVERGING ORTHOGONAL SERIES 5 0 ´ ´ Adam Paszkiewicz (Lodz) 0 2 l u Abstract. We characterize sequences of numbers (an) such that n≥1 anΦn con J verges a.e. for any orthonormal system (Φ ) in any L space. In our criterion, we n 2 2 7 use the set B = { m≥n |am | ;n ≥ 1} and its information function 2 hB (t) = − log (β − α) ] 3 P A for t ∈ (α, β], [α, β] ∩ B = {α, β }. . h t a m [ 1. Introduction. The aim of this paper is to give a complete characterization of 1 sequences (a ) for which v n 8 6 5 (*) a Φ converges a.e. for any orthonormal sequence (Φ ) 7 n n n 0 5 0 (O.N. for short) in any L2 space. / h t The celebrated Rademacher-Menshov theorem gives complete characterization a of so called Weil coefficients r , n ≥ 1, for which the convergence m n : v i r |a |2 ∞, a ∈ C X n n n r a implies (*). Namely increasing r ’s are Weil coefficients if the sequence r / log2 n n n 2 is bounded and are not Weil coefficients if r / log2 n → ∞. n 2 The theorem, ho