О рядах по системе Фра нклина

Abstract

In this paper some basis properties are proved for the series with respect to the Franklin system, which are analogous to those of the series with respect to the Haar system. In particular, the following statements hold: 1. The Franklin series\(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) converges a.e. onE if and only if\(\mathop \Sigma \limits_{n = 0}^\infty a_n^2 f_n^2 (x)< + \infty \) a.e. onE; 2. If the series\(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\), with coefficients ¦an¦↓0, converges on a set of positive measure, then it is the Fourier-Franklin series of some function from\(\bigcap\limits_{p< \infty } {L_p } \); 3. The absolute convergence at a point for Fourier—Franklin series is a local property; 4. If an integrable function (fx) has a discontinuity of the first kind atx=x0, then its Fourier-Franklin series diverges atx=x0.