Dual Orthogonal Series with Modifier Tending to Zero

Abstract

In this paper we consider the basis property for $\{ P_{\varphi _n } + c_n Q\varphi _n \} $ where $\{ c_n \} $ is a positive sequence converging to 0 and (as in [Feinerman and Kelman, this Journal, 1974]), $\{ \varphi _n \} $ is a complete orthonormal sequence in a Hilbert space H, and P and Q are orthogonal projections on H. In [Feinerman and Kelman] it was proven that if $\{ c_n \} $ converges to a positive limit then $\{ P_{\varphi _n } + c_n Q_{\varphi _n } \} $ is an $l^2 $ basis while in this paper we prove that if $c_n $ converges to 0 it is not an $l^2 $ basis. We also include an application of the result to a problem in heat transfer.