# Convergence Theorems

Conference paper

## Abstract

As indicated in Sec. 6 (see (6.19)), the theory of Carleman operators is connected with series of the form where a

$$ \sum\limits_{{n = 0}}^{\infty } {{a_{n}}} |{\phi _{n}}\left( x \right){|^{2}} $$

(10.1)

_{n}≧ {ф and (x)} is some orthonormal system in L^{2}. In fact, a series of this type emerged in the proof of Theorem 9.2 (cf. (9.7)). We shall now investigate the convergence problems of series like (10.1). We first consider two complete orthonormal systems ф and (x) and {ψ_{n}(x)} on L^{2}(a,b), the Hilbert space of complex valued functions of one real variable, defined on the interval [a,b] with -∞ ≤ > b ≤ ∞.## Preview

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## Copyright information

© Springer-Verlag Berlin · Heidelberg 1967