Convergence of regularized greedy approximations

In this paper we consider a new version of the greedy algorithm by biorthogonal systems in separable Banach spaces. We consider approximations of an element f using the m-term greedy sum: it is obtained from the expansion by choosing the first m greatest in absolute value coefficients. It is known that the greedy algorithm does not always converge to the original element. We prove a theorem showing that a new version of the greedy algorithm, which is called regularized greedy algorithm, always converges to the original element in the Efimov-Stechkin space. We also construct examples that show the significance of the conditions of the main theorem.