Convergence of regularized greedy approximations

We consider a new version of a greedy algorithm in biorthogonal systems in separable Banach spaces. We consider approximations of an element $f$ via $m$-term greedy sum, which is constructed 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 the new version of a greedy algorithm (called the regularized greedy algorithm) always converges to the original element in Efimov-Stechkin spaces. We also construct examples that show the significance of the conditions of the main theorem.