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.