We construct the blending-type modified Bernstein–Durrmeyer operators and investigate their approximation properties. First, we derive the Voronovskaya-type asymptotic theorem for this type of operator. Then, the local and global approximation theorems are obtained by using the classical modulus of continuity and K-functional. Finally, we derive the rate of convergence for functions with a derivative of bounded variation. The results show that the new operators have good approximation properties.

We propose the Weak Rescaled Pure Super Greedy Algorithm (WRPSGA) for approximation with respect to a dictionary $\mathcal{D}$ in Hilbert space. The WRPSGA is simpler than some popular greedy algorithms. We show that the convergence rate of the RPSGA on the closure of the convex hull of the $\mu$-coherent dictionary $\mathcal{D}$ is optimal. Then, we design the Rescaled Pure Super Greedy Learning Algorithm (RPSGLA) for kernel-based supervised learning. We prove that the convergence rate of the RPSGLA can be arbitrarily close to the best rate $O\left(m^{-1}\right)$ under some mild assumptions.

In this paper, we introduce the q-analogue of modified Gamma operators preserving linear functions. We establish the moments of the operators using the q-Gamma functions. Next, some local approximation for the above operators are discussed. Also, the rate of convergence and weighted approximation by these operators in terms of modulus of continuity are studied. Furthermore, we obtain the Voronovskaja type theorem.

We study the approximation capability of the orthogonal super greedy algorithm (OSGA) with respect to $\mu$-coherent dictionaries in Hilbert spaces. We establish the Lebesgue-type inequalities for OSGA, which show that the OSGA provides an almost optimal approximation on the first $[1/(18\mathsf{\mu }\mathrm{s}\left)\right]$ steps. Moreover, we improve the asymptotic constant in the Lebesgue-type inequality of OGA obtained by Livshitz E D.

In this paper, a kind of new analogue of Gamma type operators based on (p,q)-integers is introduced. The Voronovskaja type asymptotic formula of these operators is investigated. And some other results of these operators are studied by means of modulus of continuity and Peetre K-functional. Finally, some direct theorems concerned with the rate of convergence and the weighted approximation for these operators are also obtained.