# Direct Theorems in the Theory of Approximation of the Banach Space Vectors by Entire Vectors of Exponential Type

## TL;DR

This paper establishes direct theorems linking the smoothness of vectors in Banach spaces to their approximation by exponential type entire vectors, enabling Jackson-type inequalities in various function spaces.

## Contribution

It introduces new direct theorems connecting smoothness, approximation order, and continuity modules for operators generating C_0-groups on Banach spaces.

## Key findings

- Established direct theorems relating smoothness and approximation order.
- Derived Jackson-type inequalities for classical and weighted function spaces.
- Connected approximation theory with operator smoothness in Banach spaces.

## Abstract

For an arbitrary operator A on a Banach space X which is a generator of C_0-group with certain growth condition at the infinity, the direct theorems on connection between the smoothness degree of a vector $x\in X$ with respect to the operator A, the order of convergence to zero of the best approximation of x by exponential type entire vectors for the operator A, and the k-module of continuity are given. Obtained results allows to acquire Jackson-type inequalities in many classic spaces of periodic functions and weighted $L_p$ spaces.