# On the Jackson constants for algebraic approximation of continuous functions

## TL;DR

This paper derives new bounds for Jackson constants in algebraic approximation of continuous functions, improving understanding of approximation quality and constants involved.

## Contribution

It provides new estimates for the Jackson constants in the Brudnyi-Jackson inequality, refining previous bounds for algebraic approximation of continuous functions.

## Key findings

- Established bounds for Jackson constants: 1/2 < J_a(2k, α) < 10.
- Provided inequalities for approximation error in terms of modulus of smoothness.
- Improved constants for algebraic polynomial approximation of continuous functions.

## Abstract

We establish new estimates for the constant $J_a(k,\alpha)$ in the Brudnyi-Jackson inequality for approximation of $f \in C[-1,1]$ by algebraic polynomials: $$ E_{n}^a (f) \le J_a(k, \alpha) \ \omega_k (f, \alpha \pi /n ), \quad \alpha >0 $$ The main result of the paper implies the following inequalities $$ 1/2< J_a (2k, \alpha) < 10, \quad n \ge 2k(2k-1), \quad \alpha \ge 2 $$