EXPONENTIAL APPROXIMATION OF FUNCTIONS IN LEBESGUE SPACES WITH MUCKENHOUPT WEIGHT

Abstract

Using a transference result, several inequalities of approximation by entire functions of exponential type in 𝒞(R), the class of bounded uniformly continuous functions defined on R=(-∞,+∞), are extended to the Lebesgue spaces 𝐿^𝑝(𝜚𝑑𝑥) 1≤𝑝<∞ with Muckenhoupt weight 𝜚. This gives us a different proof of Jackson type direct theorems and Bernstein-Timan type inverse estimates in 𝐿^𝑝(𝜚𝑑𝑥). Results also cover the case 𝑝=1.

Keywords

• lebesgue spaces
• muckenhoupt weight
• entire functions of exponential type
• one-sided steklov operator
• best approximation
• direct theorem
• inverse theorem
• modulus of smoothness
• marchaud-type inequality