Modulus of Smoothness and Approximation in Morrey–Banach Spaces

This paper extends the approximation inequalities for the modulus of smoothness to Morrey–Banach spaces. The estimates of the modulus of smoothness are controlled by the derivatives and the approximation by the trigonometric polynomials. As special cases of the main result, we have the approximation inequalities for the modulus of smoothness for the Morrey spaces with variables exponents.