Some Problems of Approximation Theory in the Spaces $L_p$ on the Line with Power Weight

In the spaces $L_p$ on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the $K$-functional. We prove a direct and an inverse theorem of Jackson–Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the $K$-functional.