Generalized one-dimensional Dunkl transform in direct problems of approximation theory

On the real line, we study the generalized Dunkl harmonic analysis depending on a parameter $r\in\mathbb{N}$. The case of $r=0$ corresponds to the usual Dunkl harmonic analysis. All constructions depend on the parameter $r\geqslant 1$. The differences and the moduli of smoothness are defined using a generalized translation operator. The Sobolev space and the $K$-functional are defined using a differential-difference operator. An approximate Jackson-type inequality is proved. The equivalence of the $K$-functional and the modulus of smoothness is established.