Abstract
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 \(r\ge 1\) . The differences and 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.
