Generalized Dunkl transform on the line in inverse problems of approximation theory

The generalized Dunkl harmonic analysis on the line, depending on the parameter $r\in\mathbb{N}$, is studied. The case $r=0$ corresponds to the usual Dunkl harmonic analysis. All designs depend on the parameter $r\ge 1$. Using the generalized shift operator, differences and moduli of smoothness are determined. Using the differential-difference operator, the Sobolev space is defined. We study the approximation of functions from space $L^{p}(\mathbb{R},d\nu_{\lambda})$ by entire functions of exponential type not higher than $\sigma$ from the class $f\in B_{p, \lambda}^{\sigma,r}$ that have the property $f^{(2s+1)}(0)=0$, $s=0,1,\dots,r-1$. For entire functions from the class $f\in B_{p, \lambda}^{\sigma,r}$, inequalities are proved that are used in inverse problems of approximation theory. Depending on the behavior of the values of the function best approximation, an estimate is given of the modulus of smoothness of the function, as well as the modulus of smoothness on the degree of its second-order differential-difference operator. A condition is given for asymptotic equality between the best approximation of the function and its modulus of smoothness.