Sharp Bernstein inequalities for Jacobi-Dunkl operators (Q2680725)

The author finds sharp constants in the Bernstein inequality \[ \|\Lambda^r_{\alpha,\beta}f\|\le M\,\|f\| \] for the Jacobi-Dunkl differential-difference operator \(\Lambda_{\alpha,\beta}\): \[ \Lambda_{\alpha,\beta}f(x)=f'(x)+\frac{A'_{\alpha,\beta}(x)}{A_{\alpha,\beta}(x)}\, \frac{f(x)-f(-x)}{2}. \] Here \(n, r\in {\mathbb N}\), \(f\) is a trigonometric polynomial of degree \(\le n\), the norm is uniform, \(\alpha,\beta\ge -1/2\), and \( A_{\alpha,\beta}(x)= (1-\cos x)^\alpha (1-cos x)^\beta |\sin x| \) is the periodic Jacobi weight. In the spaces \(L_{p,\alpha,\beta}\) with Jacobi weight, upper bounds are obtained. In terms of the Fourier coefficients in the system of Jacobi-Dunkl polynomials, the operator \(\Lambda_{\alpha,\beta}\) is a multiplier, i.e., it acts as multiplication. The author describes the general scheme for constructing interpolation formulas and obtaining inequalities for the norms of operators of this type in the spaces \(L_{p,\alpha,\beta}\) with the Jacobi weight.