Bernstein inequality for differential operators
Some analogues to Bernstein inequality for algebraic (with corresponding weights) and hyperspherical polynomials in \(L^ p\)-norms (1\(\leq p\leq \infty\), \(L_{\infty}=C)\) replacing the differential operator \(d^ 2/dx^ 2\) by corresponding differential operators D of second order are established. These inequalities have the form \((*)\quad \| DP_ n\|_{L_ p}\leq Const\cdot n^{1+\tau}\| P_ n\|_{L_ p},\) \(1\leq p\leq \infty\). In the case of algebraic polynomials D is the operator the eigenfunctions of which are the classical orthogonal polynomials of Jacobi (in this case \(\tau =1)\), Laguerre and Hermite (in these cases \(\tau =0)\). In case of m-dimensional (m\(\geq 2)\) spherical polynomials D is the Laplace-Beltrami operator and in this case \(\tau =1.If\) in the last case \(X=L_{\infty}\), the estimate (*) is made more precise. Namely, Const is replaced by (m-1) and the obtained inequality is exact.
scientific article; zbMATH DE number 4090093
scientific article