Multivariate Bernstein inequalities for entire functions of exponential type in L^{p}(mathbb{R}^{n})(0< p< 1)

Abstract

In (Rahman and Schmeisser in Trans. Amer. Math. Soc. 320: 91–103, 1990), the authors prove that the classical Bernstein inequality also holds for 0< ple 1. We extend their result for a differential operator induced by polynomials and find the several equivalent conditions to the Paley–Wiener theorem. As applications of the results, we also derive the Paley–Wiener type theorems for some special compact sets generated by number sequences, generated by polynomial, convex compact sets, in which we show that the Bernstein type inequalities have concrete upper bounds.