Some Classical Theorems

Abstract

The classical results which provided the main impetus for the study of interpolation in se are the theorems of M. Riesz, with Thorin’s proof, and of Marcinkiewicz. Thorin’s proof of the Riesz-Thorin theorem contains the idea behind the complex interpolation method. Analogously, the way of proving the Marcinkiewicz theorem resembles the construction of the real interpolation method. We give direct proofs of these theorems (Section 1.1 and Section 1.3), and a few of their applications (Section 1.2 and Section 1.4). More recently, interpolation methods have been used in approximation theory. In Section 1.5 we rewrite the classical Bernstein and Jackson inequalities to indicate the connection with approximation theory.