Müntz-type theorems on the half-line with weights

Abstract

We consider the linear span S of the functions t a k (with some a k > 0) in weighted L 2 spaces, with rather general weights. We give one necessary and one sufficient condition for S to be dense. Some comparisons are also made between the new results and those that can be deduced from older ones in the literature.This classical result was first proved in L 2 [0, 1] and then extended to C[0, 1], as stated above. Also, it was stated only for increasing sequences λ k . Subsequently, this theorem has had several different proofs and generalizations, and there are several surveys in this topic (see for instance the papers of J. Almira and A. Pinkus [1], [23]).On C[0, 1] and L p (0, 1), "full Müntz theorems", i.e. theorems with rather general exponents, were later proved by eg. P. Borwein, T. Erdélyi, W. B. Johnson and V. Operstein ([7], [13], [12], [22]).