The “Full Müntz Theorem” inL _p[0, 1] for 0<p<∞

Abstract

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ℝ. The principal result of the paper is the following.

Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ⊂ [0, 1] with positive lower density at 0). Let A ⊂ [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) ^∞ _j=1 is a sequence of distinct real numbers greater than −(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if$\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty }$. Moreover, if$\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty }$, then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ℂ \ (−∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where$r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure).

This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.