Müntz–Szasz Theorems for Nilpotent Lie Groups

Abstract

The classic Müntz–Szasz theorem says that forf∈L2([0,1]) and {nk}∞k=1, a strictly increasing sequence of positive integers,∫10xnjf(x)dx=0∀j⇒f=0⇔∑j=1∞1nj=∞.We have generalized this theorem to compactly supported functions on Rnand to an interesting class of nilpotent Lie groups. On Rnwe rephrased the condition above on an integral against a monomial, as a condition on the derivative of the Fourier transformf. This transform, for compactly supportedf, has an entire extension to complexn-space and these derivatives are coefficients in a Taylor series expansion off. For nilpotent Lie groups, we have proven a Müntz–Szasz theorem for the matrix coefficients of the operator valued Fourier transform, on groups that have a fixed polarizer for the representations in general position. Our work here is inspired by recent work on Paley–Wiener theorems for nilpotent Lie groups by Moss (J. Funct. Anal.114(1993), 395–411) and Park (J. Funct. Anal.133(1995), 211–300), who have proven Paley–Wiener theorems on restricted classes of nilpotent Lie groups. Lipsman and Rosenberg (Trans. Amer. Math. Soc.348(1996), 1031–1050) have extended these results, for matrix coefficients, to any connected, simply connected nilpotent Lie group. As part of the proof of the Müntz–Szasz theorem for matrix coefficients, We construct a new basis in a nilpotent Lie algebra, which we call analmost strong Malcev basis. This new basis has many of the features of a strong Malcev basis, although it can be used to pass through subalgebras that are not ideals. Almost strong Malcev bases are unique up to a fixed strong Malcev basis. We will also show that, using almost strong Malcev bases, we can provide a partial answer to a question posed by Corwin and Greenleaf (“Representations of Nilpotent Lie Groups and Their Applications”, Cambridge Univ. Press, Cambridge, UK, 1990) on using additive coordinates for a cross-section.