The classic Mu ntz Szasz theorem says that for f # L 2 ([0, 1]) and [n k ] k=1 , a strictly increasing sequence of positive integers,
We have generalized this theorem to compactly supported functions on R n and to an interesting class of nilpotent Lie groups. On R n we rephrased the condition above on an integral against a monomial, as a condition on the derivative of the Fourier transform f . This transform, for compactly supported f, has an entire extension to complex n-space and these derivatives are coefficients in a Taylor series expansion of f .
For nilpotent Lie groups, we have proven a Mu ntz 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 Mu ntz Szasz theorem for matrix coefficients, We construct a new basis in a nilpotent Lie algebra, which we call an almost 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.
1998 Academic Press
1. Introduction
The Mu ntz Szasz theorem for the unit interval states that for f # L 2 [0, 1] and a strictly increasing sequence of natural numbers [n k ] k=1 , article no.