Muntz-Szasz theorems for nilpotent Lie groups

The classic Muntz-Szasz theorem says that for f is an element of L-2([0, 1]) and {n(k)}(k = 1)(infinity), a strictly increasing sequence of positive integers, (integral(0)(1) x(j)(n)f(x)dx = 0 For All j double right arrow f = 0) double left right arrow Sigma(j = 1)(infinity) 1/n(j) = infinity. 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 rendition above on-an integral against a monomial. as a condition on the derivative of the Fourier transform (f) over cap. 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) over cap. For nilpotent Lit groups, we have proven a Muntz-Szasz theorem for the matrix coefficients of the operator valued Fourier transform, on groups that have a fixed polarizer ibr the representations ill 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 Muntz-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. (C) 1998 Academic Press.