## LSU Historical Dissertations and Theses

1996

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

Leonard Richardson

#### Abstract

The classical Szasz-Muntz theorem says that for $f\ \in\ L\sp2(\lbrack 0, 1\rbrack )$ and $\{n\sb{k}\}\sbsp{k=1}{\infty}$ a strictly increasing sequence of positive integers,$$\int\limits\sbsp{0}{1}x\sp{n\sb{j}}f(x)dx=0\ \forall j\Rightarrow f=0\Leftrightarrow\sum\sbsp{j=1}{\infty}{{1}\over{n\sb{j}}}=\infty.$$We have generalized this theorem to compactly supported functions on $\Re\sp{n}$ and to an interesting class of nilpotent Lie groups. On $\Re\sp{n}$ we rephrased the condition above on an integral against a monomial as a condition on the derivative of the Fourier transform $\ f$. For compactly supported f this transform has an entire extension to complex n-space, and these derivatives are coefficients in a Taylor series expansion of $\ f$. In the nilpotent Lie groups case there are several possible choices for the equivalent of a Fourier transform: the operator valued transform, the matrix coefficients for the operator transform relative to a basis of an infinite dimensional Hilbert space, and finally the trace transform. We have proven a Szasz-Muntz theorem for the matrix coefficients on groups that have a fixed polarizer for the representations in general position. For groups with flat orbits and a fixed radical for the representations in general position, we have proven a Szasz-Muntz theorem for the trace transform. Our work here is inspired by recent work on Paley-Wiener theorems on nilpotent Lie groups. Moss (1) proved a Paley-Wiener theorem on groups with a fixed polarizer for the generic representations. Park (2) extended these results to two and three step groups. Lipsman and Rosenberg (3) have proven a Paley-Wiener theorem for the matrix coefficients on any simply connected nilpotent Lie group. As part of the proof of the Szasz-Muntz 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 which are not ideals. Almost strong Malcev basis have the nice property that they are unique up to the original strong Malcev basis.

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