Date of Award
Doctor of Philosophy (PhD)
This thesis is composed of two parts, each part treating a different problem from the theory of Harmonic Analysis. In the first part we present an inequality in White Noise Analysis similar to the classical Heisenberg Inequality for functions in L2Rn . To do this we replace the finite dimensional space R n and its Lebesgue measure by the infinite dimensional space E' , which is the dual of a nuclear space E , and its Gaussian measure. Choosing an arbitrary element eta in E , we may define the multiplication operator Q&d5;h , which is the sum between the differentiation operator D&d5;h and its adjoint D*h . We then use Q&d5;h as a substitute for the multiplication operator by x from the finite dimensional case. Because the Fourier transform is a unitary operator on L2Rn , whose eigenvalues are powers of -i and the second quantization operator of -iI, denoted by G-iI , from (L2) into itself has the same properties, we replace the Fourier transform by ( G-iI ). Here, (L2) is the space of all complex valued, square integrable functions on E' . The proof of our Heisenberg Inequality relies on the Schwartz Inequality and the commutation relationships between any two of the following three operators: D&d5;h, D*h , and G-iI . In the second part of this work, we show some results in White Noise Analysis analogous to the classical Paley-Wiener Theorem that describes functions on Rn with compact support in terms of their Fourier transform. Because the Fourier transform for a Schwartz function is defined pointwise as an inner product between that function and an exponential function and the S-transform from White Noise Analysis is defined similarly, we have chosen this time the S-transform as the natural replacement for the Fourier transform. After giving a thorough description of the weakly and strongly compact subsets of E' , we finally characterize some classes of (L 2) functions with compact support in terms of their S-transform. As in the classical Paley-Wiener theorem two conditions are essential, namely: an analyticity and a growth condition.
Stan, Aurel Iulian, "On Harmonic Analysis for White Noise Distribution Theory." (1999). LSU Historical Dissertations and Theses. 7016.