Special Functions Commons^™

The Commutant Of The Fourier–Plancherel Transform, Brianna Cantrall

Honors Theses

One can see that this matrix is unitary and has eigenvalues {1,−i,−1, I}, each of infinite multiplicity. Throughout the remainder of this thesis, we will convince the reader that the above linear transformation is actually the Fourier transform. We will compute the commutant, as well as its invariant subspaces. The key to do this relies on the Hermite polynomials. Why do we recast the Fourier transform from its well-known and well studied integral form to the matrix form shown above? As we will see, the matrix form allows us to efficiently discover the operator theory of the Fourier transform obfuscated …

Interpolating With Outer Functions, Javad Mashreghi, Marek Ptak, William T. Ross

Department of Math & Statistics Faculty Publications

The classical theorems of Mittag-Leffler and Weierstrass show that when (λn)n≥1 is

a sequence of distinct points in the open unit disk D, with no accumulation points in

D, and (wn)n≥1 is any sequence of complex numbers, there is an analytic function

ϕ on D for which ϕ(λn) = wn. A celebrated theorem of Carleson [2] characterizes

when, for a bounded sequence (wn)n≥1, this interpolating problem can be solved with

a bounded analytic function. A theorem of Earl [5] goes further and shows that when

Carleson’s condition is satisfied, the interpolating function ϕ can be …