# The Commutant of the Fourier–Plancherel Transform

4-20-2023

Thesis

Mathematics

William Ross

Emory F. Bunn

## Abstract

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 behind an integral that is difficult to compute. In the Chapter 1, we establish some basic notation about Hilbert spaces and introduce the two Hilbert spaces central to the ideas developped in this thesis, L^2(R) and l^2. We then define the Hermite polynomials and the Hermite functions in Chapter 2, which we will show form a convenient orthonormal basis for L2(R). The Hermite polynomials are further employed in Chapter 3, where we establish the Fourier-Plancherel transform F on L^2(R). A central step in this is to compute the eigenbasis of the Fourier transform, which we show is the set of Hermite functions. After establishing the Fourier transform, we further characterize it in Chapter 4 by defining the set {F}0, which contains all bounded linear operators on L^2(R) that commute with F. We continue this characterization in Chapter 5 by defining root F, the set of bounded linear operators on L^2(R) that are square roots of F. Finally, we conclude our analysis of F in Chapter 6 by describing all of the invariant subspaces of the Fourier transform.

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