[Paper Review] Fourier series on compact symmetric spaces

Gestur Ólafsson, Henrik Schlichtkrull|arXiv (Cornell University)|Apr 23, 2008

Advanced Algebra and Geometry26 references3 citations

TL;DR

This paper establishes a Paley-Wiener-type theorem for $K$-finite smooth functions on compact symmetric spaces $U/K$, showing that the Fourier coefficients of such functions, valued in a Hilbert space $\mathcal{H} = L^2(K/M)$, extend holomorphically to a complexified parameter space with exponential type $r$ if and only if the function is supported in a geodesic ball of radius $r$. The result generalizes previous work on $K$-invariant functions by reducing the problem to the $K$-invariant case via Kostant's theory of spherical principal series.

ABSTRACT

The Fourier coefficients F(t) of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter t, which determines the representation, and they can be represented by elements F(t) in a common Hilbert space H. We obtain a theorem of Paley-Wiener type which describes the size of the support of f by means of the exponential type of a holomorphic H-valued extension of F, provided f is K-finite and of sufficiently small support. The result was obtained previously for K-invariant functions, to which case we reduce.

Motivation & Objective

• To extend the local Paley-Wiener theorem from $K$-invariant functions to $K$-finite functions on compact symmetric spaces $U/K$.
• To characterize the support of $K$-finite smooth functions in terms of the holomorphic extension properties of their Fourier coefficients.
• To establish a bijection between $K$-finite smooth functions of small support and holomorphic $\mathcal{H}$-valued functions of exponential type $r$.
• To reduce the general $K$-finite case to the previously solved $K$-invariant case using representation-theoretic tools such as the spherical principal series.

Proposed method

• Use the Hilbert space $\mathcal{H} = L^2(K/M)$ to represent Fourier coefficients of $K$-finite functions, replacing scalar-valued coefficients in the $K$-invariant case.
• Apply Kostant’s description of the spherical principal series to decompose $K$-finite functions into $K$-invariant components via projection operators.
• Reduce the $K$-finite problem to the $K$-invariant case by analyzing the action of $\mathcal{U}(\mathfrak{g})$-invariant differential operators on $K$-invariant functions.
• Use holomorphic extension of the Fourier transform to characterize functions supported in geodesic balls of radius $r$, leveraging the Weyl group invariance and exponential type estimates.
• Construct the inverse Fourier transform by summing over $K$-types using operators $L(u_i)$ acting on $K$-invariant functions of exponential type $r$, ensuring smoothness and correct support.
• Leverage results from [19] on $K$-invariant functions and extend them via representation theory to the $K$-finite setting, proving a bijective correspondence for small $r$.

Experimental results

Research questions

• RQ1 Can the Paley-Wiener theorem for $K$-invariant functions on compact symmetric spaces be extended to $K$-finite functions with small support?
• RQ2 What conditions on the Fourier coefficients (as $\mathcal{H}$-valued functions) characterize $K$-finite smooth functions supported in a geodesic ball of radius $r$?
• RQ3 How can the $K$-finite case be reduced to the $K$-invariant case using representation-theoretic tools such as the spherical principal series?
• RQ4 Is the Fourier transform a bijection between $K$-finite smooth functions of small support and holomorphic $\mathcal{H}$-valued functions of exponential type $r$?
• RQ5 Can the support of a $K$-finite function be characterized by the holomorphic extension and growth rate of its Fourier transform?

Key findings

• There exists a radius $R > 0$ such that for all $r < R$, the Fourier transform induces a bijection between the space $C^\infty_{K,r}(U/K)$ of $K$-finite smooth functions supported in a geodesic ball of radius $r$ and the space $\mathrm{PW}_{K,r}(\mathfrak{a})$ of holomorphic $\mathcal{H}$-valued functions of exponential type $r$.
• Each $K$-finite function of small support arises as a finite sum of $\mathcal{U}(\mathfrak{g})$-derivatives of $K$-invariant functions of small support, with coefficients in $H^*_\delta$.
• The Fourier transform of a $K$-finite function extends holomorphically to the complexified dual space $\mathfrak{a}^*_\mathbb{C}$ and satisfies a growth condition of exponential type $r$, which characterizes the support radius.
• The reduction to the $K$-invariant case is achieved via Kostant’s theory of spherical principal series, allowing the use of known results from [19] on $K$-invariant functions.
• For each $K$-type $\delta$, the Fourier transform $\tilde{f}_\delta$ of the $\delta$-component of $f$ extends holomorphically with exponential type $r$, and this condition is both necessary and sufficient for $f$ to be supported in a ball of radius $r$.
• The result is sharp in the sense that the radius $R$ depends on the geometry of $U/K$, and no such bijection exists for $r \geq R$.