Measure, Integration & Real Analysis · Problem 11C.5

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · fourier-transform-polynomial-gaussian

Problem

Prove that if

p

is a polynomial on

\mathbf{R}

with complex coefficients and

f: \mathbf{R} \to \mathbf{C}

is defined by

f(x) = p(x)e^{-\pi x^2}

, then there exists a polynomial

q

\mathbf{R}

with complex coefficients such that

\deg q = \deg p

and

\hat{f}(t) = q(t)e^{-\pi t^2}

for all

t \in \mathbf{R}