Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2014 • Question 8

Question

Math 6010 • Linear Models Fall 2014 • Question 8

Let \(f\) be a continuous element of \(L^1(\mathbb {R})\), and recall that its Fourier transform is \[ \hat {f}(t):=\int _{-\infty }^{\infty }\exp (itx)f(x)\,dx. \]

(a) Suppose also that \(\hat {f}\in L^1(\mathbb {R})\). Prove the following [inversion formula]: \[ f(x)=\frac {1}{2\pi }\int _{-\infty }^{\infty }e^{-ixt}\hat {f}(t)\,dt \qquad \text {for all }x\in \mathbb {R}. \] In other words, \(\widehat {\widehat {f}}(-x)=2\pi f(x)\) for all \(x\in \mathbb {R}\).

(b) Choose and fix some \(\theta >0\). Use the preceding to prove that the Fourier transform of the probability density function \[ f(x):=\frac {\theta }{\pi (\theta ^2+x^2)} \qquad [-\infty <x<\infty ] \] is \(\hat {f}(t)=\exp (-\theta |t|)\).

Let \(f\) be a continuous element of \(L^1(\R)\), and recall that its
Fourier transform is
\[
  \hat{f}(t):=\int_{-\infty}^{\infty}\exp(itx)f(x)\,dx.
\]

\textbf{(a)} Suppose also that \(\hat{f}\in L^1(\R)\). Prove the following
[inversion formula]:
\[
  f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-ixt}\hat{f}(t)\,dt
  \qquad\text{for all }x\in\R.
\]
In other words, \(\widehat{\widehat{f}}(-x)=2\pi f(x)\) for all \(x\in\R\).

\textbf{(b)} Choose and fix some \(\theta>0\). Use the preceding to prove that
the Fourier transform of the probability density function
\[
  f(x):=\frac{\theta}{\pi(\theta^2+x^2)}
  \qquad[-\infty<x<\infty]
\]
is \(\hat{f}(t)=\exp(-\theta|t|)\).

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Topic Tags

Characteristic Functions and Fourier Inversion Classical Probability and Inclusion-Exclusion Measure-Theoretic Probability Uniform Integrability and Truncation Methods