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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2009 • Question 8

Question

Math 6420 • PDE Spring 2009 • Question 8

The windowed Fourier transform of a function \(f(x)\in L^2(-\infty ,\infty )\) is defined by \[ Gf(\omega ,\mu )=\int _{-\infty }^{\infty }g(x-\mu )f(x)e^{-\omega x}\,d\mu \,dx. \] Suppose \(g(x)\) is a real valued function. State and verify the formula for the reconstruction of \(f(x)\) from \(Gf\), including any additional conditions on \(g(x)\). You may assume the validity of the Fourier transform.

Raw LaTeX

The windowed Fourier transform of a function \(f(x)\in L^2(-\infty,\infty)\) is defined by
\[
  
  Gf(\omega,\mu)=\int_{-\infty}^{\infty}g(x-\mu)f(x)e^{-\omega x}\,d\mu\,dx.
\]
Suppose \(g(x)\) is a real valued function. State and verify the formula for the reconstruction
of \(f(x)\) from \(Gf\), including any additional conditions on \(g(x)\). You may assume the validity
of the Fourier transform.

Topic Tags

Fourier Transform and Sampling Theory Linear Programming and Optimization Real Integral Evaluation by Complex Methods Zero-Stability and Root Conditions