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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2016 • Question 9

Question

Math 6420 • PDE Spring 2016 • Question 9

Show that Fourier transform \[ f(x)\to F(\mu )=\int _{-\infty }^{\infty } f(x)e^{i\mu x}\,dx \] preserves the inner product and the \(L_2\) norm (Plancherel’s equations): \[ \int _{-\infty }^{\infty }\bar {f}(x)g(x)\,dx = \int _{-\infty }^{\infty }\bar {F}(\mu )G(\mu )\,\frac {d\mu }{2\pi }, \] \[ \text {and}\qquad \int _{-\infty }^{\infty }\lvert f(x)\rvert ^2\,dx = \int _{-\infty }^{\infty }\lvert F(\mu )\rvert ^2\,\frac {d\mu }{2\pi }. \] (\(\overline {\phantom {x}}\) denotes complex conjugation).

Raw LaTeX

Show that Fourier transform
\[
  f(x)\to F(\mu)=\int_{-\infty}^{\infty} f(x)e^{i\mu x}\,dx
\]
preserves the inner product and the \(L_2\) norm (Plancherel's equations):
\[
  \int_{-\infty}^{\infty}\bar{f}(x)g(x)\,dx
  =
  \int_{-\infty}^{\infty}\bar{F}(\mu)G(\mu)\,\frac{d\mu}{2\pi},
\]
\[
  \text{and}\qquad
  \int_{-\infty}^{\infty}\lvert f(x)\rvert^2\,dx
  =
  \int_{-\infty}^{\infty}\lvert F(\mu)\rvert^2\,\frac{d\mu}{2\pi}.
\]
(\(\overline{\phantom{x}}\) denotes complex conjugation).

Topic Tags

Fourier Transform and Sampling Theory Hilbert Spaces and Orthogonality Linear Programming and Optimization Matrix Norms and Induced Norms