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Introduction to Linear Algebra with Mathematica

Glossary

Preface

In opposite to to the Banach space 𝔏¹(ℝ) and Hilbert space 𝔏²(ℝ), the Fourier transform and its inverse are both well-defined in the Schwartz space 𝒮(ℝ) = S(ℝ).

We always use the notation ∫ f for \( \displaystyle \int_{-\infty}^{+\infty} f(x)\.{\text d}x \) throughout this chapter; especially, the limits of integration are always ±∞ if nothing is said to the contrary.

 

Fourier integral in 𝒮

Recall that we denote by 𝒮(ℝ) = S = S(ℝ, ℂ) class of rapidly decreasing complex-valued functions on ℝ that all their derivatives are bounded upon multiplication by any polynomial.

Theorem 1: The Fourier transform in the Schwartz space S(ℝ) \[ ℱ(f) = f^F (\xi ) = \int_{-\infty}^{+\infty} f(x)\, e^{-{\bf j}x\xi} {\text d}x \] possesses the following properties.
  1. The Fourier transform ℱ maps the Schwartz space onto itself.
  2. The inverse transform ℱ−1 does what it should: \[ ℱ^{-1} \left( f^F \right) = f . \]
  3. The Plancherel identity holds: \[ (2\pi ) \{ f \|_2 = \{ f^F \|_2 . \]
  1. For any function f from 𝒮 = S, we have using integration by pars that \[ ℱ_{x\to\xi}\left[ f' \right] = \int f' (x)\,e^{-{\bf j}x\xi} {\text d}x = -\int f(x) \left( e^{-{\bf j}x\xi} \right)' {\text d} x = {\bf j}\xi \int f(x) \, e^{-{\bf j}x\xi} {\text d}x . \] Also, by the rapid decrease of f ∈ 𝒮, \[ ℱ_{x\to\xi}\left[ -{\bf j}x \,f \right] = \frac{\text d}{{\text d}x} \left[ f^F \right] , \] and so, by induction, \[ ℱ_{x\to\xi}\left[ \texttt{D}^p \left( -{\bf j} x \right)^q f \right] = \left( {\bf j} \xi \right)^p \texttt{D}^p f^F , \] for any nonnegative integers p and q. Therefore, \[ \left\vert \xi \right\vert^p \left\vert \texttt{D}^q f^F \right\vert \le \left\| \texttt{D}^p x^q f \right|_1 < \infty . \] Hence, the Fourier transform of f ∈ 𝒮 also belongs to 𝒮.
  2. Now let f be a function with compact support and regard it as an infinitely differentiable function on the circle −T/2 ≤ x ≤ T/2, as is periodic and extended by zero outside this interval [−T/2, T/2]. Then you can express f for |x| < T/2 as a rapidly convergent Fourier series of period T: \begin{align*} f(x) &= \sum_{n=-\infty}^{\infty} e^{{\bf j}nx/T} \frac{1}{T} \int_{-T/2}^{T/2} f(y)\,e^{-{\bf j}ny/T} {\text d}y \\ &= \sum_{n=-\infty}^{\infty} e^{{\bf j}nx/T} \frac{1}{T} \, f^F \left( \frac{n}{T} \right) . \end{align*} But this is just a Riemann sum approximating to the integral \[ ℱ_{y\to x}\left[ f^F (y) \right] = \int_{-\infty}^{\infty} f^F (y) \,e^{{\bf j}yx} {\text d}y . \] In order to prove that \[ ℱ^{-1} \left[ f^F \right] = f , \] for functions with compact support, you have only to check that the sum converges to the integral as T ↑ ∞
  3. Similar reasoning leads to the formula \[ \| f \|_2 = \| f^F \|_2^2 = \int_{-T/2}^{T/2} | f |^2 = \sum_{n=-\infty}^{+\infty| \frac{1}{T} \, \left\vert f^F \left( \frac{n}{T} \right)\right\vert^2 \] from which we derive the Plancherel identity: \[ \| f \|_2 = \frac{1}{2\pi} \,\| f^F \|_2 = \frac{1}{2\pi} \left( \int_{-\infty}^{\infty} | f^F (y) |^2 {\text d}y \right)^{1/2} . ]