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Introduction to Linear Algebra with Mathematica

Glossary

Preface

The Schwartz space ๐’ฎ(ℝ) was introduced by the French mathematician Laurent Schwartz (1915--2002), a different person from a German mathematician Hermann Schwarz, whose name is associated with the Cauchy-Bunyakovsky-Schwarz inequality. Actually, -Schwarz's contribution to this inequality is quite subtle---his 1888 paper contains no new information except excellent translation into German.

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 ๐’ฎ(ℂ) = S(ℂ). 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 ๐’ฎ

The Schwartz space ๐’ฎ (or S) consists of those infinitely differentiable complex-valued functions f ∈ ๐’ฎ on ℝ such that, for all non-negative integers 𝑎, b, \[ \lim_{|x| \to \infty} |x|^a \left( \frac{\text d}{{\text d}x} \right)^b f(x) = 0 . \]
Functions in the Schwartz space decay so rapidly at infinity that, even after differentiating or multiplying by x an arbitrary (finite) number of times, the resulting functions still decay at infinity. Therefore, ๐’ฎ(ℝ) = S = S(ℝ, ℂ) is called the class of rapidly decreasing complex-valued functions on ℝ because all their derivatives are bounded upon multiplication by any polynomial. For any ϵ > 0, the Gaussian \( \displaystyle \quad e^{- \epsilon x^2} \quad \) is in ๐’ฎ. Smooth functions of compact support provide additional examples. For convenience, the existence of such functions is presented below.    

Example 1: First we define a function h on ℝ by \[ h(t) = \begin{cases} 0, & \quad\mbox{for} t \le 9 , \\ e^{-1/t} &\quad\mbox{for} t > 0 . \end{cases} \] This function is infinitely differentiable on all of ℝ, and all of its derivatives vanish at 0. Put g(t) = h(t) h(1 โˆ’ t). Then g is also infinitely differentiable. Furthermore, g(t) > 0 for 0 < t < 1 and g(t) = 0 otherwise. Thus, function g is a smooth function with compact support.    ■

End of Example 1
This example can be extended:

Proposition 1: Let I denote any closed bounded interval on ℝ and let J denote any open interval containing I. Then there is an infinitely differentiable function g : ℝ โ†’ [0, 1] such that g = 1 on I and g = 0 off J.

Theorem 1: The Schwartz space ๐’ฎ is a complex vector space. It is closed under differentiation and under multiplication by x.
Left to the reader.
The Fourier transform of a function f ∈ ๐’ฎ is defined as the improper integral: \begin{equation} \label{EqSchwartz.1} ℱ(f) = f^F (\xi ) = \int_{-\infty}^{+\infty} f(x)\, e^{-{\bf j}x\xi} {\text d}x , \end{equation} where j is the imaginary unit on the complex plane ℂ so j² = −1.
Sometimes the Fourier transform is defined with the factor โˆš2ฯ€:
\[ ℱ(f) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x)\, e^{-{\bf j}x\xi} {\text d}x , \]
and sometimes it is defined with a factor of 2ฯ€ in the exponent:
\[ ℱ(f) = \int_{-\infty}^{+\infty} f(x)\, e^{-{\bf j}2\pi\,x\xi} {\text d}x . \]
Some authors from    West coast    define the Fourier transformation with oposite sign in the exponent:
\[ ℱ(f) = f^F (\xi ) = \int_{-\infty}^{+\infty} f(x)\, e^{{\bf j}x\xi} {\text d}x . \]
The inverse Fourier transform of a function fF ∈ ๐’ฎ is \begin{equation} \label{EqSchwartz.2} ℱ^{-1} \left[ f^F (\xi ) \right] (x) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} f^F (\xi )\, e^{{\bf j}x\xi} {\text d}\xi = \frac{1}{2\pi}\,\lim_{N,M\to +\infty} \int_{-M}^{N} f^F (\xi )\, e^{{\bf j}x\xi} {\text d}\xi . \end{equation}
The Wolfram Mathematica web page on the Fourier transform defines this transformation using two parameters, and symemtric definition of the Fourier transform:
\[ ℱ \left[ f (x) \right] (\xi ) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f (x )\, e^{-{\bf j}x\xi} {\text d}x . \]
FourierTransform[Exp[-t^2]*Sin[t], t, s]
Mathematica also has a numerical approximation to the Fourier transform with build-in command: NFourierTransform. The inverse Fourier transform is also implemented in Wolfram language:
InverseFourierTransform[Exp[-Abs[s]]*Sin[s], s, t]
   

Example 2:    ■

End of Example 2

Observation 1: The Fourier transform is a linear transformation in the Schwartz space, ℱ : ๐’ฎ(ℝ) ⇾ ๐’ฎ(ℂ) and ℱ : ๐’ฎ(ℂ) ⇾ ๐’ฎ(ℂ).

The definition of ๐’ฎ regards differentiation and multiplication on an equal footing. Let \( \displaystyle \quad \mathtt{D} = {\text d}/{\text d}x \quad \) denote differentiation and let M = Mjฮพ denote multiplication operator by jξ. Working on ๐’ฎ is convenient for several reasons; in particular, the Fourier transform exchanges these operations. Furthermore, as we will show in the following Theorem, the Fourier transform maps ๐’ฎ to itself bijectively. We can interpret the last two items of the following Proposition as saying that the Fourier transform gives the spectral decomposition for the derivative operator.

Theorem 2: The Fourier transform in the Schwartz space ๐’ฎ \[ ℱ(f) = f^F (\xi ) = \hat{f} (\xi ) = \int_{-\infty}^{+\infty} f(x)\, e^{-{\bf j}x\xi} {\text d}x \] 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. \( \displaystyle \quad \hat{f}_h (\xi ) = e^{{\bf j}h\xi} \hat{f} (\xi ) , \quad \) where fh(x) = f(x + h).
  4. \( \displaystyle \quad \texttt{D}\,\hat{f}(\xi ) = \frac{\text d}{{\text d}\xi}\,\hat{f}(\xi ) = - {\bf j}\,ℱ\left( M_x f \right) , \quad \) taht is, \( \displaystyle \quad \texttt{D}_{{\bf j}\xi} \,ℱ = ℱ\,M_x . \)
  5. \( \displaystyle \quad ℱ\,\texttt{D}_x = M_{{\bf j}\xi}ℱ. \)
  6. \( \displaystyle \quad \texttt{D} = ℱ^{-1} M \,ℱ \quad \) and \( \displaystyle \quad M = ℱ\,\texttt{D}\,ℱ^{-1} . \)
  1. For any function f from ๐’ฎ = 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 ∈ ๐’ฎ, \[ ℱ_{x\to\xi}\left[ -{\bf j}x \,f \right] = \frac{\text d}{{\text d}x} \left[ f^F \right] , \] 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 ∈ ๐’ฎ also belongs to ๐’ฎ.
  2. Now let f be a function with compact support and regard it as an in๏ฌnitely 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: \[ \| f \|_2 = \frac{1}{2\pi} \,\| f^F \|_2 = \frac{1}{2\pi} \left( \int_{-\infty}^{\infty} | f^F (y) |^2 {\text d}y \right)^{1/2} . \]
   

Example 3:    ■

End of Example 3

 

 

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