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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Let ℝ be the real line parameterized by x We consider complex-valued functions on ℝ that is integrable in Lebesue sence. The Fourier transform of a function f is denoted by
\begin{equation} \label{EqFunction.1} ℱ_{x\to \xi}\left[ f(x) \right] (\xi ) = \hat{f} (\xi ) = f^F (\xi ) = \int_{-\infty}^{+\infty} f(x) \,e^{-{\bf j}x\cdot\xi} {\text d} x , \end{equation}
where j is the imaginary unit on the complex plane ℂ, so j² = −1. This integral \eqref{EqFunction.1} is a functional on a space of functions (called dual) parameterized by ξ. The goal is to show that f has a representation as an inverse Fourier transform
\begin{equation} \label{EqFunction.2} f(x) = ℱ_{\xi\to x}^{-1} \left[ \hat{f}(\xi ) \right] (x ) = \mbox{V.P.} \frac{1}{2\pi} \int_{-\infty}^{+\infty} \hat{f}(\xi ) \,e^{{\bf j}x\cdot\xi} {\text d} \xi , \end{equation}
where «V.P.» is the abbreviation for the Cauchy principal value regularization ("valeur principale" in French).

There are two problems. One is to interpret the sense in which these integrals converge. The second is to show that the inversion formula actually holds. <

 

 

Functions

Before the 19th century, “function" in mathematical science and its applications understood as "formula”. From the Bernoullis, Euler, and Lagrange, a function was essentially:
  • an analytic expression,
  • built from algebraic operations, exponentials, logarithms, trig functions, etc.
This excluded:
  • piecewise definitions,
  • discontinuous functions,
  • arbitrary assignments.
So the concept was intuitive but not rigorous. Joseph Fourier analysis forced mathematicians to abandon the classical “function = formula” view and adopt the modern “function = assignment rDirichlet stated (paraphrasing his German text):
A function on an interval is a rule that assigns to each a single real number , without requiring any analytic expression.
This was the first time a function was defined:
  • independently of formulas,
  • allowing arbitrary assignments,
  • allowing discontinuities,
  • allowing pathological examples.
Dirichlet gave a famous example of the indicator function of the rational numbers:
\[ f(x) = \begin{cases} 1, &\quad\mbox{for rational } x, \\ 0, &\quad\mbox{for irrational } x. \end{cases} \]
This example would have been unthinkable under Euler’s or Lagrange’s definition.

Later, other mathematicians reinforced Dirichlet's viewpoint.

  • Weierstrass (1860s--1870s) pushed generality further.
  • Cantor (1879s--1890s) provided a set-theoretical foundation.
  • Lebesgue (1902) extended the definition for measure functions.

 

Space of bounded functions

The set of all real or complex numbers is denoted by ℝ or ℂ. Since these sets are unbounded, we consider a special class of functions.

A function f defined on some set X with real or complex values is called bounded if the set of its values (its image) is bounded. In other words, there exists a real number M such that \[ | f(x) | \le M, \qquad \forall x \in X. \] A function that is not bounded is said to be unbounded.

If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ B for all x in X, then the function is said to be bounded (from) below by B. A real-valued function is bounded if and only if it is bounded from above and below.

The space of bounded almost everywhere (a.e.) functions, often denoted as 𝔏∞(Ω), consists of measurable functions defined on a measure space Ω that are bounded outside a set of measure zero. The relevant norm is the essential supremum, \[ \| f \|_{\infty} = \mbox{ess}\,\sup |f| = \inf \left\{ M > 0\ : \ \mu\left( \{ x\ : \ |f(x) | > M \} \right) = 0 \right\} . \]

The Banach space 𝔏∞ consists of equivalence classes of essentially bounded measurable functions.

 

 

Absolutely Integrable Functions

The space of lebesque integrable functions 𝔏¹(ℝ) is a Banach space with norm Its dual space is 𝔏∞, the space of essentially bounded functions. The Fourier transform \eqref{EqFunction.1} defines a linear functional that assigns to a bounded function e−ξx ∈ 𝔏∞ a function that belongs to 𝔏¹(ℝ).

The continuity of the Fourier teansform fF follows from the dominated convergence theorem. Unlike on the torus 𝕋, 𝔏¹(ℝ) does not contain 𝔏²(ℝ), so formula \eqref{EqFunction.2} does not recover f from its Fourier transform: we simply don't know whether fF is integrable---in fact, it is generally not integrable.

 

Square Integrable Functions

Let (X, μ) be a measure space and let p ∈ [1, ∞). An 𝔏p function on X is a measurable function f : X → ℝ for which 𝔏p norm is finite: \[ \| f \|_p = \left( \int_X | f |^p \,{\text d}\mu \right)^{1/p} < \infty . \]

We leave definition of the root in the definition of 𝔏p norm to mathematicians because in this tutirial we will use only three cases: p = 1, 2, ∞. It is convenient to use 𝔏p notation without repeating statements for these three cases. Note that the 𝔏p-norm of a function f may ne either finite or infinite. The 𝔏p functions are those for which the p-norm is finite.

Like any measurable function, 𝔏p function is allowed to take values of ±∞. However, it takes finite values almost everywhere. It is well-known (from any course in functional analysis) that 𝔏p(X) is a Banach space for any p ∈ [1, ∞]. Moreover, for any 𝔏p functions f and g, the Minkowski inequality holds:

\[ \| f + g \|_p \le \| f \|_p + \| g \|_p < \infty . \]
   
Example 1: Any bounded function on [0, 1] is automatically 𝔏p for every value of p. However, it is possible for the p-norm of a measurable function on [0, 1] to be infinite. For example, let f : [0, 1] → ℝ be the function \[ f(x) = \frac{1}{x} , \] where the value of f(0) is immaterial. Then, by the monotone convergence theorem \[ \int_0^1 | f |\,{\text d}\mu = \lim_{a\downarrow 0} \,\int_a^1 \frac{1}{x}\,{\text d}x = - \lim_{a \downarrow 0} \,\ln a = \infty , \] so f is not 𝔏¹. Indeed, f is not 𝔏p is not 𝔏p for any p ∈ [1, ∞).

A function with vertical asymptote does not automatically have infinite p-norm. For instance, function \[ f(x) = x^{-1/2} \, = \frac{1}{\sqrt{x}} \] has a vertical asymptote at x = 0, but its 1-norm \[ \int_0^1 |f|\,{\text d} x = \lim_{a\downarrow 0} \,\int_a^1 \frac{{\text d}x}{\sqrt{x}} = \lim_{a\downarrow 0} \,\left[ 2\,\sqrt{x} \right]_{x=a}^{x=1} = 2 . \] In general, \[ \int_0^1 \frac{{\text d}x}{x^r} = \,\begin{cases} \infty , &\quad \mbox{if }\ r\ge 1 , \\ 1/(1-r) , &\quad \mbox{if }\ f < 1 . \end{cases} \] It follows that function f(x) = 1/xr is 𝔏p if and only if p r < 1, i.e., if and only if p < 1/r.    ■

End of Example 1
Theorem 1: Let (X, μ) be a measure space, and let 1 ≤ p ≤ q < ∞. If μ(X) = 1, then \[ \| f \|_p \le \| f \|_q \] for every measurable function f. More generally, if 0 < μ(X) < ∞, then \[ \| f \|_p \le \mu (X)^r\, \| f \|_q \] for every measurable function f, where r = (1/p) − (1/q), and hence every 𝔏q function is also 𝔏p.
The case where µ(X) = 1 is the generalized mean inequality for the p-mean and the q-mean. For 0 < μ(X) < ∞, let C = μ(X) and let ν be the measure \[ {\text d}\nu = \frac{1}{C}\,{\text d}\mu . \] Then ν(X) = 1, so by the generalized mean inequality \begin{align*} \left( \int_X | f |^p \,{\text d}\mu \right)^{1/p} &= C^{1/p} \left( \int_X | f |^p \,{\text d}\nu \right)^{1/p} \\ &\le C^{1/p} \left( \int_X | f |^q \,{\text d}\nu \right)^{1/q} \\ &= C^{1/p} C^{-1/q} \left( \int_X | f |^q \,{\text d}\nu \right)^{1/q} . \end{align*}    ▣
Note that this proposition only applies in the case where µ(X) is finite. As the following example shows, the relationship between 𝔏p and 𝔏q functions can be more complicated when μ(X) = ∞.

https://e.math.cornell.edu/people/belk/measuretheory/LpFunctions.pdf

Since 𝔏²(ℝ) is a Hilbert space, it provides the most elegant and simple theory of the Fourier transform.

Theorem 4: If f ∈ 𝔏¹(ℝ) ∩ 𝔏²(ℝ), then its Fourier transform fF ∈ 𝔏²(ℝ) and
\[ \| \hat{f} \|_2^2 = \| f \|_2^2 . \]

Theorem 5: Let f ∈ 𝔏²(ℝ). For every positive number 𝑎, let \( f_a = \chi_{[-a, a]} f \) be its product with characteristic function of the interval [−𝑎, 𝑎]. Then f𝑎 is in 𝔏¹(ℝ) ∩ 𝔏²(ℝ) and f𝑎 → f in 𝔏²(ℝ) as 𝑎 → ∞. Furthermore, there exists \( \hat{f} \) from 𝔏²(ℝ) such that \( \hat{f}_a \to \hat{f} \) as 𝑎 → ∞.
The map from a function into its Fourier transform gives a continuous map from 𝔏¹(ℝ) to the part of ℭ0(ℝ). That is, the Fourier transform of an integrable function is continuous and bounded (this is obvious) and approach zero at infinity. Furthermore, this map is one-to-one.

The inverse Fourier transform gives a continuous map from 𝔏¹(ℝ) to ℭ0(ℝ). This is also a one-to-one transformation

Theorem 6: If f ∈ 𝔏¹(ℝ) ∩ 𝔏²(ℝ) and its derivative f' = df/dx exists (in the sense that f is an integral of its derivative) and if f' is also in 𝔏²(ℝ), then the Fourier transform of df/dx is in 𝔏¹(ℝ). As a consequence, f is in ℭ0(ℝ).
\[ \| \hat{f} \|_2^2 = \| f \|_2^2 . \]

 

Space of continuous functions

 

Schwartz Space

The following function space is named after French mathematician Laurent Schwartz (1915--2002). This space, denoted by 𝒮 or S, has the important property that the Fourier transform is an automorphism on this space.

 

Let ℕ = {0, 1, 2, … ) be the set of non-negative integers. The Schwartz space or space of rapidly decreasing functions on ℝ is the function space \[ 𝒮(ℝ) = S(\mathbb{R}) = \left\{ f \in C^{\infty}(\mathbb{R}, \mathbb{C} \, : \ \| f \|_{\alpha , \beta} < \infty , \quad \forall \alpha , \beta \in \mathbb{N} \right\} , \] where C∞(ℝ, ℂ) is the function space of smooth functions from ℝ into ℂ, and \[ \| f \|_{\alpha , \beta} = \sup_{x\in\mathbb{R}} \left\vert x^{\alpha} \texttt{D}^{\beta} f(x) \right\vert . \] Here, sup denotes the \( \displaystyle \texttt{D} = {\text d}/{\text d}x \) is the derivative operator. This spaceis usually denoted as 𝒮(ℝ) or S(ℝ) or just 𝒮.

 

\[ f(r) = \int_0^{\infty} F_{\nu} (k)\,J_{\nu} (kr)\,k\,{\text d}k . \]
For Hankel transformations, we have
\[ \int_0^{\infty} r\left( \frac{{\text d}^2 f}{{\text d} r^2} + \frac{1}{r} \, \frac{{\text d} f}{{\text d} r} - \frac{\nu^2}{r^2} \, f \right) J_{\nu} (rk)\,{\text d}r = - k^2 F_{\nu} (k) = - k^2 \int_0^{\infty} f(r)\,J_{\nu} (kr)\,r\,{\text d}r . \]
■

 

  1. Dirichlet, P. G. L. (1837). "Ueber die Darstellung ganz willkürlicher Functionen durch Sinus- und Cosinusreihen" [On the Representation of Entirely Arbitrary Functions by Sine- and Cosine-Series], Abhandlungen der Königlichen Preußischen Akademie der Wissenschaften zu Berlin [Transactions of the Royal Prussian Academy of Sciences of Berlin], 48: 45–71.

 

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