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Introduction to Linear Algebra with Mathematica

Glossary

Preface

The practical use of distributions can be traced back to the use of Green's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. Although the English scientist Paul Dirac formaly used generalized functions in his work at the beginning of twentith century, their strict mathematical definition were introduced in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s.

 

Dual Space ๐’ฎ ′

The Schwartz space ๐’ฎ concists of Schwartz functions defined on real axis ℝ that is also called the space of test functions of rapid decreas. It is not a normed space; however, a notion of convergence can be introduced. Then the topology is introduced in ๐’ฎ with the aid of semi-norms.
Let 𝑎, b be non-negative integers. For any function f from the Schwartz space, we define a semi-norm of f by \[ \| f \|_{a,b} = \sup_x \left\{ |x|^a \left\vert \left( \frac{\text d}{{\text d}x} \right)^b f(x) \right\vert \right\} . \]
These measurements are not norms because ∥f∥𝑎,b can be zero without f being 0. If f∥𝑎,0 vanishes for some 𝑎, however, then f is the zero function. Note that we could replace supremum by maximum in the definition of the semi-norm, because functions in ๐’ฎ are continuous and decay rapidly at infinity. The number of semi-norms is countable, and hence we can make ๐’ฎ into a metric space. The distance between two functions is given by the formula
\[ \mbox{dist}(f, g) = \sum_{a,b} c_{a,b} \frac{\| f-g \|_{a,b}}{1 + \| f-g \|_{a,b}} , \]
where c𝑎,b > 0 and is chosen to make the sum converge. For example, c𝑎,b = 2โˆ’𝑎โˆ’b is often used. With this distance function, ๐’ฎ is a complete metric space. It is adequate to state the notion of convergence in terms of the semi-norms, rather than in an equivalent manner using this distance function.
A sequence { fn } converges to f in ๐’ฎ if, for all 𝑎, b, \[ \| f - f_n \|_{a,b} \to 0 . \]

Theorem 1: With the distance function defined above, the Schwartz space ๐’ฎ is becomes a complete metric space (in fact it is a Frรฉchet space).
Since ๐’ฎ is a vector space, it would have sufficed to give the notion of convergence to 0. To say that a sequence { fn } converges to 0 means that, any derivative of any polynomial multiple of fn tends to 0 uniformly.
Let L : ๐’ฎ โ†’ ℂ be a linear functional. Then L is called continuous if, whenever fn converges to f in ๐’ฎ, then L(fn ) converges to L(f) in ℂ.
The dual space ๐’ฎ ′ to ๐’ฎ is the vector space consisting of all continuous linear functionals on ๐’ฎ. Elements of ๐’ฎ′ are called tempered distributions.
It is often convenient to write the action of a linear functional using Dirac's bra-ket notation:
\[ \phi (f) = \left\langle \phi \mid f \right\rangle . \]
Each element g of ๐’ฎ can be regarded as a distribution by the formula
\begin{equation} \label{EqDual.1} g (\psi ) = \left\langle g \mid \psi \right\rangle = \int_{\mathbb{R}} g(x)\,\psi (x)\,{\text d}x . \end{equation}
Every integral in Eq.\eqref{EqDual.1} defines a distribution, so it is an element of ๐’ฎ ′.
\begin{equation} \label{EqDual.2} T_g : ๐’ฎ(\mathbb{R}) \ni \psi \mapsto \int_{-\infty}^{+\infty}g(x)\,\psi (x)\,{\text d}x \in \mathbb{C} . \end{equation}
For example, when g is bounded and continuous, \eqref{EqDual.1} makes sense and defines g as an element of ๐’ฎ ′. When g is any function such that \eqref{EqDual.1} makes sense for all ψ โˆˆ ๐’ฎ, we regard g as the element of ๐’ฎ ′ defined by \eqref{EqDual.1}. Distributions are more general than functions.

Lemma 1: The map \[ ๐’ฎ(\mathbb{R}) \ni g \mapsto T_g \in ๐’ฎ'(\mathbb{R}) \] is an injection (one-to-one).
For any g ∈ ๐’ฎ, we have \[ T_d (g) = \int_{\mathbb{R}} |g(x)|^2 {\text d} x, \] so Tg = 0 implies g ≡ 0.
   

Example 1:    ■

End of Example 1
We define the differentiation of distributions by formula
\[ \left(\texttt{D}g \right) \psi = \int_{-\infty}^{+\infty} \left(\texttt{D}g \right) \psi (x)\,{\text d}x = - \int_{-\infty}^{+\infty} g \left(\texttt{D}\psi (x) \right) = - \int_{-\infty}^{+\infty} g \,\psi' (x)\,{\text d}x . \]

Theorem 2 (Schwartz representation theorem): For any u ∈ ๐’ฎ ′(ℝ) there is a finite collection u𝑎,b : ℝ ↦ ℂ of bounded continuous functions, |𝑎| + |b| ≤ k, such that \[ u = \sum_{|a| + |b| \le k} x^a \texttt{D}^b u_{a,b} \]
If u : ℝ ↦ ℂ is a bounded and continuous function, then Eq.(1) defines a distribution.

For any nonnegative integers 𝑎, b, \( \displaystyle \quad x^a \texttt{D}^b u \in \)๐’ฎ ′(ℝ) if u : ℝ ↦ ℂ is bounded and continuous.

Thus tempered distributions are just products of polynomials and derivatives of bounded continuous functions

Lemma 2: Suppose u ∈ ๐’ฎ ′(ℝ) and x u = 0, then u = cδ for some constant c.
We use the de๏ฌnition of multiplication and a dual result for test functions. Namely, choose ฯ โˆˆ ๐’ฎ(ℝ) with ฯ(x) = 1 in |x| < ½, ฯ(x) = 0 in |x| โ‰ฅ 3/4. Then any g โˆˆ ๐’ฎ(ℝ) can be written \[ g = g(0) \cdot \rho (x) + x\,\psi (x) , \quad \psi \in ๐’ฎ . \tag{L2.1} \] This in turn can be proved using Taylorโ€™s formula \[ g = d(0) + x\,\phi \qquad \mbox{for} \quad |x| \le 1 , \ \ \phi \in V^{\infty} . \] Then, \[ \rho (x) g(x) = g(0)\,\rho (x) + x\,\phi (x) \] and &rho ϕ ∈ ๐’ฎ(ℝ). Thus, it su๏ฌƒces to check (L2.1) for (1 โˆ’ ฯ)g, which vanishes identically near 0. Then \( \displaystyle \quad \phi = |x|^{-2} \left( 1 - \rho \right) g \in \) ๐’ฎ(ℝ) and so \[ \left( 1 - \rho \right) g = |x|^2 \phi = x \left( x\phi \right) , \] which gives (L2.1) with ψ(x) = ρ(x)ϕ(x) + x ϕ(x). Having proved the existence of such a decomposition we see that if x u = 0, then \[ u(g) = u(g(0))\,\rho (x) + u(x\,\psi ) = c\,g(0), \quad c = u(\rho (x)), \] i.e., u = cδ(x).
   

Example 2:    ■

End of Example 2

 

Fourier Transform of distributions

When Fourier transform
\begin{equation} \label{EqDual.3} ℱ[f](\xi ) = \hat{f}(\xi ) = f^F (\xi ) = \int_{-\infty}^{+\infty} e^{-{\bf j}x\xi} f(x)\,{\text d}x \end{equation}
is applied to functionals from ๐’ฎ(ℝ), the integral \eqref{EqDual.3} absolutely converges. Its inverse Fourier transform
\begin{equation} \label{EqDual.4} ℱ^{-1}[\hat{f}](x ) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{{\bf j}x\xi} \hat{f}(\xi )\,{\text d}\xi \end{equation}
also converges absolutely. Both ℱ and ℱ−1 give continuous linear maps
\[ ℱ,\ ℱ^{-1} \,:\ ๐’ฎ(\mathbf{R}) \to ๐’ฎ(\mathbf{R}) . \]
This conclusion follows from inequalities:
\[ \left\vert ℱ[f(x)]_{x\to\xi} \right\vert \le \int_{-\infty}^{+\infty} |f(x)|\,{\text d}x \le \int_{-\infty}^{+\infty} \frac{{\text d}x}{1+ x^2} \times \sup_{x\in\mathbb{R}} \left( 1 + |x|^2 \right) |f(x)| . \]
This inequality shows that sup |ℱ[f]| < ∞ if f ∈ ๐’ฎ(ℝ).

There is no universal notation for complex conjugate numbers (or functions). Mathematicians prefer overline notation: \( \displaystyle \quad \overline{z} = \overline{a + {\bf j}\,b} = a - {\bf j}\,b . \quad \) On the other hand, in physics and engineering complex conjugate is denoted by asterisk: \( \displaystyle \quad z^{\ast} = \left( a + {\bf j}\,b \right)^{\ast} = a - {\bf j}\,b . \quad \) Here j is the unite vector in positive vertical direction on the complex plane ℂ, so j² = −1, A similar descrepansy in notation is observed when inner product is defined. In mathematical literature, an inner product of two vectors is defined as

\[ \left\langle \phi , {\bf u} \right\rangle = \phi \bullet \overline{\bf u} . \]
In physics and engineering, the same definition of inner product is denoted as
\[ \left\langle \phi \mid {\bf u} \right\rangle = \phi^{\ast} \bullet {\bf u} . \]
We mostly follow physics' notation because it is impossible to please both parties. When linear functional ϕ is defined via integral, the inner product becomes (in one-dimensional case)
\[ \left\langle \phi \mid u \right\rangle = \int_{-\infty}^{+\infty} \phi^{\ast} (x) \,u(x)\,{\text d} x . \]
Let ฯ† โˆˆ ๐’ฎ ′(ℝ). We define its Fourier transform ℱ(ฯ†) by duality as follows. For each f โˆˆ ๐’ฎ(ℝ) we decree that \[ \left\langle ℱ(\phi ) \mid f \right\rangle = \left\langle ℱ(\phi ) \mid ℱ(f) \right\rangle \qquad\mbox{or} \qquad \left\langle \hat{\phi} \mid f \right\rangle = \left\langle \phi \mid \hat{f} \right\rangle . \]
   

Example 3:    ■

End of Example 3

Theorem 3: The Fourier transform extends by continuity to an isomorphism \[ ℱ \]
   

Example 4:    ■

End of Example 4
   

 

 

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