11.2. Distributions: more

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11.2. Distributions: more

Supports

Definition 1. Let us consider ordinary function $f$. Observe that if $f=0$ on open sets $\Omega_\iota$ (where $\iota$ runs any set of indices––finite, infinite or even non-enumerable) then $f=0$ on $\bigcup_\iota \Omega_\iota$. Therefore there exists a largest open set $\Omega$ such that $f=0$ on $\Omega$. Complement to this set is called support of $f$ and denoted as $\supp(f)$.

Definition 2.

1. Let us consider distribution $f$. We say that $f=0$ on open set $Omega$ if $f(\varphi)=0$ for any test function $\varphi$ such that $\supp \varphi \subset \Omega$.
2. Then the same observation as in (a) holds and therefore there exists a largest open set $\Omega$ such that $f=0$ on $\Omega$. Complement to this set is called support of $f$ and denoted as $\supp(f)$.

Definition 3. Observe that $\supp(f)$ is always a closed set. If it is also bounded we say that $f$ has a compact support.

Exercise 1.

1. Prove that for two functions $f,g$ and for $f\in \mathcal{D}'$, $g\in \mathcal{E}$ \begin{gather} \supp (gf)\subset \supp(f) \cap \supp(g),\label{eq-11.2.1}\\ \supp (\partial f)\subset \supp(f)\label{eq-11.2.2} \end{gather} where $\partial$ is a differentiation;
2. Prove that $\supp(f)=\emptyset$ iff $f=0$ identiacally;
3. Prove that $\supp(\delta_a)=\{a\}$. Prove that the same is true for any of its derivatives.

Remark 1. In fact, the converse to Exercise 1(c) is also true: if $\supp(f)={a}$ then $f$ is a linear combination of $\delta (x-a)$ and its derivatives (up to some order).

Remark 2. In the previous section we introduced spaces of test functions $\mathcal{D}$ and $\mathcal{E}$ and the corresponding spaces of distributions $\mathcal{D}'$ and $\mathcal{E}'$. However for domain $\Omega\subset \mathbb{R}^d$ one can introduce $\mathcal{D}(\Omega):= \{ \varphi \in \mathcal{D}:\, \supp \varphi \subset \Omega\}$ and $\mathcal{E}=C^\infty (\Omega)$. Therefore one can introduce corresponding spaces of distributions $\mathcal{D}'(\Omega)$ and $\mathcal{E}'(\Omega)=\{ f\in \mathcal{E}:\, \supp f\subset \Omega\}$. As $\Omega=\mathbb{R}^d$ we get our "old spaces".

Non-linear change of variables

Definition 4. Let $f$ be a distribution with $\supp f\subset \Omega_1$ and let $\Phi:\Omega_1\to \Omega_2$ be one-to-one correspondence, infinitely smooth and with non-vanishing Jacobian $\det \Phi'$. Then $\Phi_* f$ is a distribution: \begin{equation} (\Phi_* f)(\varphi) = f( |\det \Phi'| \cdot \Phi^*\varphi ) \label{eq-11.2.3} \end{equation} where $(\Phi^*\varphi)(x)=\varphi(\Phi(x))$.

Remark 3.

1. This definition generalizes Definition 11.1.6 and Definition 11.1.7.
2. Mathematicians call $\Phi^*\varphi$ pullback of $\varphi$ and $\Phi_*f$ pushforward of $f$.

Exercise 2. Check that for ordinary function $f$ we get $(\Phi_*f)(x)=f (\Phi^{-1}(x))$.

Fourier transform

Definition 5. Let $f\in \mathcal{S}'$. Then Fourier transform $\hat{f}\in \mathcal{S}'$ is defined as \begin{equation} \hat{f}(\varphi) = f(\hat{\varphi}) \label{eq-11.2.4} \end{equation} for $\varphi \in \mathcal{S}$. Similarly, inverse Fourier transform $\check{f}\in \mathcal{S}'$ is defined as \begin{equation} \check{f}(\varphi) = f(\check{\varphi}) \label{eq-11.2.5} \end{equation}

Exercise 3.

1. Check that for ordinary function $f$ we get a standard definition of $\hat{f}$ and $\check{f}$.
2. To justify Definition 5 one need to prove that $f\in \mathcal{S}\iff \hat{f}\in \mathcal{S}$. Do it!
3. Prove that for $f\in \mathcal{E}'$ both $\hat{f}$ and $\check{f}$ are ordinary smooth functions \begin{gather} \hat{f}(k) = (2\pi)^{-d} f(e^{-ix\cdot k}), \label{eq-11.2.6}\\ \check{f}(k) = f(e^{ix\cdot k}).\label{eq-11.2.7} \end{gather}
4. Check that all properties of Fourier transform (excluding with norms and inner products which may not exist are preserved.

Exercise 4.

1. Prove that Fourier transforms of $\delta (x-a)$ is $(2\pi)^{-d}e^{-ix\cdot a}$.
2. Prove that Fourier transforms of $e^{ix\cdot a}$ is $\delta (x-a)$.

Exercise 5. In dimension $1$

1. Prove that Fourier transforms of $\theta(x-a)$ and $\theta(-x+a)$ are respectively $(2\pi i)^{-1} (k-a-i0)^{-1}$ and $-(2\pi i)^{-1} (k-a+i0)^{-1}$ which are understood as limits in the sense of distributions of $(2\pi i)^{-1}(k-a\mp i\varepsilon)^{-1}$ as $\varepsilon\to+0$. Recall that $\theta(x)$ is a Heaviside function.
2. As a corollary conclude that Fourier transform of $\operatorname{sgn}(x):=\theta(x)-\theta(-x)=x/|x|$ is $(2\pi i)^{-1} \bigl((k-a-i0)^{-1}+ (k-a+i0)\bigr)^{-1}= \pi^{-1}(k-a)^{-1}$ with the latter understood in as principal value (see Exercise 11.1.4(f)).
3. As another corollary conclude that Fourier transform of $\theta(x)+\theta(-x)=1$ is $(2\pi i)^{-1} \bigl((k-a-i0)^{-1}- (k-a+i0)\bigr)^{-1}$ and therefore \begin{equation} (2\pi i)^{-1} \bigl((k-a-i0)^{-1}- (k-a+i0)\bigr)^{-1}=\delta(x-a). \label{eq-11.2.8} \end{equation}

Convolution

Recall convolution (see Definition 5.2.1) and its connection to Fourier transform.

Definition 6. Let $f,g\in \mathcal{D}'$ (or other way around), $\varphi\in \mathcal{D}$ Then we can introduce $h(y) \in \mathcal{E}$ as \begin{equation*} h(y) = g ( T_y\varphi ),\qquad T_y \varphi(x):= \varphi (x-y). \end{equation*} Observe that $h \in \mathcal{D}$ provided $g\in \mathcal{E}'$. In this case we can introduce $h \in \mathcal{E}$ for $\varphi \in \mathcal{E}$.

Therefore if either $f\in \mathcal{E}'$ or $g\in \mathcal{E}'$ we introduce $f*g$ as \begin{equation*} (f*g)(\varphi) = f ( h ). \end{equation*}

Exercise 6.

1. Check that for ordinary function $f$ we get a standard definition of the convolution;
2. Prove that convolution convolution has the same properties as multiplication;
3. Prove that Theorem 5.2.4 holds;
4. Prove that $f*\delta=\delta*f=f$;
5. Prove that $\partial (f*g)=(\partial f)*g = f*(\partial g)$;
6. Prove that $T_a (f*g)=(T_a f)*g = f*(T_a g)$ for operator of shift $T_a$;
7. Prove that $\supp (f*g) \subset \supp(f)+\supp(g)$ where arithmetic sum of two sets is defined as $A+B:=\{x+y:\, x\in A,\, y\in B\}$.

Remark 4.

1. One can prove that if a linear map $L:\mathcal{E}'\to \mathcal{D}'$ commutes with all shifts: $T_a (L f )=L(T_a f)$ for all $f\in \mathcal{E}'$ then there exists $g\in \mathcal{D}'$ such that $L$ is an operator of convolution: $Lf= g*f$;
2. One can extend convolution if none of $f,g$ has a compact support but some other assumption is fulfilled. For example, in one–dimensional case we can assume that either $\supp(f)\subset [a,\infty)$, $\supp(g)\subset [a,\infty)$ or that $\supp(f)\subset (-\infty,a]$, $\supp(g)\subset (-\infty,a]$.

Similarly in multidimensional case we can assume that $\supp(f)\subset C$, $\supp(g)\subset C$ where $C$ is a cone with angle $<\pi$ at its vertex $a$.

Fourier series

Definition 7.

1. We call one-dimensional distribution $f$ periodic with period $L$ if $f(x-L)=f(x)$.
2. More generally, let $\Gamma$ be a lattice of periods (see Definition 4.B.1). We call distribution $f$ $\Gamma$-periodic if $f(x-n)=f(x)$ for all $n\in \Gamma$.

Periodic distributions could be decomposed into Fourier series: in one-dimensional case we have \begin{equation} f= \sum_{-\infty< m<\infty} c_n e^{\frac{2\pi imx}{L} } \label{eq-11.2.9} \end{equation} and in multidimensional case \begin{equation} f= \sum_{ m\in \Gamma^*} c_m e^{im\cdot } \label{eq-11.2.10} \end{equation} where $\Gamma^*$ is a dual lattice (see Definition 4.B.3).

To define coefficients $c_m$ we cannot use ordinary formulae since integral over period (or elementary cell, again see the same definition) is not defined properly. Instead we claim that there exists $\varphi\in \mathcal{D}$ such that \begin{equation} \sum_{n\in \Gamma} \varphi(x-n) =1. \label{eq-11.2.11} \end{equation} Indeed, let $\psi \in \mathcal{D}$ be non-negative and equal $1$ in some elementary cell. Then $\varphi (x)= \psi (x)/ \bigl(\sum_{n\in \Gamma} \psi(x-n)\bigr)$ is an appropriate function.

Then \begin{equation} c_m= |\Omega|^{-1} (\varphi f)( e^{-im\cdot x}) \label{eq-11.2.12} \end{equation} where $|\Omega|$ is a volume of the elementary cell.

Exercise 7.

1. Find decomposition in Fourier series of one-dimensional distribution $f=\sum_{-\infty< n<\infty} \delta (x-n L)$;
2. Find Fourier transform of $f$ defined in (a);
3. Find the connection to Poisson summation formula (see Theorem 5.2.5).
4. Find decomposition in Fourier series of $d$-dimensional distribution $f=\sum_{n\in \Gamma } \delta (x-n)$;
5. Find Fourier transform of $f$ defined in (d);
6. Find the connection to multidimensional Poisson summation formula (see Remark 5.2A.3).