Derivatives of distributions

Let $f \in \mathcal{D}^{\prime }(\mathbb {R})$ and $f(x+a) = f(x)$ for all $a \in \mathbb {R}$. Prove that $f =$ const, i.e., there exists $C \in \mathbb {C}$ such that $\left\langle f, \varphi \right\rangle = C \int_{\mathbb {R}} \varphi (x) d x, \varphi \in \mathcal{D}(\mathbb {R})$.

Prove that the equality $\left|\sin x\right|^{\prime \prime }+\left|\sin x\right| = 2 \sum_{k \in \mathbb {Z}} \delta_{k \pi }$ holds in $\mathcal{D}^{\prime }(\mathbb {R})$.

Let $\theta \left(x_{1}, \ldots , x_{m}\right) = \theta \left(x_{1}\right) \theta \left(x_{2}\right) \ldots \theta \left(x_{m}\right)$, where $\theta \left(x_{i}\right)$ is the Heaviside function. Prove that in the space $\mathcal{D}^{\prime }\left(\mathbb {R}^{m}\right)$ the following equality holds:

where $x = \left(x_{1}, \ldots , x_{m}\right)$.

Find the derivatives in $\mathcal{D}^{\prime }(\mathbb {R})$ of the functions

Prove that for any $f \in \mathcal{D}^{\prime }(\mathbb {R}), a \in \mathbb {R}$ and $n \in \mathbb {N}$ the following equality is true: $(f(x+a))^{(n)} = f^{(n)}(x+a)$.

Calculate the derivatives in $\mathcal{D}^{\prime }(\mathbb {R})$ for $n \geq 1$ :

Find the derivatives of the first three orders in $\mathcal{D}^{\prime }(\mathbb {R})$ of the following functions:

Prove that

Prove that in $\mathcal{D}^{\prime }(\mathbb {R})$ the following equalities hold:

Let $\alpha \in C^{\infty }(\mathbb {R})$ and $f \in \mathcal{D}^{\prime }(\mathbb {R})$. Prove that $(\alpha f)^{\prime } = \alpha^{\prime } f+\alpha f^{\prime }$. Calculate $(\alpha \theta )^{\prime }$, where $\theta$ is the Heaviside function.

Prove in $\mathcal{D}^{\prime }(\mathbb {R})$ the following equalities

Prove in $\mathcal{D}^{\prime }(\mathbb {R})$ the equality $\left|\cos x\right|^{\prime \prime }+\left|\cos x\right| = 2 \sum_{k \in \mathbb {Z}} \delta_{\left(k+\frac{1}{2}\right) \pi }$.

Prove that if a function $f$ is absolutely continuous on $\mathbb {R}$, then $\left(F_{f}\right)^{\prime } = F_{f^{\prime }}$. Prove the Leibniz rule: if $f \in \mathcal{E}$, and $F \in \mathcal{D}^{\prime }$, then $(f F)^{\prime } = f^{\prime } F+f F^{\prime }$.

Let $\left\{ x_{k}\right\}_{k \in \mathcal{K}}$ be a sequence of real numbers, numbered in increasing order, $\mathcal{K} = \left\{ 1, \ldots , n\right\}$, or $\mathcal{K} = \mathbb {N}$, where the sequence $\left\{ x_{k}\right\}_{k = 1}^{\infty }$ has no finite accumulation points. Let the function $f$ be absolutely continuous on every interval $\left(x_{k}, x_{k+1}\right)$ and have jump discontinuities at the points $x_{k}$, $h_{k}: = f\left(x_{k}+0\right)-f\left(x_{k}-0\right)$. Prove that $f_{\text{gen.}}^{\prime } = f_{\text{classical}}^{\prime }+\sum_{k \in \mathcal{K}} h_{k} \delta_{x_{k}}$.

Compute the derivative of the following generalized functions:

Compute the second derivative of the following generalized functions:

Compute the $n$-th derivative of $x_{+}^{\lambda }$ for $-1 < \operatorname {Re} \lambda < 0$ (for the definition of $x_{+}^{\lambda }$, see problem 16.8).

Prove that the support of the derivative of any generalized function $F$ is a subset of the support of $F$.

Prove that every generalized function $G$ has an antiderivative, that is, a generalized function $F$ such that $F^{\prime } = G$. Consider the cases:

Let $f : \Omega \to \mathbb {F}$ be a function with continuous partial derivatives and let $L_f : \mathscr {D}(\Omega ) \to \mathbb {F}$ be the distribution $L_f(\phi ) = \int f \phi \, d\lambda$ (Example 5.20). Show that for every $\phi$ in $\mathscr {D}(\Omega )$ and $1 \le j \le d$, $L_f(\partial \phi / \partial x_j) = -L_g(\phi )$, where $g = \partial f / \partial x_j$.

Exercise 12 motivates the following definition. If $L$ is a distribution on $\Omega$, define $\partial L / \partial x_j : \mathscr {D}(\Omega ) \to \mathbb {F}$ by $\partial L / \partial x_j(\phi ) = -L(\partial \phi / \partial x_j)$ for all $\phi$ in $\mathscr {D}(\Omega )$. Show that $\partial L / \partial x_j$ is a distribution.

Using the fact that a locally integrable function $f$ on $\mathbb {R}$ defines a distribution $L_f(\phi ) = \int f\phi \, d\lambda$ (Example 5.20), and that the derivative of a distribution $L$ is the distribution $\partial L/\partial x(\phi ) = -L(\phi ')$ (Exercise 13), one is justified to talk of the derivative of any locally integrable function as a distribution. By Exercise 11 we can differentiate measures. Let $f : \mathbb {R} \to \mathbb {R}$ be the characteristic function of $[0, \infty )$ and show that its derivative as a distribution is $\delta_0$, the unit point mass at $0$. [That is, $\delta_0$ is the measure such that $\delta_0(\Delta ) = 1$ if $0 \in \Delta$ and $\delta_0(\Delta ) = 0$ if $0 \notin \Delta$.]

If a function $f: R \rightarrow C$ is continuously differentiable, then for the generalized function generated by its derivative $f^{\prime }$ the relation $\left(f^{\prime }, \varphi \right) = -\left(f, \varphi^{\prime }\right), \quad \varphi \in D$, holds.

The derivative $f^{\prime }$ of a generalized function $f \in D^{\prime }$ is also a generalized function in $D^{\prime }$.

The derivative of any order of a generalized function from the space $D^{\prime }$ is a generalized function in the same space.

If $f$ and $g$ are generalized functions, and $\lambda , \mu \in C$, then $(\lambda f + \mu g)^{\prime } = \lambda f^{\prime }+\mu g^{\prime }$.

Compute the derivative of the generalized function. $y = \theta \left(x-x_{0}\right) = \begin{cases} 1 & \text{ for } x \geqslant x_{0}, \\ 0 & \text{ for } x < x_{0}. \end{cases}$

Compute the derivative of the generalized function. $y = \delta \left(x-x_{0}\right)$.

Compute the derivative of the generalized function. $y = \left|x\right|$.

Compute the derivative of the generalized function. $y = x_{+}$, where $x_{+} = \begin{cases} x & \text{ for } & x \geqslant 0, \\ 0 & \text{ for } & x < 0. \end{cases}$

Compute the derivative of the generalized function. $y = \ln \left|x\right|$.

Compute the derivative of the generalized function. $y = x_{+}^{\lambda } = \begin{cases} x^{\lambda } & \text{ for } x > 0, \\ 0 & \text{ for } x < 0, \end{cases} \quad -1 < \lambda < 0$.

Compute the derivative of the generalized function. $y = \ln x_{+} = \begin{cases} \ln x & \text{ for } & x > 0, \\ 0 & \text{ for } & x < 0. \end{cases}$

Find the derivatives of order $n$ of the generalized function. $y = \delta (x)$.

Find the derivatives of order $n$ of the generalized function. $y = x_{+}^{k}, k = 0,1,2, \ldots$

Prove the stated assertion. $\left(\frac{d}{d x}+\lambda \right) \theta (x) e^{-\lambda x} = \delta (x)$.

Prove the stated assertion. $\left(\frac{d^{2}}{d x^{2}}+\omega^{2}\right) \frac{\theta (x) \sin \omega x}{\omega } = \delta (x)$.

Prove the stated assertion. If $\delta_{\varepsilon }(x) = \begin{cases} 1 / \varepsilon & \text{ for } & \left|x\right| < \varepsilon / 2, \\ 0 & \text{ for } & \left|x\right| \geqslant \varepsilon / 2, \end{cases}$ then in the space $D^{\prime }$ the limit

exists.

Prove the stated assertion. If

where the functions $f_{1}$ and $f_{2}$ are continuous and piecewise continuously differentiable on $R$ (so that, in particular, the limits $f(x_{0} \pm 0)$ exist), and

(see problem 71), then the function $g(x)$ is continuous on $R$, has a locally integrable derivative $g^{\prime }$, and

Prove the stated assertion. If $f$ is a piecewise smooth function on $R$, having at the points $x_{1}, \ldots , x_{n}$ jump discontinuities of the first kind with jumps $p_{1}, p_{2}, \ldots , p_{n}$, then

where $f^{\prime }$ is the generalized derivative of the function $f$, and $\frac{d f(x)}{d x}$ is the generalized function generated by the ordinary derivative of the function $f$ for $x \neq x_{k}, k = 1,2, \ldots , n$.

Prove the stated assertion. If $f_{n} \in D^{\prime }, n = 1,2, \ldots , f \in D^{\prime }$ and $\lim_{n \rightarrow \infty } f_{n} = f$, then for any $k = 1,2, \ldots$ in the space $D^{\prime }$ the equality $\lim_{n \rightarrow \infty } f_{n}^{(k)} = f^{(k)}$ holds.

Prove the stated assertion. In the space of generalized functions, convergent series can be differentiated term by term any number of times: if $f = \sum_{n = 1}^{\infty } f_{n}$, then $f^{(k)} = \sum_{n = 1}^{\infty } f_{n}^{(k)}$ in $D^{\prime }, k = 1,2, \ldots$

Prove the stated assertion. For derivatives of generalized functions of slow growth, complete analogues of the assertions of problems $69, 70, 85$ and 86 hold.

Prove that the numbers $c_{\alpha \beta }$ in the Leibniz formula are

Suppose $\left\{ f_{i}\right\}$ is a sequence of locally integrable functions in $\Omega$ (an open set in $\mathbb {R}^{n}$) and

for every compact $K \subset \Omega$. Prove that then $D^{\alpha } f_{i} \rightarrow 0$ in $\mathscr {D}^{\prime }(\Omega )$, as $i \rightarrow \infty$, for every multi-index $\alpha$.

Recall that $\delta$ (the Dirac measure) is the distribution defined by $\delta (\phi ) = \phi (0)$, for $\phi \in \mathscr {D}(R)$. For which $f \in C^{\infty }(R)$ is it true that $f \delta^{\prime } = 0$? Answer the same question for $f \delta^{\prime \prime }$. Conclude that a function $f \in C^{\infty }(R)$ may vanish on the support of a distribution $\Lambda \in \mathscr {D}^{\prime }(R)$ although $f \Lambda \neq 0$.

For $u \in \mathscr {D}^{\prime }(R)$, show that

as $x \rightarrow 0$. (The derivative of $u$ may thus still be regarded as a limit of quotients).

If $f \in L^{1}((-\infty ,-\delta ) \cup (\delta , \infty ))$ for every $\delta > 0$, define its principal value integral to be

if the limit exists. For $\phi \in \mathscr {D}(R)$, put

Show that

Suppose $u$ is a distribution in $\mathbb {R}^{n}$ whose first derivatives $D_{1} u, \ldots , D_{n} u$ are functions in $L^{2}\left(\mathbb {R}^{n}\right)$. Prove that $u$ is also a function and that $u$ is locally $L^{2}$. (Show that "locally" cannot be omitted in the conclusion). Hint: $u$ is in fact the sum of an $L^{2}$-function and an entire function.

When $n = 1$, show that $u$ is actually a continuous function. Show that this stronger conclusion is false when $n = 2$. For example, the gradient of the function

is in $L^{2}\left(\mathbb {R}^{2}\right)$. See Exercise 11, Chapter 8, for the same result under weaker hypotheses.