Generalized functions

On the subspace $\left\{ \varphi \in \mathcal{D}: \varphi (0) = 0\right\}$ a linear functional $\varphi \rightarrow \int_{\mathbb {R}} \frac{\varphi (t)}{t} d t$ is defined. Prove that this functional is continuous, and find all its continuous extensions to the whole space $\mathcal{D}$.

On the subspace $\left\{ \varphi \in \mathcal{D}: \varphi (x) \equiv 0\text{ for }\left|x\right| < \varepsilon \right\}$ (the number $\varepsilon = \varepsilon (\varphi )$ its own for each function $\varphi$) a linear functional $\varphi \rightarrow \int_{\mathbb {R}} \exp \left(1 / t^{2}\right) \varphi (t) d t$ is defined. Prove that this functional cannot be extended to a continuous functional on the whole space $\mathcal{D}$.

Give examples of generalized functions belonging to

Prove that the functionals $\delta_{a}, \mathcal{P} \frac{1}{x}$ and $\frac{1}{x \pm i 0}$ lie in the space $\mathcal{D}^{\prime }$.

Prove that the generalized functions $\delta_{a}, \mathcal{P} \frac{1}{x}$ and $\frac{1}{x \pm i 0}$ are singular.

Prove the Sokhotski formulas:

Define the generalized function $(x+i 0)^{\lambda }, \operatorname {Re} \lambda > -1$, as the limit of the regular generalized functions

(where $\arg z \in [0,2 \pi )$) as $\varepsilon \rightarrow +0$. Prove that

Let $\operatorname {Re} \lambda \leqslant -1, \lambda \neq -1,-2, \ldots$, and let $n$ be the integer part of the number $-\operatorname {Re} \lambda$. For each function $\varphi \in \mathcal{D}$ set

Prove that these formulas define generalized functions on D. Prove that in every strip $-n-1 < \operatorname {Re} \lambda < -n$ these functionals can be given by the formulas

Prove that for fixed $\varphi$ the function $g(\lambda ): = \left\langle x_{+}^{\lambda }, \varphi \right\rangle$ is holomorphic on the domain $\mathbb {C} \backslash \left\{ -1,-2, \ldots \right\}$ and, in general, has simple poles at the points $-1,-2, \ldots$ Find the residues at these points.

Prove that for fixed $\varphi$ the function $g(\lambda ): = \left\langle \left|x\right|^{\lambda }, \varphi \right\rangle$ is holomorphic on the domain $\mathbb {C} \backslash \left\{ -1,-3,-5, \ldots \right\}$ and, in general, has simple poles at the points $-1,-3,-5, \ldots$ Find the residues at these points.

Prove that for fixed $\varphi$ the functions $g_{+}(\lambda ): = : = \left\langle (x+i 0)^{\lambda }, \varphi \right\rangle$ and $g_{-}(\lambda ): = \left\langle (x-i 0)^{\lambda }, \varphi \right\rangle$ are holomorphic on $\mathbb {C}$.

Prove that if two locally summable functions $f, g \in L_{1, \text{ loc }}(\mathbb {R})$ define one and the same regular generalized function ($\left\langle F_{f}, \varphi \right\rangle = \left\langle F_{g}, \varphi \right\rangle$ for every $\varphi \in \mathcal{D}$), then $f(x) = g(x)$ almost everywhere on $\mathbb {R}$.

Let $F \in \mathcal{D}^{\prime }$ and $\langle F, \varphi \rangle \geqslant 0$ for every nonnegative function $\varphi \in \mathcal{D}$. Prove that there exists a nondecreasing function $\mu$ such that $\langle F, \varphi \rangle = \int_{\mathbb {R}} \varphi (t) d \mu$, where the integral is understood in the Riemann-Stieltjes sense.

Prove that every generalized function is regular on the complement of its singular support.

Find the support and singular support of the following generalized functions:

Prove that a generalized function $F$ belongs to $\mathcal{E}^{\prime }$ if and only if the support of $F$ lies on some segment (that is, is compact).

Prove that if a function $f$ has the form $f(x) = g(x)\left(1+\left|x\right|^{n}\right)$, where $g \in L_{1}(\mathbb {R})$ and $n \in \mathbb {N} \cup \left\{ 0\right\}$, then $f$ generates a generalized function $F_{f} \in S^{\prime }$.

Let $W_{2}^{-1}[-\pi , \pi ]$ be the completion of the space $L_{2}[-\pi , \pi ]$ with respect to the norm

Show that $\delta_{0} \in W_{2}^{-1}[-\pi , \pi ]$, and find $\left\| \delta_{0}\right\|_{W_{2}^{-1}[-\pi , \pi ]}$ (cf. problem 5.20 а)).

Prove that if a sequence of functions $\left\{ f_{n}\right\}_{n = 1}^{\infty } \subset L_{1,\text{ loc }}(\mathbb {R})$ satisfies conditions (1) and (2) for every segment $[a, b]$, then $f_{n}$ is a $\delta$-shaped sequence:

1. for any $a_{1}, b_{1},\left[a_{1}, b_{1}\right] \subset [a, b]$, the inequality

   $$\left|\int _{a_{1}}^{b_{1}} f_{n} d x\right| < C,$$

   holds, where the constant $C$ does not depend on $a_{1}, b_{1}$ and $n$, but depends only on $[a, b]$;

2. $$$$

\int {a}^{b} f{n} d x \rightarrow \begin{cases} 1, & \text{ if } 0 \in (a, b), \ 0, & \text{ if } 0 \notin [a, b]. \end{cases}

Let $f \in L_{1}(\mathbb {R})$ be a nonnegative function and $\int_{\mathbb {R}} f d x = 1$. Prove that $f_{n}(x) = n f(n x)$ is a $\delta$-shaped sequence in $\mathcal{D}^{\prime }$.

Find the limit in $\mathcal{D}^{\prime }$ of the following sequences:

Find the limit $F = \lim_{n \rightarrow \infty } F_{n}$ in $\mathcal{D}^{\prime }$ of the following sequences:

Is it true that any sequence of continuous functions $f_{n}$ converging pointwise on $\mathbb {R}$ to zero also converges to zero in $\mathcal{D}^{\prime }$?

Prove that regular generalized functions are dense in $\mathcal{D}^{\prime }$.

Prove that regular generalized functions generated by compactly supported infinitely differentiable functions are dense in $\mathcal{D}^{\prime }$ (that is, $\mathcal{D}$ is dense in $\mathcal{D}^{\prime }$).

Prove that for any generalized function $F \in \mathcal{D}^{\prime }$ there exists a sequence of regular functions, generated by functions from $\mathcal{D}$, which converges to $F$ in $\mathcal{D}^{\prime }$. ¹

Let $\varphi \in \mathcal{D}, F \in \mathcal{D}^{\prime }$. Prove that if $\varphi \cdot F = 0$, then $\langle F, \varphi \rangle = 0$. Is the converse statement true?

Prove that every generalized function is the limit of generalized functions with compact support (in other words, $\mathcal{E}^{\prime }$ is sequentially dense in $\mathcal{D}^{\prime }$).

Prove that $x^{m} x_{+}^{\lambda } = x_{+}^{m+\lambda }, \lambda \neq -1,-2, \ldots , m \in \mathbb {N}$.

Prove that $x^{2 m}\left|x\right|^{\lambda } = \left|x\right|^{2 m+\lambda }$, and $x^{2 m+1}\left|x\right|^{\lambda } = \left|x\right|^{2 m+1+\lambda } \operatorname {sign} x$, $\lambda \neq -1,-2, \ldots , m \in \mathbb {N}$.

Solve in $\mathcal{D}^{\prime }$ the equations:

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 $F$.

Find the order of singularity of the generalized functions:

Find the support and order of singularity of the generalized functions:

Prove Theorem 16.1.

Theorem 16.1: For every generalized function $F \in \mathcal{E}^{\prime }$ (a generalized function with compact support) there exist $f \in L_{1, \text{ loc }}(\mathbb {R})$ and $n \in \mathbb {Z}_{+}$ such that $F = F_{f}^{(n)}$ (here $F_{f}$ is the regular generalized function generated by $f$).

Prove that any generalized function $F \in \mathcal{D}^{\prime }$ can be represented as an at most countable sum $\sum F_{f_{k}}^{\left(n_{k}\right)}$, where all $f_{k} \in L_{1, \text{ loc }}(\mathbb {R})$ and every segment on the line intersects only a finite number of supports of the functions $f_{k}$.

Give an example of a function $F \in \mathcal{D}^{\prime }$ with infinite order of singularity.

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

Solve in $\mathcal{D}^{\prime }$ the differential equations:

Prove that a generalized function $F$ is invariant with respect to shifts ($\langle F, \varphi (x)\rangle = \langle F, \varphi (x+a)\rangle$ for all $\varphi \in \mathcal{D}$ and $a \in \mathbb {R})$ if and only if it coincides with the regular function generated by some constant.

Prove that any generalized function satisfying the equation

with constant coefficients $a_{k} \in \mathbb {C}$ is a regular generalized function, generated by some ordinary solution of this equation.

Prove that a generalized function $F$ has one-point support $\left\{ a\right\} : \operatorname {supp} F = \left\{ a\right\}$ if and only if $F$ is representable as a finite linear combination of the function $\delta_{a}$ and its derivatives.

Prove the equality

(the sums of the series on both sides are taken in $\mathcal{D}^{\prime }$).

Prove the equality $\sum_{n = 1}^{\infty } \sin n x = \frac{1}{2} \operatorname {ctg} \frac{x}{2}$ (the series converges in $\mathcal{D}^{\prime }$; $\left\langle \operatorname {ctg} \frac{x}{2}, \varphi \right\rangle : =$ v.p. $\int \operatorname {ctg} \frac{x}{2} \varphi (x) d x$).

Prove that the series $\sum_{n = 1}^{\infty } a_{n} \cos n x$ converges in $\mathcal{D}^{\prime }$ if and only if there exist $C > 0$ and $k \in \mathbb {N}$ such that $\left|a_{n}\right| < C n^{k}, n = 1,2, \ldots$

Compute the composition $\delta (a x+b)$, where $a \neq 0$ and $b$ are some real numbers.

Let $F_{g}$ be a regular generalized function, generated by a locally integrable function $g$. Prove that for any diffeomorphism $\gamma : \mathbb {R} \rightarrow \mathbb {R}$ the equality $F_{g} \circ \gamma = F_{g \circ \gamma }$ holds, where $g \circ \gamma$ is the ordinary composition of the functions $g$ and $\gamma$.