Main concepts

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