Test (basic) function spaces

Prove that the compactly supported function

is infinitely differentiable on the real axis (we shall call this function a "bump").

Prove that for every $\delta > 0$ there exists a function $\varphi_{\delta } \in \mathcal{D}$ with the properties: $\operatorname {supp} \varphi_{\delta } = [-1-\delta , 1+\delta ]$, $\varphi_{\delta }(t) \in [0,1]$ for all $t$, $\varphi_{\delta }(t) = 1$ for $t \in [-1,1]$ (we shall call this function a "hat").

Prove that a function $\varphi \in \mathcal{D}$ is the derivative of some other function from $\mathcal{D}$ if and only if

Prove that a function $\varphi \in \mathcal{D}$ satisfies the equality $\varphi (0) = 0$ if and only if $\varphi (t) = t \cdot \psi (t)$, where $\psi$ is some other function from $\mathcal{D}$.

Give examples of functions $\varphi \in \mathcal{S} \backslash \mathcal{D}$ and $\psi \in \mathcal{E} \backslash \mathcal{S}$.

To which of the spaces $\mathcal{E}, \mathcal{S}, \mathcal{D}$ do the functions

Prove that for any numerical sequence $\left\{ a_{n}\right\}_{0}^{\infty }$ there exists a function $\varphi \in \mathcal{D}$ such that $\varphi^{(n)}(0) = a_{n}$ for all $n = 0,1,2, \ldots$

Let $\varphi \in C^{\infty }[a, b]$ (at the endpoints of the segment the existence of the corresponding one-sided derivatives of every order is required). Prove that there exists a function $\psi \in \mathcal{D}$ such that $\left.\psi \right|_{[a, b]} \equiv \varphi$.

Prove that any function $\psi \in \mathcal{E}$ can be represented as a series $\psi = \sum_{k = 1}^{\infty } \varphi_{k}$, where all $\varphi_{k} \in \mathcal{D}$ and every point $t \in \mathbb {R}$ belongs only to a finite number of the supports $\operatorname {supp} \varphi_{k}$.

Does there exist a function $\varphi \in \mathcal{D}$ such that $\operatorname {supp} \varphi \subset [0, \infty )$ and such that the function $\varphi$ itself and all its derivatives are positive on the interval $(0,1)$?

Prove that the following seminorms are admissible on $\mathcal{D}$:

Let $p$ be an admissible seminorm on $\mathcal{D}$. Prove that the following expressions are admissible seminorms:

Give an example of a seminorm, defined on the whole space $\mathcal{D}$, that is not admissible.

Prove that the following three systems of seminorms on $\mathcal{S}$ are equivalent (i.e., define the same topology):

Define a system of seminorms $\mathcal{P}^{\prime }$ on the space $\mathcal{D}$ as follows: $\mathcal{P}^{\prime } = \left\{ p_{\alpha , c}\right\}$, where $\alpha = \left(\alpha_{1}, \alpha_{2}, \ldots \right)$ is an arbitrary sequence of nonnegative integers, $c = \left(c_{1}, c_{2}, \ldots \right)$ an arbitrary sequence of positive numbers, and $p_{\alpha , c}(\varphi ): = : = \sum_{m = 1}^{\infty } c_{m} p_{m, \alpha_{m}}(\varphi )$, where $p_{m, k}(\varphi ) = \max_{\substack {0 \leqslant j \leqslant k, m-1 \leqslant \left|t\right| \leqslant m}}\left|\varphi^{(j)}(t)\right|$. Prove that the systems $\mathcal{P}$ and $\mathcal{P}^{\prime }$ are equivalent.

Prove Theorem 15.1.

Definition 15.3: The space $\mathcal{D}$ of compactly supported functions is the linear subspace of all functions $\varphi \in C^{\infty }(\mathbb {R})$ with bounded support, equipped with the system $\mathcal{P}$ of admissible seminorms: $p \in \mathcal{P}$ $\Leftrightarrow$ for every $N$ there exist a constant $C > 0$ and an integer $n \geqslant 0$ such that for every function $\varphi \in \mathcal{D}$ with support $\operatorname {supp} \varphi \subset [-N, N]$ the inequality $p(\varphi ) \leqslant C \cdot p_{N, n}(\varphi )$ holds, where $p_{N, n}(f) := \max \left\{ \left|f^{(k)}(t)\right|: k = 0,1, \ldots , n ; t \in [-N, N]\right\}$.

Theorem 15.1: Convergence $\varphi_{n} \rightarrow \varphi$ in $\mathcal{D}$, defined by the system of seminorms $\mathcal{P}$, is equivalent to the following: all the supports $\operatorname {supp} \varphi_{n}$ and $\operatorname {supp} \varphi$ lie in one segment $I$, and for every $k = 0,1,2, \ldots$ the derivatives $\varphi_{n}^{(k)}$ converge uniformly on $I$ to $\varphi^{(k)}$.

Prove that if a set $M \subset \mathcal{D}$ is bounded with respect to every admissible seminorm, then the supports of all functions from $M$ lie on one and the same segment, and the set $M$ is sequentially precompact in $\mathcal{D}$: from any sequence contained in it one can select a convergent subsequence.

Find the limit (or prove divergence) of the sequences in the space $\mathcal{E}$:

Let $\varphi_{0} \in \mathcal{D}$ be a nonzero function. Find the limit (or prove divergence) of the sequences in the space $\mathcal{D}$:

Give an example of a sequence of compactly supported functions that converges in $\mathcal{E}$, but does not converge in $\mathcal{S}$.

Give an example of a sequence of compactly supported functions that converges in $\mathcal{S}$, but does not converge in $\mathcal{D}$.

Prove that the spaces $\mathcal{E}$ and $\mathcal{S}$ are metrizable (and, moreover, are Fréchet spaces), while the space $\mathcal{D}$ is not metrizable.

Prove that the natural embeddings $\mathcal{D} \subset \mathcal{S} \subset \mathcal{E}$ are continuous.

Show that:

Prove that $\mathcal{D}$ is dense in each of the spaces

Prove that the set of all polynomials is dense in $\mathcal{E}$.

Prove that for any function $\varphi \in \mathcal{D}$ and any other function $\psi \in \mathcal{D}$, not vanishing at any point of $\operatorname {supp} \varphi$, there exists a sequence of polynomials $p_{n}$ such that $\psi p_{n} \rightarrow \varphi$ in $\mathcal{D}$.

Prove that the space $\mathcal{D}$ is separable.

Prove that the differentiation operator is continuous in each of the spaces $\mathcal{D}, \mathcal{S}, \mathcal{E}$.

Let $\alpha : \mathbb {R} \rightarrow \mathbb {R}$ be an infinitely smooth function. Under what conditions on the function $\alpha$ is the operator $\varphi \longmapsto \alpha \varphi$ continuous

Prove the continuity of the integration operator

acting in the space $\mathcal{D}$ and defined on the subspace $\mathcal{D}_{0} = \left\{ \varphi \in \mathcal{D}: \int_{\mathbb {R}} \varphi (t) d t = 0\right\}$.

Let $\alpha : \mathbb {R} \rightarrow \mathbb {R}$ be a monotonically increasing infinitely differentiable function.

Let $\alpha : \mathbb {R} \rightarrow \mathbb {R}$ be a compactly supported Lebesgue-summable function. Prove the continuity of the convolution operator

in the spaces $\mathcal{D}$ and $\mathcal{E}$. Prove the continuity of this operator in the space $\mathcal{S}$ in the case when $\alpha$ is a measurable rapidly decreasing (i.e., $\sup \left\{ (1+\left|t\right|)^{n}\left|\alpha (t)\right|\right\} < \infty$ for every $n = 0,1, \ldots$) function.

Let $\omega_{-1,1}$ be the function from problem 15.1 (the "bump"), $c = \int_{\mathbb {R}} \omega_{-1,1}(t) d t$, and let $f \in C(\mathbb {R})$ and $\operatorname {supp} f \subset [-N, N]$. Prove that the convolution $\varphi_{\varepsilon }(t) = (c \varepsilon )^{-1} \int_{\mathbb {R}} f(s) \omega ((t-s) / \varepsilon ) d s$ lies in the space $\mathcal{D}$, with $\operatorname {supp} \varphi_{\varepsilon } \subset [-N-\varepsilon , N+\varepsilon ]$. Prove that $\varphi_{\varepsilon } \rightarrow f$ as $\varepsilon \rightarrow 0$ in the space $C(\mathbb {R})$. Is it true that $\varphi_{\varepsilon } \rightarrow f$ in the space $\mathcal{D}$ (in the case $f \in \mathcal{D})$?