Convergence of sequences in D, S and E

[12/58%]

Show

LaTeX

Show LaTeX

Problem 14.1 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 14.1 "Examples of Problems with Solutions"

Let $\varphi$ be a nonzero function from $\mathcal{D}(\mathbb {R})$ . Does the sequence $\varphi_{n}(x) = \frac{1}{n} \varphi \left(\frac{x}{n}\right)$ , $x \in \mathbb {R}, n \geq 1$ , converge in $\mathcal{D}(\mathbb {R})$ ?

Problem 14.17 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 14.2 "Problems to Solve"

(1)(2)(3)

Let $\varphi \in \mathcal{D}(\mathbb {R})$ be a nonzero function. Do the following sequences converge in $\mathcal{D}(\mathbb {R})$ :

(1)

$\left\{ \frac{1}{n} \varphi (x), x \in \mathbb {R}: n \geq 1\right\}$ ;

(2)

$\left\{ \frac{1}{n} \varphi (n x), x \in \mathbb {R}: n \geq 1\right\}$ ;

(3)

$\left\{ \varphi (x+n), x \in \mathbb {R}: n \geq 1\right\}$ ?

Problem 14.65 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 14.2 "Problems to Solve"

(1)(2)(3)

Let $\varphi$ be a nonzero function from $\mathcal{S}(\mathbb {R})$ . Do the following sequences converge in $\mathcal{S}(\mathbb {R})$ :

(1)

$\left\{ \frac{1}{n} \varphi (x), x \in \mathbb {R}: n \geq 1\right\}$ ;

(2)

$\left\{ \frac{1}{n} \varphi (n x), x \in \mathbb {R}: n \geq 1\right\}$ ;

(3)

$\left\{ \frac{1}{n} \varphi \left(\frac{x}{n}\right), x \in \mathbb {R}: n \geq 1\right\}$ ?

Problem 15.18 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Chapter 15 "Test (basic) function spaces"

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

(a)

$f_{n}(t) = \frac{n t}{n^{2}+t^{2}}$ ;

(b)

$f_{n}(t) = \frac{n^{3} t^{2}}{\left(1+n^{2} t^{2}\right)^{2}}$ ;

(c)

$f_{n}(t) = e^{-n t^{2}}$ ;

(d)

$f_{n}(t) = \sum_{k = 0}^{n} \frac{t^{k}}{k!}$ ;

(e)

$f_{n}(t) = \frac{1}{t+n+i / n}$ ;

(f)

$f_{n}(t) = \frac{\sin \left(n t+n^{2}\right)}{t+n}$ .

Problem 15.19 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Chapter 15 "Test (basic) function spaces"

(a)(b)(c)

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

(a)

$f_{n}(t) = \frac{1}{n} \varphi_{0}(t / n)$ ;

(b)

$f_{n}(t) = \frac{1}{n^{2}} \varphi_{0}(n t)$ ;

(c)

$f_{n}(t) = \frac{1}{n} \varphi_{0}(n+t)$ .

Problem 15.20 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Chapter 15 "Test (basic) function spaces"

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

Problem 15.21 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Chapter 15 "Test (basic) function spaces"

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

Problem 424 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.4.3 "Spaces of Smooth Functions"

(a)(b)

(a)

Let $f\in \mathcal D(\mathbb {R}^{n})$ . Prove that for any $y\in \mathbb {R}^{n}$ the net $f_t(x)=[f(x+ty)-f(x)]/t$ has a limit in $\mathcal D(\mathbb {R}^{n})$ as $t\to 0$ .

(b)

Let $f\in \mathcal E(\mathbb {R}^{n})$ . Prove that for any $y\in \mathbb {R}^{n}$ the net $f_t(x)=[f(x+ty)-f(x)]/t$ has a limit in $\mathcal E(\mathbb {R}^{n})$ as $t\to 0$ .

Problem 3.21.89 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.21.2 "Problems"

Prove the stated assertion. For $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ to hold, it is necessary and sufficient that

\lim _{n \rightarrow \infty } \sup _{R}\left|x^{k}\left[\varphi _{n}^{(m)}(x)-\varphi ^{(m)}(x)\right]\right| = 0, \quad k, m = 0,1,2, \ldots

Problem 3.21.90 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.21.2 "Problems"

Prove the stated assertion. If $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ in $S$ , then for any $m = 1,2, \ldots$ and any $x \in R$

\lim _{n \rightarrow \infty } \varphi _{n}^{(m)}(x) = \varphi ^{(m)}(x)

holds.

Problem 3.21.91 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.21.2 "Problems"

Prove the stated assertion. The space of test functions $D$ is contained in the space of test functions of slow growth $S$ , and if $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ in $D$ , then $\lim_{n \rightarrow \infty } \varphi_{n} = \varphi$ in $S$ .

Problem 3.7.5 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.7 "Fourier Transforms"

(a)(b)

(a)

Construct a sequence in $\mathscr {D}\left(\mathbb {R}^{n}\right)$ which converges to 0 in the topology of $\mathscr {S}_{n}$ but not in that of $\mathscr {D}\left(\mathbb {R}^{n}\right)$ .

(b)

Construct a sequence of polynomials which converges in the topology of $\mathscr {D}^{\prime }\left(\mathbb {R}^{1}\right)$ but not in that of $\mathscr {S}_{1}^{\prime }$ .

References

[BCK+24] Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. Functional Analysis and Operator Theory. Springer, 2024.

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.

[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.