Math Problems Vault

Problem 14.1

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 $\mathcal{D}(\mathbb {R})$ ?

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Problem 14.2

Let $n \in \mathbb {N}, \xi \in \mathcal{D}(\mathbb {R})$ be such that $\xi (x) = 1$ in some neighborhood of zero, and $\varphi \in \mathcal{D}(\mathbb {R})$ . Prove that the function

\psi (x) = \frac{1}{x^{n}}\left(\varphi (x)-\xi (x) \sum _{k = 0}^{n-1} \frac{\varphi ^{(k)}(0)}{k!} x^{k}\right), x \in \mathbb {R} \backslash \left\{ 0\right\} ,

which is defined by continuity at $x = 0$ , belongs to the space $\mathcal{D}(\mathbb {R})$ .

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Remark.

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The function $\xi \in \mathcal{D}(\mathbb {R})$ , for which $\xi (x) = 1$ in some neighborhood of zero, exists, see problem $14.18^{\circ }$ .

Problem 14.3

(1)(2)

Using the definition, prove that the following functionals are the generalized functions:

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(1)

$f(\varphi ) = \int_{\mathbb {R}} x \varphi (x) d x$

(2)

$g(\varphi ) = \sum_{n = 1}^{\infty } \varphi^{(n)}(n), \varphi \in \mathcal{D}(\mathbb {R})$ .

Are they regular? Find supports of these generalized functions.

Problem 14.4

Prove that the functional $\mathcal{P} \frac{1}{x}$ , which is defined by the formula

\left(\mathcal{P} \frac{1}{x}\right)(\varphi ) = \text{ v.p. } \int _{\mathbb {R}} \frac{\varphi (x)}{x} d x: = \lim _{\varepsilon \rightarrow 0+} \int _{\mathbb {R} \backslash (-\varepsilon , \varepsilon )} \frac{\varphi (x)}{x} d x, \varphi \in \mathcal{D}(\mathbb {R})

is a singular generalized function.

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Problem 14.5

Prove that the series $\sum_{n = 1}^{\infty } a_{n} \delta_{n}$ converges in $\mathcal{D}^{\prime }(\mathbb {R})$ for any $a_{n} \in \mathbb {C}$ .

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Problem 14.6

Let for $n \in \mathbb {N}$ the function $\mathcal{P} \frac{\cos n x}{x}$ be defined by the formula

\left\langle \mathcal{P} \frac{\cos n x}{x}, \varphi \right\rangle = \text{ v.p. } \int _{\mathbb {R}} \frac{\cos n x}{x} \varphi (x) d x: = \lim _{\varepsilon \rightarrow 0+} \int _{\mathbb {R} \backslash (-\varepsilon , \varepsilon )} \frac{\cos n x}{x} \varphi (x) d x

$\varphi \in \mathcal{D}(\mathbb {R})$ . Prove that $\mathcal{P} \frac{\cos n x}{x} \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} 0, n \rightarrow \infty$ .

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Problem 14.7

Prove that the generalized function $f = \sum_{k = 0}^{n-1} c_{k} \delta_{0}^{(k)}, c_{0}, \ldots , c_{n-1} \in \mathbb {C}$ is the general solution in $\mathcal{D}^{\prime }(\mathbb {R})$ of the equation $x^{n} f(x) = 0, n \in \mathbb {N}$ .

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Problem 14.8

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 $\left\langle f, \varphi \right\rangle = C \int_{\mathbb {R}} \varphi (x) d x, \varphi \in \mathcal{D}(\mathbb {R})$ .

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Problem 14.9

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 $\mathcal{D}^{\prime }(\mathbb {R})$ .

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Problem 14.10

Find the general solution of the equation $(x+1) y^{\prime \prime } = 0$ in $\mathcal{D}^{\prime }(\mathbb {R})$ .

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Problem 14.11

Let $a \in \mathbb {R} \backslash \left\{ 0\right\} , \varphi \in \mathcal{D}(\mathbb {R})$ . Find the general solution of the equation $\frac{d^{2} u}{d x^{2}}+ a^{2} u = \varphi$ .

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Problem 14.12

Prove that $\mathcal{D}(\mathbb {R}) \subset \mathcal{S}(\mathbb {R})$ , and $\mathcal{D}(\mathbb {R})$ is dense in $\mathcal{S}(\mathbb {R})$ .

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Problem 14.13

Prove that the series $\sum_{k \in \mathbb {Z}} e^{k^{2}} \delta_{k}$ converges in $\mathcal{D}^{\prime }(\mathbb {R})$ , but diverges in $\mathcal{S}^{\prime }(\mathbb {R})$ .

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Problem 14.14

Prove that for all $f \in L_{2}(\mathbb {R})$ the equality $\left(F^{2}\left[f\right]\right)(x) = f(-x)$ holds for almost all $x \in \mathbb {R}$ .

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Problem 14.15

Prove that $F\left[\delta_{0}\right] = \frac{1}{\sqrt{2 \pi }}$ and $F\left[1\right] = \sqrt{2 \pi } \delta_{0}$ , where $F$ acts in $\mathcal{S}^{\prime }(\mathbb {R})$ .

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Problem 14.16

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:

\frac{\partial ^{m} \theta }{\partial x_{1} \partial x_{2} \ldots \partial x_{m}} = \delta _{0}

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

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