Chapter 14

Generalized Functions

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Section 14.1Examples of Problems with Solutions[16/100%]
Problem 14.1

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

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Problem 14.2

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

ψ(x)=1xn(φ(x)−ξ(x)∑k=0n−1φ(k)(0)k!xk),x∈R\{0}, \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=0x = 0, belongs to the space D(R)\mathcal{D}(\mathbb {R}).

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

Problem 14.3

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

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(1)

f(φ)=∫Rxφ(x)dxf(\varphi ) = \int_{\mathbb {R}} x \varphi (x) d x

(2)

g(φ)=∑n=1∞φ(n)(n),φ∈D(R)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 P1x\mathcal{P} \frac{1}{x}, which is defined by the formula

(P1x)(φ)= v.p. ∫Rφ(x)xdx:=lim⁡ε→0+∫R\(−ε,ε)φ(x)xdx,φ∈D(R) \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 ∑n=1∞anδn\sum_{n = 1}^{\infty } a_{n} \delta_{n} converges in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) for any an∈Ca_{n} \in \mathbb {C}.

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Problem 14.6

Let for n∈Nn \in \mathbb {N} the function Pcos⁡nxx\mathcal{P} \frac{\cos n x}{x} be defined by the formula

⟨Pcos⁡nxx,φ⟩= v.p. ∫Rcos⁡nxxφ(x)dx:=lim⁡ε→0+∫R\(−ε,ε)cos⁡nxxφ(x)dx \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

φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}). Prove that Pcos⁡nxx→D′(R)0,n→∞\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=∑k=0n−1ckδ0(k),c0,…,cn−1∈Cf = \sum_{k = 0}^{n-1} c_{k} \delta_{0}^{(k)}, c_{0}, \ldots , c_{n-1} \in \mathbb {C} is the general solution in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) of the equation xnf(x)=0,n∈Nx^{n} f(x) = 0, n \in \mathbb {N}.

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Problem 14.8

Let f∈D′(R)f \in \mathcal{D}^{\prime }(\mathbb {R}) and f(x+a)=f(x)f(x+a) = f(x) for all a∈Ra \in \mathbb {R}. Prove that f=f = const, i.e., there exists C∈CC \in \mathbb {C} such that ⟨f,φ⟩=C∫Rφ(x)dx,φ∈D(R)\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 ∣sin⁡x∣′′+∣sin⁡x∣=2∑k∈Zδkπ\left|\sin x\right|^{\prime \prime }+\left|\sin x\right| = 2 \sum_{k \in \mathbb {Z}} \delta_{k \pi } holds in D′(R)\mathcal{D}^{\prime }(\mathbb {R}).

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Problem 14.10

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

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Problem 14.11

Let a∈R\{0},φ∈D(R)a \in \mathbb {R} \backslash \left\{ 0\right\} , \varphi \in \mathcal{D}(\mathbb {R}). Find the general solution of the equation d2udx2+a2u=φ\frac{d^{2} u}{d x^{2}}+ a^{2} u = \varphi.

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Problem 14.12

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

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Problem 14.13

Prove that the series ∑k∈Zek2δk\sum_{k \in \mathbb {Z}} e^{k^{2}} \delta_{k} converges in D′(R)\mathcal{D}^{\prime }(\mathbb {R}), but diverges in S′(R)\mathcal{S}^{\prime }(\mathbb {R}).

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Problem 14.14

Prove that for all f∈L2(R)f \in L_{2}(\mathbb {R}) the equality (F2[f])(x)=f(−x)\left(F^{2}\left[f\right]\right)(x) = f(-x) holds for almost all x∈Rx \in \mathbb {R}.

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Problem 14.15

Prove that F[δ0]=12πF\left[\delta_{0}\right] = \frac{1}{\sqrt{2 \pi }} and F[1]=2πδ0F\left[1\right] = \sqrt{2 \pi } \delta_{0}, where FF acts in S′(R)\mathcal{S}^{\prime }(\mathbb {R}).

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Problem 14.16

Let θ(x1,…,xm)=θ(x1)θ(x2)…θ(xm)\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 θ(xi)\theta \left(x_{i}\right) is the Heaviside function. Prove that in the space D′(Rm)\mathcal{D}^{\prime }\left(\mathbb {R}^{m}\right) the following equality holds:

∂mθ∂x1∂x2…∂xm=δ0 \frac{\partial ^{m} \theta }{\partial x_{1} \partial x_{2} \ldots \partial x_{m}} = \delta _{0}

where x=(x1,…,xm)x = \left(x_{1}, \ldots , x_{m}\right).

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Section 14.2Problems to Solve[73/56%]
Problem 14.17

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

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(1)

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

(2)

{1nφ(nx),x∈R:n≥1}\left\{ \frac{1}{n} \varphi (n x), x \in \mathbb {R}: n \geq 1\right\};

(3)

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

Problem 14.18

Let ε>0\varepsilon > 0. For a locally integrable function f:R→Cf: \mathbb {R} \rightarrow \mathbb {C} consider the function fε:R→Cf_{\varepsilon }: \mathbb {R} \rightarrow \mathbb {C}, defined as follows:

fε(x)=∫Rωε(x−y)f(y)dy,x∈R f_{\varepsilon }(x) = \int _{\mathbb {R}} \omega _{\varepsilon }(x-y) f(y) d y, x \in \mathbb {R}

Prove the following statements:

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(1)

if f=χ[−a,a]f = \chi_{\left[-a, a\right]} and ε<a<+∞\varepsilon < a < +\infty, then fε∈D(R)f_{\varepsilon } \in \mathcal{D}(\mathbb {R}), moreover 0≤fε(x)≤10 \leq f_{\varepsilon }(x) \leq 1 for all x∈R,fε(x)=1x \in \mathbb {R}, f_{\varepsilon }(x) = 1 for ∣x∣≤a−ε\left|x\right| \leq a-\varepsilon and fε(x)=0f_{\varepsilon }(x) = 0 for ∣x∣≥a+ε\left|x\right| \geq a+\varepsilon;

(2)

if f∈C(R)f \in C(\mathbb {R}) and supp⁡f⊂[−a,a]\operatorname {supp} f \subset [-a, a], then fε∈D(R)f_{\varepsilon } \in \mathcal{D}(\mathbb {R}), and fε(x)=0f_{\varepsilon }(x) = 0 for ∣x∣≥a+ε\left|x\right| \geq a+\varepsilon.

Problem 14.19

Let fεf_{\varepsilon } be a function defined in problem 14.18∘14.18^{\circ }. Prove the following statements:

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(1)

if f∈Lp(R),1≤p<+∞f \in L_{p}(\mathbb {R}), 1 \leq p < +\infty, then for arbitrary ε>0\varepsilon > 0 it holds that fε∈Lp(R)∩C∞(R)f_{\varepsilon } \in L_{p}(\mathbb {R}) \cap C^{\infty }(\mathbb {R}), and ∥fε∥p≤∥f∥p\left\| f_{\varepsilon }\right\|_{p} \leq \left\| f\right\|_{p};

(2)

if there exists a>0a > 0 such that f=0f = 0 almost everywhere for ∣x∣≥a\left|x\right| \geq a, then for f∈C(R)f \in C(\mathbb {R}), we have the convergence fε→ff_{\varepsilon } \rightarrow f as ε→0+\varepsilon \rightarrow 0+ in C(R)C(\mathbb {R}), and for f∈Lp(R),1≤p<∞f \in L_{p}(\mathbb {R}), 1 \leq p < \infty, we have the convergence fε→ff_{\varepsilon } \rightarrow f in Lp(R)L_{p}(\mathbb {R});

(3)

the set D(R)\mathcal{D}(\mathbb {R}) is dense in Lp(R),1≤p<+∞L_{p}(\mathbb {R}), 1 \leq p < +\infty.

Problem 14.20

Let f∈L∞(R)f \in L_{\infty }(\mathbb {R}), and there exists a>0a > 0 such that f=0f = 0 almost everywhere for ∣x∣≥a\left|x\right| \geq a. Prove that there exists a sequence {gn}⊂D(R)\left\{ g_{n}\right\} \subset \mathcal{D}(\mathbb {R}) such that ∥gn∥∞≤∥f∥∞,n≥1\left\| g_{n}\right\|_{\infty } \leq \left\| f\right\|_{\infty }, n \geq 1, and gn→f( mod m),n→∞g_{n} \rightarrow f(\bmod m), n \rightarrow \infty.

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Problem 14.21

The set M⊂D(R)M \subset \mathcal{D}(\mathbb {R}) is said to be bounded in D(R)\mathcal{D}(\mathbb {R}) if the supports of all functions from MM belong to the same segment [a,b]\left[a, b\right], and ∀k∈N∪{0}∃Ck>0∀φ∈M:max⁡a≤x≤b∣φ(k)(x)∣≤Ck\forall k \in \mathbb {N} \cup \left\{ 0\right\} \exists C_{k} > 0 \forall \varphi \in M: \max_{a \leq x \leq b}\left|\varphi^{(k)}(x)\right| \leq C_{k}.

Prove that every bounded in D(R)\mathcal{D}(\mathbb {R}) set is relatively compact, i.e., every sequence of its elements contains a subsequence which converges in D(R)\mathcal{D}(\mathbb {R}).

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Problem 14.22

Prove that the operations of differentiation and multiplication by a function α∈C∞(R)\alpha \in C^{\infty }(\mathbb {R}) are continuous in D(R)\mathcal{D}(\mathbb {R}), i.e. the convergence φn→D(R)φ\varphi_{n} \xrightarrow {\mathcal{D}(\mathbb {R})} \varphi implies that

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(1)

φn′→D(R)φ′\varphi_{n}^{\prime } \xrightarrow {\mathcal{D}(\mathbb {R})} \varphi^{\prime };

(2)

αφn→D(R)αφ,n→∞\alpha \varphi_{n} \xrightarrow {\mathcal{D}(\mathbb {R})} \alpha \varphi , n \rightarrow \infty.

Problem 14.23

Let ξ∈D(R),ξ(x)=1\xi \in \mathcal{D}(\mathbb {R}), \xi (x) = 1 in a neighborhood of zero, and the function α∈C∞(R)\alpha \in C^{\infty }(\mathbb {R}) has a unique zero of order 1 at the point x=0x = 0. Let φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}). Prove that the function

ψ(x)=φ(x)−ξ(x)φ(0)α(x),x∈R\{0} \psi (x) = \frac{\varphi (x)-\xi (x) \varphi (0)}{\alpha (x)}, x \in \mathbb {R} \backslash \left\{ 0\right\}

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

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Problem 14.24

Prove that the function φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}) is a derivative of some other function ψ∈D(R)\psi \in \mathcal{D}(\mathbb {R}) if and only if it satisfies the condition ∫Rφ(x)dx=0\int_{\mathbb {R}} \varphi (x) d x = 0.

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Problem 14.25

Prove that every function φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}) can be represented in the form

φ(x)=η(x)∫Rφ(y)dy+ψ′(x),x∈R \varphi (x) = \eta (x) \int _{\mathbb {R}} \varphi (y) d y+\psi ^{\prime }(x), x \in \mathbb {R}

where ψ∈D(R)\psi \in \mathcal{D}(\mathbb {R}), and η\eta is an arbitrary fixed function from D(R)\mathcal{D}(\mathbb {R}), which satisfies the condition ∫Rη(x)dx=1\int_{\mathbb {R}} \eta (x) d x = 1.

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Problem 14.26

Prove that the following functionals f:D(R)→Cf: \mathcal{D}(\mathbb {R}) \rightarrow \mathbb {C} are generalized function, i.e., f∈D′(R)f \in \mathcal{D}^{\prime }(\mathbb {R}).

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(1)

f(φ)=φ(0)+φ(1)f(\varphi ) = \varphi (0)+\varphi (1);

(2)

f(φ)=φ′(0)f(\varphi ) = \varphi^{\prime }(0);

(3)

f(φ)=∫Rφ(x)dxf(\varphi ) = \int_{\mathbb {R}} \varphi (x) d x;

(4)

f(φ)=∫−11∣x∣φ′(x)dx;f(\varphi ) = \int_{-1}^{1}\left|x\right| \varphi^{\prime }(x) d x ;

(5)

f(φ)=∫−22sign⁡xφ′(x)dxf(\varphi ) = \int_{-2}^{2} \operatorname {sign} x \varphi^{\prime }(x) d x;

(6)

f(φ)=∫Rln⁡∣x∣φ(x)dxf(\varphi ) = \int_{\mathbb {R}} \ln \left|x\right| \varphi (x) d x;

(7)

f(φ)=∫Rexφ′(x)dxf(\varphi ) = \int_{\mathbb {R}} e^{x} \varphi^{\prime }(x) d x;

(8)

f(φ)=∫Re−xφ(n)(x)dx,n∈Nf(\varphi ) = \int_{\mathbb {R}} e^{-x} \varphi^{(n)}(x) d x, n \in \mathbb {N};

(9)

f(φ)=φ′(1)+∫02πsin⁡xφ(x)dxf(\varphi ) = \varphi^{\prime }(1)+\int_{0}^{2 \pi } \sin x \varphi (x) d x;

(10)

f(φ)=φ(0)+∫13xφ′′(x)dxf(\varphi ) = \varphi (0)+\int_{1}^{3} x \varphi^{\prime \prime }(x) d x.

In which of these cases is the function regular? Find the support of ff.

Problem 14.27

Prove that the following functionals are singular generalized functions from D′(R)\mathcal{D}^{\prime }(\mathbb {R}).

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(1)

⟨P1x2,φ⟩=\left\langle \mathcal{P} \frac{1}{x^{2}}, \varphi \right\rangle = v.p. ∫Rφ(x)−φ(0)x2dx\int_{\mathbb {R}} \frac{\varphi (x)-\varphi (0)}{x^{2}} d x;

(2)

⟨P1x3,φ⟩=\left\langle \mathcal{P} \frac{1}{x^{3}}, \varphi \right\rangle = v.p. ∫Rφ(x)−φ(0)−φ′(0)xx3dx\int_{\mathbb {R}} \frac{\varphi (x)-\varphi (0)-\varphi^{\prime }(0) x}{x^{3}} d x;

(3)

⟨P1xn,φ⟩=v.p.∫Rφ(x)−φ(0)−φ′(0)x−…−φ(n−1)(0)(n−1)!xn−1xndx,n∈N\left\langle \mathcal{P} \frac{1}{x^{n}}, \varphi \right\rangle = \mathrm{v}. \mathrm{p}. \int_{\mathbb {R}} \frac{\varphi (x)-\varphi (0)-\varphi^{\prime }(0) x-\ldots -\frac{\varphi^{(n-1)}(0)}{(n-1)!} x^{n-1}}{x^{n}} d x, n \in \mathbb {N};

(4)

⟨Pcos⁡axx,φ⟩=\left\langle \mathcal{P} \frac{\cos a x}{x}, \varphi \right\rangle = v. p. ∫Rcos⁡axxφ(x)dx,a>0\int_{\mathbb {R}} \frac{\cos a x}{x} \varphi (x) d x, a > 0, where v. p. ∫Rf(x)dx:=lim⁡ε→0+(∫−∞−εf(x)dx+∫ε∞f(x)dx)\int_{\mathbb {R}} f(x) d x: = \lim_{\varepsilon \rightarrow 0+}\left(\int_{-\infty }^{-\varepsilon } f(x) d x+\int_{\varepsilon }^{\infty } f(x) d x\right).

Problem 14.28

Let f∈D′(R)f \in \mathcal{D}^{\prime }(\mathbb {R}) and ⟨f,φ⟩≥0\left\langle f, \varphi \right\rangle \geq 0 for every nonnegative φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}). Prove that there exists a measure μ\mu, defined on the σ\sigma-algebra of Borel sets B(R)\mathcal{B}(\mathbb {R}), such that ⟨f,φ⟩=∫Rφ(x)dμ(x)\left\langle f, \varphi \right\rangle = \int_{\mathbb {R}} \varphi (x) d \mu (x) for all φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}).

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Problem 14.29

Prove that δa→D′(R)0,a→+∞\delta_{a} \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} 0, a \rightarrow +\infty.

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Problem 14.30

Prove that

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(1)

ηε→D′(R)δ0,ε→0+\eta_{\varepsilon } \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \delta_{0}, \varepsilon \rightarrow 0+, where ηε(x)={12ε,∣x∣≤ε,0,∣x∣>ε;\eta_{\varepsilon }(x) = \begin{cases} \frac{1}{2 \varepsilon }, & \left|x\right| \leq \varepsilon , \\ 0, & \left|x\right| > \varepsilon ; \end{cases}

(2)

fε→D′(R)δ0∫Rf(x)dx,ε→0+f_{\varepsilon } \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \delta_{0} \int_{\mathbb {R}} f(x) d x, \varepsilon \rightarrow 0+, where f∈L1(R)f \in L_{1}(\mathbb {R}) and fε(x):=1εf(xε)f_{\varepsilon }(x): = \frac{1}{\varepsilon } f\left(\frac{x}{\varepsilon }\right).

Problem 14.31

Prove that

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(1)

εx2+ε2→D′(R)πδ0,ε→0+\frac{\varepsilon }{x^{2}+\varepsilon^{2}} \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \pi \delta_{0}, \varepsilon \rightarrow 0+;

(2)

1εe−x2ε→D′(R)πδ0,ε→0+\frac{1}{\sqrt{\varepsilon }} e^{-\frac{x^{2}}{\varepsilon }} \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \sqrt{\pi } \delta_{0}, \varepsilon \rightarrow 0+;

(3)

1xsin⁡xεχ[0,+∞)(x)→D′(R)π2δ0,ε→0+\frac{1}{x} \sin \frac{x}{\varepsilon } \chi_{[0,+\infty )}(x) \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \frac{\pi }{2} \delta_{0}, \varepsilon \rightarrow 0+;

(4)

12x+εχ(ε,4ε)(x)→D′(R)ln⁡32δ0,ε→0+;\frac{1}{2 x+\varepsilon } \chi_{(\varepsilon , 4 \varepsilon )}(x) \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \frac{\ln 3}{2} \delta_{0}, \varepsilon \rightarrow 0+;

(5)

tneixt→D′(R)0,t→+∞t^{n} e^{i x t} \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} 0, t \rightarrow +\infty, for every n≥0n \geq 0.

Problem 14.32

Prove that

1x±iε→D′(R)P1x∓iπδ0,ε→0+. \frac{1}{x \pm i \varepsilon } \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \mathcal{P} \frac{1}{x} \mp i \pi \delta _{0}, \varepsilon \rightarrow 0+.
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Problem 14.33

Prove that

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(1)

∀α∈C∞(R):αδ0=α(0)δ0\forall \alpha \in C^{\infty }(\mathbb {R}): \alpha \delta_{0} = \alpha (0) \delta_{0};

(2)

∀a∈R\{0}:δ0(ax)=1∣a∣δ0(x)\forall a \in \mathbb {R} \backslash \left\{ 0\right\} : \delta_{0}(a x) = \frac{1}{\left|a\right|} \delta_{0}(x).

Problem 14.34

Let α∈C∞(R)\alpha \in C^{\infty }(\mathbb {R}) have a unique zero of order 1 at the point x=0x = 0. Find the general solution of the equation αf=0\alpha f = 0 in D′(R)\mathcal{D}^{\prime }(\mathbb {R}).

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Problem 14.35

Let f∈D′(R)f \in \mathcal{D}^{\prime }(\mathbb {R}), supp f⊂[−a,a],η∈C∞(R)f \subset [-a, a], \eta \in C^{\infty }(\mathbb {R}), and there exists ε>0\varepsilon > 0 such that η(x)=1\eta (x) = 1 for all x∈[−a−ε,a+ε]x \in [-a-\varepsilon , a+\varepsilon ]. Prove that ηf=f\eta f = f.

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Problem 14.36

Prove that in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) the following equalities hold:

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(1)

xnP1x=xn−1,n∈Nx^{n} \mathcal{P} \frac{1}{x} = x^{n-1}, n \in \mathbb {N};

(2)

xP1x2=P1xx \mathcal{P} \frac{1}{x^{2}} = \mathcal{P} \frac{1}{x};

(3)

xnP1xk=xn−k,n,k∈N,n≥kx^{n} \mathcal{P} \frac{1}{x^{k}} = x^{n-k}, n, k \in \mathbb {N}, n \geq k;

(4)

xnP1xk=P1xk−n,n,k∈N,n<kx^{n} \mathcal{P} \frac{1}{x^{k}} = \mathcal{P} \frac{1}{x^{k-n}}, n, k \in \mathbb {N}, n < k, where P1xn,n≥1\mathcal{P} \frac{1}{x^{n}}, n \geq 1, are generalized functions, defined in problems 14.4 and 14.27.

Problem 14.37

Find the derivatives in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) of the functions

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(1)

f(x)=∣x∣,x∈Rf(x) = \left|x\right|, x \in \mathbb {R};

(2)

f(x)=sign⁡(sin⁡x),x∈Rf(x) = \operatorname {sign}(\sin x), x \in \mathbb {R}.

Problem 14.38

Prove that for any f∈D′(R),a∈Rf \in \mathcal{D}^{\prime }(\mathbb {R}), a \in \mathbb {R} and n∈Nn \in \mathbb {N} the following equality is true: (f(x+a))(n)=f(n)(x+a)(f(x+a))^{(n)} = f^{(n)}(x+a).

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Problem 14.39

Calculate the derivatives in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) for n≥1n \geq 1 :

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(1)

θ(n)(x−a)\theta^{(n)}(x-a);

(2)

(θ(x)eax)(n)\left(\theta (x) e^{a x}\right)^{(n)};

(3)

(sign⁡x)(n)(\operatorname {sign} x)^{(n)};

(4)

∣x∣(n)\left|x\right|^{(n)};

(5)

(sign⁡(cos⁡x))(n)(\operatorname {sign}(\cos x))^{(n)};

(6)

[x](n)\left[x\right]^{(n)}, where θ(x)=χ[0,+∞)(x)\theta (x) = \chi_{[0,+\infty )}(x) is the Heaviside function, [x]\left[x\right] is the integer part of a number xx.

Problem 14.40

Find the derivatives of the first three orders in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) of the following functions:

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(1)

f(x)=θ(x)sin⁡xf(x) = \theta (x) \sin x;

(2)

f(x)=θ(x)cos⁡xf(x) = \theta (x) \cos x;

(3)

f(x)={x2,∣x∣≤2,0,∣x∣>2;f(x) = \begin{cases} x^{2}, & \left|x\right| \leq 2, \\ 0, & \left|x\right| > 2 ; \end{cases}

(4)

f(x)={1,x<0,x+1,0≤x<1,x2+1,x≥1;f(x) = \begin{cases} 1, & x < 0, \\ x+1, & 0 \leq x < 1, \\ x^{2}+1, & x \geq 1 ; \end{cases}

(5)

f(x)={0,x<−1,(x+1)2,−1≤x<0,x2+1,x≥0;f(x) = \begin{cases} 0, & x < -1, \\ (x+1)^{2}, & -1 \leq x < 0, \\ x^{2}+1, & x \geq 0 ; \end{cases}

(6)

f(x)=sin⁡xχ[−π,π](x)f(x) = \sin x \chi_{\left[-\pi , \pi \right]}(x);

(7)

f(x)={x2,x<1,(x−2)2,1≤x<2,0,x≥2;f(x) = \begin{cases} x^{2}, & x < 1, \\ (x-2)^{2}, & 1 \leq x < 2, \\ 0, & x \geq 2 ; \end{cases}

(8)

f(x)=∣cos⁡x∣χ[0,2π](x)f(x) = \left|\cos x\right| \chi_{\left[0,2 \pi \right]}(x).

Problem 14.41

Prove that

?
(1)

1ε2f′(xε)→D′(R)δ0′∫Rf(x)dx,ε→0+\frac{1}{\varepsilon^{2}} f^{\prime }\left(\frac{x}{\varepsilon }\right) \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})} \delta_{0}^{\prime } \int_{\mathbb {R}} f(x) d x, \varepsilon \rightarrow 0+, where f∈L1(R)∩C1(R)f \in L_{1}(\mathbb {R}) \cap C^{1}(\mathbb {R});

(2)

xε(x2+ε2)2→D′(R)−π2δ0′,ε→0+\frac{x \varepsilon }{\left(x^{2}+\varepsilon^{2}\right)^{2}} \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})}-\frac{\pi }{2} \delta_{0}^{\prime }, \varepsilon \rightarrow 0+;

(3)

xεεe−x2ε→D′(R)−π2δ0′,ε→0+\frac{x}{\varepsilon \sqrt{\varepsilon }} e^{-\frac{x^{2}}{\varepsilon }} \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})}-\frac{\sqrt{\pi }}{2} \delta_{0}^{\prime }, \varepsilon \rightarrow 0+;

(4)

1x2χ[ε,2ε)(x)−1εδε+12εδ2ε→D′(R)−ln⁡2δ0′,ε→0+\frac{1}{x^{2}} \chi_{[\varepsilon , 2 \varepsilon )}(x)-\frac{1}{\varepsilon } \delta_{\varepsilon }+\frac{1}{2 \varepsilon } \delta_{2 \varepsilon } \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})}-\ln 2 \delta_{0}^{\prime }, \varepsilon \rightarrow 0+;

(5)

eε2e−xεχ[ε,+∞)(x)−1εδε→D′(R)−δ0′,ε→0+\frac{e}{\varepsilon^{2}} e^{-\frac{x}{\varepsilon }} \chi_{[\varepsilon ,+\infty )}(x)-\frac{1}{\varepsilon } \delta_{\varepsilon } \xrightarrow {\mathcal{D}^{\prime }(\mathbb {R})}-\delta_{0}^{\prime }, \varepsilon \rightarrow 0+.

Problem 14.42

Prove that in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) the following equalities hold:

?
(1)

(ln⁡∣x∣)′=P1x(\ln \left|x\right|)^{\prime } = \mathcal{P} \frac{1}{x};

(2)

(P1x)′=−P1x2\left(\mathcal{P} \frac{1}{x}\right)^{\prime } = -\mathcal{P} \frac{1}{x^{2}};

(3)

(P1x2)′=−2P1x3\left(\mathcal{P} \frac{1}{x^{2}}\right)^{\prime } = -2 \mathcal{P} \frac{1}{x^{3}};

(4)

(P1xn)′=−nP1xn+1,n≥1\left(\mathcal{P} \frac{1}{x^{n}}\right)^{\prime } = -n \mathcal{P} \frac{1}{x^{n+1}}, n \geq 1, where P1xn,n≥1\mathcal{P} \frac{1}{x^{n}}, n \geq 1, are generalized functions defined in problems 14.4 and 14.27.

Problem 14.43

Let α∈C∞(R)\alpha \in C^{\infty }(\mathbb {R}) and f∈D′(R)f \in \mathcal{D}^{\prime }(\mathbb {R}). Prove that (αf)′=α′f+αf′(\alpha f)^{\prime } = \alpha^{\prime } f+\alpha f^{\prime }. Calculate (αθ)′(\alpha \theta )^{\prime }, where θ\theta is the Heaviside function.

?
Problem 14.44

Prove in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) the following equalities

?
(1)

xδ0(n)=−nδ0(n−1),n∈Nx \delta_{0}^{(n)} = -n \delta_{0}^{(n-1)}, n \in \mathbb {N};

(2)

exδ0′′′=δ0′′′−3δ0′′+3δ0′−δ0e^{x} \delta_{0}^{\prime \prime \prime } = \delta_{0}^{\prime \prime \prime }-3 \delta_{0}^{\prime \prime }+3 \delta_{0}^{\prime }-\delta_{0};

(3)

xkδ0(n)=(−1)k(nk)δ0(n−k),n,k∈N,n≥kx^{k} \delta_{0}^{(n)} = (-1)^{k}\binom {n}{k} \delta_{0}^{(n-k)}, n, k \in \mathbb {N}, n \geq k;

(4)

xkδ0(n)=0,n,k∈N,n<kx^{k} \delta_{0}^{(n)} = 0, n, k \in \mathbb {N}, n < k;

(5)

αδ0(n)=∑k=0n(−1)k(nk)α(k)(0)δ0(n−k),α∈C∞(R)\alpha \delta_{0}^{(n)} = \sum_{k = 0}^{n}(-1)^{k}\binom {n}{k} \alpha^{(k)}(0) \delta_{0}^{(n-k)}, \alpha \in C^{\infty }(\mathbb {R}).

Problem 14.45

Prove that the generalized functions δ0,δ0′,…,δ0(n)\delta_{0}, \delta_{0}^{\prime }, \ldots , \delta_{0}^{(n)} are linearly independent in D′(R)\mathcal{D}^{\prime }(\mathbb {R}).

?
Problem 14.46

Prove in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) the equality ∣cos⁡x∣′′+∣cos⁡x∣=2∑k∈Zδ(k+12)π\left|\cos x\right|^{\prime \prime }+\left|\cos x\right| = 2 \sum_{k \in \mathbb {Z}} \delta_{\left(k+\frac{1}{2}\right) \pi }.

?
Problem 14.47

Using the expansion

x2−x24π=π6−12π∑k∈Z\{0}1k2eikx,x∈[0,2π] \frac{x}{2}-\frac{x^{2}}{4 \pi } = \frac{\pi }{6}-\frac{1}{2 \pi } \sum _{k \in \mathbb {Z} \backslash \left\{ 0\right\} } \frac{1}{k^{2}} e^{i k x}, x \in [0,2 \pi ]

prove that in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) the equality 12π∑k∈Zeikx=∑k∈Zδ2πk\frac{1}{2 \pi } \sum_{k \in \mathbb {Z}} e^{i k x} = \sum_{k \in \mathbb {Z}} \delta_{2 \pi k} holds.

?
Problem 14.48

Prove in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) the equality 2π∑k=0∞cos⁡(2k+1)x=∑k∈Z(−1)kδπk\frac{2}{\pi } \sum_{k = 0}^{\infty } \cos (2 k+1) x = \sum_{k \in \mathbb {Z}}(-1)^{k} \delta_{\pi k}.

?
Problem 14.49

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

?
Problem 14.50

Prove that the general solution of the equation xu′=1x u^{\prime } = 1 in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) has the form u=c1+c2θ(x)+ln⁡∣x∣u = c_{1}+c_{2} \theta (x)+\ln \left|x\right|, where c1,c2∈Cc_{1}, c_{2} \in \mathbb {C} are arbitrary and θ\theta is the Heaviside function.

?
Problem 14.51

Find the general solutions in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) of the following equations

?
(1)

(x−1)u=0(x-1) u = 0;

(2)

(x2−1)u=0\left(x^{2}-1\right) u = 0;

(3)

xu=P1xx u = \mathcal{P} \frac{1}{x};

(4)

x2u′=0x^{2} u^{\prime } = 0;

(5)

(x+1)2u′′=0(x+1)^{2} u^{\prime \prime } = 0;

(6)

x(x−1)u=0x(x-1) u = 0;

(7)

xu=1x u = 1;

(8)

x2u=0x^{2} u = 0;

(9)

xu′=P1xx u^{\prime } = \mathcal{P} \frac{1}{x};

(10)

x2u′=1x^{2} u^{\prime } = 1;

(11)

(x+1)u′′′=0(x+1) u^{\prime \prime \prime } = 0.

Problem 14.52

Find the general solution in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) of the equations

?
(1)

sin⁡x⋅y=0\sin x \cdot y = 0;

(2)

cos⁡x⋅y=0\cos x \cdot y = 0.

Problem 14.53

Prove that for n>mn > m the general solution in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) of the equation xnu(m)=x^{n} u^{(m)} = 0 is the generalized function

u=∑k=0m−1akxk+∑k=0m−1bkθ(x)xk+∑k=mn−1bkδ0(k−m) u = \sum _{k = 0}^{m-1} a_{k} x^{k}+\sum _{k = 0}^{m-1} b_{k} \theta (x) x^{k}+\sum _{k = m}^{n-1} b_{k} \delta _{0}^{(k-m)}

where ak,bka_{k}, b_{k} are arbitrary complex numbers.

?
Problem 14.54

Prove that f1∗f2=f2∗f1f_{1} * f_{2} = f_{2} * f_{1} and (f1∗f2)∗f3=f1∗(f2∗f3)\left(f_{1} * f_{2}\right) * f_{3} = f_{1} *\left(f_{2} * f_{3}\right) for all f1,f2,f3∈L1(R)f_{1}, f_{2}, f_{3} \in L_{1}(\mathbb {R}).

?
Problem 14.55

Prove that for all f,g∈L1(R)f, g \in L_{1}(\mathbb {R}) and y∈Ry \in \mathbb {R}

F[f∗g](y)=2πF[f](y)F[g](y) F\left[f * g\right](y) = \sqrt{2 \pi } F\left[f\right](y) F\left[g\right](y)
?
Problem 14.56

Prove that

?
(1)

δ0∗φ=φ,δa∗φ=φ(x−a)\delta_{0} * \varphi = \varphi , \delta_{a} * \varphi = \varphi (x-a) for all φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}) and a∈Ra \in \mathbb {R}.

(2)

f∗φ∈C∞(R)f * \varphi \in C^{\infty }(\mathbb {R}) and (f∗φ)(n)=f∗φ(n)=f(n)∗φ(f * \varphi )^{(n)} = f * \varphi^{(n)} = f^{(n)} * \varphi for all n≥1,f∈D′(R)n \geq 1, f \in \mathcal{D}^{\prime }(\mathbb {R}) and φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}).

(3)

δ0(n)∗φ=φ(n)\delta_{0}^{(n)} * \varphi = \varphi^{(n)} for all n≥1n \geq 1 and φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}).

Problem 14.57

Find the fundamental solution of the operator D+aI,a∈RD+a I, a \in \mathbb {R}, and the general solution of the equation dudx+au=φ,φ∈D(R)\frac{d u}{d x}+a u = \varphi , \varphi \in \mathcal{D}(\mathbb {R}).

?
Problem 14.58

Find the fundamental solution of the operator LL and the general solution of the equation Lu=φ,φ∈D(R)L u = \varphi , \varphi \in \mathcal{D}(\mathbb {R}).

?
(1)

L=D2−IL = D^{2}-I;

(2)

L=(D−aI)n,a∈R,n≥2L = (D-a I)^{n}, a \in \mathbb {R}, n \geq 2;

(3)

L=D2+4DL = D^{2}+4 D;

(4)

L=D2−2D+IL = D^{2}-2 D+I;

(5)

L=D2+3D+2IL = D^{2}+3 D+2 I;

(6)

L=D2−4D+5IL = D^{2}-4 D+5 I;

(7)

L=D3−IL = D^{3}-I;

(8)

L=D3−3D2+2DL = D^{3}-3 D^{2}+2 D;

(9)

L=D4−IL = D^{4}-I;

(10)

L=D4−2D2+1L = D^{4}-2 D^{2}+1.

Problem 14.59

Let a1,…,an∈C∞(R),t∈R,z(x,t)a_{1}, \ldots , a_{n} \in C^{\infty }(\mathbb {R}), t \in \mathbb {R}, z(x, t) be the solution of the linear homogeneous differential equation

Lz=z(n)+a1(x)z(n−1)+…+an(x)z=0 L z = z^{(n)}+a_{1}(x) z^{(n-1)}+\ldots +a_{n}(x) z = 0

for which z(t)=z′(t)=…=z(n−2)(t)=0,z(n−1)(t)=1z(t) = z^{\prime }(t) = \ldots = z^{(n-2)}(t) = 0, z^{(n-1)}(t) = 1. Prove that

?
(1)

for any t∈Rt \in \mathbb {R} the function E(x,t)=θ(x−t)z(x,t),x∈R\mathcal{E}(x, t) = \theta (x-t) z(x, t), x \in \mathbb {R}, satisfies in D′(R)\mathcal{D}^{\prime }(\mathbb {R}) the equation LE=δtL \mathcal{E} = \delta_{t} (the function E\mathcal{E} is called the fundamental solution of the operator LL);

(2)

for φ∈D(R)\varphi \in \mathcal{D}(\mathbb {R}) the inhomogeneous differential equation Lu=φ,u∈C∞(R)L u = \varphi , u \in C^{\infty }(\mathbb {R}), has a particular solution u(x)=⟨E(x,t),φ(t)⟩,x∈Ru(x) = \left\langle \mathcal{E}(x, t), \varphi (t) \right\rangle , x \in \mathbb {R};

(3)

if all the coefficients of LL are constant, then E(x,t)=E(x−t,0)\mathcal{E}(x, t) = \mathcal{E}(x-t, 0).

Problem 14.60

Find the fundamental solution E\mathcal{E} of the operator LL.

?
(1)

L=D−2x1+x2L = D-\frac{2 x}{1+x^{2}};

(2)

L=D−2xL = D-2 x;

(3)

L=D+3x2L = D+3 x^{2};

(4)

L=D−sin⁡xL = D-\sin x;

(5)

L=D+cos⁡xL = D+\cos x;

(6)

L=D+4xsin⁡x2L = D+4 x \sin x^{2};

(7)

L=D−arctan⁡x1+x2L = D-\frac{\arctan x}{1+x^{2}};

(8)

L=D−4x3+1L = D-4 x^{3}+1;

(9)

L=D−2xL = D-2^{x};

(10)

L=D+x23x3+1L = D+x^{2} 3^{x^{3}+1}.

Problem 14.61

Do the functions

?
(1)

e−x2e^{-x^{2}},

(2)

e−xe^{-x},

(3)

xe−x2x e^{-x^{2}},

(4)

11+x2\frac{1}{1+x^{2}},

x∈Rx \in \mathbb {R} belong to the space S(R)\mathcal{S}(\mathbb {R})?

Problem 14.62
?
(1)

Let φ∈S(R)\varphi \in \mathcal{S}(\mathbb {R}) and PP be a polynomial. Prove that Pφ∈S(R)P \varphi \in \mathcal{S}(\mathbb {R}).

(2)

Let φ∈S(R)\varphi \in \mathcal{S}(\mathbb {R}) and k≥1k \geq 1. Prove that φ(k)∈S(R)\varphi^{(k)} \in \mathcal{S}(\mathbb {R}).

Problem 14.63

Assume that the function ψ\psi belongs to C∞(R)C^{\infty }(\mathbb {R}), equals zero on (−∞,0)(-\infty , 0) and is bounded together with all its derivatives. Prove that for σ>0\sigma > 0 the function ψ(x)e−σx,x∈R\psi (x) e^{-\sigma x}, x \in \mathbb {R}, belongs to the space S(R)\mathcal{S}(\mathbb {R}).

?
Problem 14.64

Prove that the set S(R)\mathcal{S}(\mathbb {R}) is dense in Lp(R),1≤p<+∞L_{p}(\mathbb {R}), 1 \leq p < +\infty.

?
Problem 14.65

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

?
(1)

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

(2)

{1nφ(nx),x∈R:n≥1}\left\{ \frac{1}{n} \varphi (n x), x \in \mathbb {R}: n \geq 1\right\};

(3)

{1nφ(xn),x∈R:n≥1}\left\{ \frac{1}{n} \varphi \left(\frac{x}{n}\right), x \in \mathbb {R}: n \geq 1\right\}?

Problem 14.66

Prove that δ0∈S′(R)\delta_{0} \in \mathcal{S}^{\prime }(\mathbb {R}) and δ0(k)∈S′(R)\delta_{0}^{(k)} \in \mathcal{S}^{\prime }(\mathbb {R}) for all k≥1k \geq 1.

?
Problem 14.67

Prove that ex2∈D′(R)e^{x^{2}} \in \mathcal{D}^{\prime }(\mathbb {R}), but ex2e^{x^{2}} does not generate a regular tempered distribution from the space S′(R)S^{\prime }(\mathbb {R}).

?
Problem 14.68

Prove that S′(R)⊂D′(R)\mathcal{S}^{\prime }(\mathbb {R}) \subset \mathcal{D}^{\prime }(\mathbb {R}) and the embedding operator is continuous.

?
Problem 14.69

Let f∈S′(R)f \in \mathcal{S}^{\prime }(\mathbb {R}) and PP be a polynomial. Define the functional ⟨Pf,φ⟩=⟨f,Pφ⟩,φ∈S(R)\left\langle P f, \varphi \right\rangle = \left\langle f, P \varphi \right\rangle , \varphi \in \mathcal{S}(\mathbb {R}). Prove that Pf∈S′(R)P f \in \mathcal{S}^{\prime }(\mathbb {R}).

?
Problem 14.70

Prove that if ∣ak∣≤C(1+∣k∣)m\left|a_{k}\right| \leq C(1+\left|k\right|)^{m} for all k∈Zk \in \mathbb {Z}, where C>0,m≥0C > 0, m \geq 0 are some constants, then the series ∑k∈Zakδk\sum_{k \in \mathbb {Z}} a_{k} \delta_{k} converges in S′(R)S^{\prime }(\mathbb {R}),

?
Problem 14.71
?
(1)

Let f:R→Cf: \mathbb {R} \rightarrow \mathbb {C} be a Lebesgue measurable function for which there exists m≥0m \geq 0 such that ∫R∣f(x)∣1+∣x∣mdx<∞\int_{\mathbb {R}} \frac{\left|f(x)\right|}{1+\left|x\right|^{m}} d x < \infty. Prove that ff defines a regular tempered distribution from S′(R)\mathcal{S}^{\prime }(\mathbb {R}) by the formula

⟨f,φ⟩=∫Rf(x)φ(x)dx,φ∈S(R) \left\langle f, \varphi \right\rangle = \int _{\mathbb {R}} f(x) \varphi (x) d x, \varphi \in \mathcal{S}(\mathbb {R})
(2)

Prove that the function f(x)=excos⁡ex,x∈Rf(x) = e^{x} \cos e^{x}, x \in \mathbb {R} does not satisfy the conditions of item (1), but defines a tempered distribution from S′(R)S^{\prime }(\mathbb {R}) by the formula ⟨f,φ⟩=∫−∞+∞excos⁡exφ(x)dx,φ∈S(R)\left\langle f, \varphi \right\rangle = \int_{-\infty }^{+\infty } e^{x} \cos e^{x} \varphi (x) d x, \varphi \in \mathcal{S}(\mathbb {R}). Here the integral should be understood as an improper Riemann integral of the first kind.

Problem 14.72

Prove that for all φ∈S(R)\varphi \in \mathcal{S}(\mathbb {R}) the following equations hold

?
(1)

F[φ(x−a)]=eiayF[φ],a∈RF\left[\varphi (x-a)\right] = e^{i a y} F\left[\varphi \right], a \in \mathbb {R};

(2)

F[φ(k)]=(−iy)kF[φ],k≥1F\left[\varphi^{(k)}\right] = (-i y)^{k} F\left[\varphi \right], k \geq 1;

(3)

(F[φ])(k)=F[(ix)kφ],k≥1(F\left[\varphi \right])^{(k)} = F\left[(i x)^{k} \varphi \right], k \geq 1;

(4)

yk(F[φ])(l)=ik+lF[(xlφ)(k)],k≥1,l≥1y^{k}(F\left[\varphi \right])^{(l)} = i^{k+l} F\left[\left(x^{l} \varphi \right)^{(k)}\right], k \geq 1, l \geq 1.

Problem 14.73

Find the Fourier transforms of the functions

?
(1)

e−x22e^{-\frac{x^{2}}{2}};

(2)

xe−x22x e^{-\frac{x^{2}}{2}};

(3)

x2e−x22x^{2} e^{-\frac{x^{2}}{2}};

(4)

x3e−x22x^{3} e^{-\frac{x^{2}}{2}}

Problem 14.74

Let FF be the Fourier transform in L2(R)L_{2}(\mathbb {R}). Prove that σp(F)=σ(F)={1,i,−1,−i}\sigma_{p}(F) = \sigma (F) = \left\{ 1, i,-1,-i\right\}.

?
Problem 14.75

Let a>0a > 0. Find the Fourier transforms of the functions

?
(1)

θ(x)e−ax\theta (x) e^{-a x};

(2)

e−a∣x∣e^{-a\left|x\right|};

(3)

xe−a∣x∣x e^{-a\left|x\right|};

(4)

2aπ(x2+a2)\frac{\sqrt{2} a}{\sqrt{\pi }\left(x^{2}+a^{2}\right)};

(5)

χ[0,a](x);\chi_{\left[0, a\right]}(x) ;

(6)

χ[−a,a](x)\chi_{\left[-a, a\right]}(x);

(7)

e−x22cos⁡(ax)e^{-\frac{x^{2}}{2}} \cos (a x);

(8)

e−x22sin⁡(ax)e^{-\frac{x^{2}}{2}} \sin (a x).

Problem 14.76

Let FF be the Fourier transform in S′(R)\mathcal{S}^{\prime }(\mathbb {R}). Show that for all f∈S′(R)f \in \mathcal{S}^{\prime }(\mathbb {R}) the following equalities hold

?
(1)

(F[f])(k)=F[(ix)kf],k≥1(F\left[f\right])^{(k)} = F\left[(i x)^{k} f\right], k \geq 1;

(2)

F[f(k)]=(−iy)kF[f],k≥1F\left[f^{(k)}\right] = (-i y)^{k} F\left[f\right], k \geq 1.

Problem 14.77

Prove that

?
(1)

F[δa]=12πeiay,y∈R,a∈RF\left[\delta_{a}\right] = \frac{1}{\sqrt{2 \pi }} e^{i a y}, y \in \mathbb {R}, a \in \mathbb {R};

(2)

F[xk]=2π(−i)kδ0(k),k≥1F\left[x^{k}\right] = \sqrt{2 \pi }(-i)^{k} \delta_{0}^{(k)}, k \geq 1;

(3)

F[δ0(k)]=(−iy)k2π,k≥1F\left[\delta_{0}^{(k)}\right] = \frac{(-i y)^{k}}{\sqrt{2 \pi }}, k \geq 1.

Problem 14.78

Prove that in S′(R)S^{\prime }(\mathbb {R}) the following equalities hold

?
(1)

F[12(δa+δ−a)]=12πcos⁡(ay),a∈RF\left[\frac{1}{2}\left(\delta_{a}+\delta_{-a}\right)\right] = \frac{1}{\sqrt{2 \pi }} \cos (a y), a \in \mathbb {R};

(2)

F[12(δa−δ−a)]=12πsin⁡(ay),a∈RF\left[\frac{1}{2}\left(\delta_{a}-\delta_{-a}\right)\right] = \frac{1}{\sqrt{2 \pi }} \sin (a y), a \in \mathbb {R};

(3)

F[3x2+2x−1]=2π(−3δ0′′+2iδ0′−δ0)F\left[3 x^{2}+2 x-1\right] = \sqrt{2 \pi }\left(-3 \delta_{0}^{\prime \prime }+2 i \delta_{0}^{\prime }-\delta_{0}\right).

Problem 14.79

Find the Fourier transforms of the following tempered distributions

?
(1)

θ(x)\theta (x);

(2)

θ(−x)\theta (-x);

(3)

sign⁡x\operatorname {sign} x;

(4)

∣x∣\left|x\right|;

(5)

P1x\mathcal{P} \frac{1}{x};

(6)

P1x2\mathcal{P} \frac{1}{x^{2}}.

Problem 14.80
?
(1)

Operators A,BA, B are defined by the formulae

(Af)(y)=lim⁡t→+∞∫0tcos⁡(yx)f(x)dx,(Bf)(y)=lim⁡t→+∞∫0tsin⁡(yx)f(x)dx (A f)(y) = \lim _{t \rightarrow +\infty } \int _{0}^{t} \cos (y x) f(x) d x,(B f)(y) = \lim _{t \rightarrow +\infty } \int _{0}^{t} \sin (y x) f(x) d x

f∈L2([0,+∞))f \in L_{2}([0,+\infty )), where the limits are taken in the sense of convergence in L2([0,+∞))L_{2}([0,+\infty )). Prove that A,B∈L(L2([0,+∞)))A, B \in \mathcal{L}\left(L_{2}([0,+\infty ))\right).

(2)

Find the spectra and the eigenvalues of operators AA and BB.

(3)

Prove that all functions of the form f=g±2πAg,g∈L2([0,+∞))f = g \pm \sqrt{\frac{2}{\pi }} A g, g \in L_{2}([0,+\infty )), satisfy the equation f=±2πAff = \pm \sqrt{\frac{2}{\pi }} A f.

(4)

Prove that all functions of the form f=g±2πBg,g∈L2([0,+∞))f = g \pm \sqrt{\frac{2}{\pi }} B g, g \in L_{2}([0,+\infty )), satisfy the equation f=±2πBff = \pm \sqrt{\frac{2}{\pi }} B f.

Problem 14.81

Let f∈L1(R)f \in L_{1}(\mathbb {R}) and g∈L2(R)g \in L_{2}(\mathbb {R}). Prove that

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(1)

the formula (f∗g)(x)=∫Rf(x−y)g(y)dy(f * g)(x) = \int_{\mathbb {R}} f(x-y) g(y) d y, where the integral converges for almost all x∈Rx \in \mathbb {R}, defines a function f∗g∈L2(R)f * g \in L_{2}(\mathbb {R});

(2)

F[f∗g](y)=2πF[f](y)⋅F[g](y)F\left[f * g\right](y) = \sqrt{2 \pi } F\left[f\right](y) \cdot F\left[g\right](y) for almost all y∈Ry \in \mathbb {R}.

Problem 14.82

Let f∈L1(R)f \in L_{1}(\mathbb {R}). The operator A∈L(L2(R))A \in \mathcal{L}\left(L_{2}(\mathbb {R})\right) is defined by the formula Ag=f∗gA g = f * g (the convolution operator with function ff). Find σ(A)\sigma (A).

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Problem 14.83

Prove that the generalized function

⟨f,φ⟩=∫{x2+y2<1}φ(x,y)−φ(0,0)x2+y2dxdy+∫{x2+y2>1}φ(x,y)x2+y2dxdy, \left\langle f, \varphi \right\rangle = \int _{\left\{ x^{2}+y^{2} < 1\right\} } \frac{\varphi (x, y)-\varphi (0,0)}{x^{2}+y^{2}} d x d y+\int _{\left\{ x^{2}+y^{2} > 1\right\} } \frac{\varphi (x, y)}{x^{2}+y^{2}} d x d y,

φ∈D(R2)\varphi \in \mathcal{D}\left(\mathbb {R}^{2}\right), satisfies the equation (x2+y2)f=1\left(x^{2}+y^{2}\right) f = 1 in D′(R2)\mathcal{D}^{\prime }\left(\mathbb {R}^{2}\right).

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Problem 14.84
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(1)

Consider the square [0,1]2\left[0,1\right]^{2} in the plane (x,y)(x, y). Let the function ff be equal to 1 inside this square and to 0 outside of it. Calculate fxy′′f_{x y}^{\prime \prime } in D′(R2)\mathcal{D}^{\prime }\left(\mathbb {R}^{2}\right).

(2)

Consider the square SS with vertices (−1,0),(0,−1),(1,0)(-1,0),(0,-1),(1,0) and (0,1)(0,1). Let the function ff be equal to 1 inside SS and to 0 outside of it. Calculate fyy′′−fxx′′f_{y y}^{\prime \prime }-f_{x x}^{\prime \prime } in D′(R2)\mathcal{D}^{\prime }\left(\mathbb {R}^{2}\right).

Problem 14.85

Do the functions

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(1)

e−∥x∥2e^{-\left\| x\right\|^{2}},

(2)

e−∥x∥e^{-\left\| x\right\| },

(3)

11+∥x∥2\frac{1}{1+\left\| x\right\|^{2}},

(4)

(x12+…+xm2)e−∥x∥2,x∈Rm\left(x_{1}^{2}+\ldots +x_{m}^{2}\right) e^{-\left\| x\right\|^{2}}, x \in \mathbb {R}^{m} belong to the space S(Rm)\mathcal{S}\left(\mathbb {R}^{m}\right)?

Problem 14.86
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(1)

Let μ\mu be a measure on B(Rm)\mathcal{B}\left(\mathbb {R}^{m}\right), which attains finite values on compact sets. Prove that the equality αμ(φ)=∫Rmφ(t)dμ(t),φ∈D(Rm)\alpha_{\mu }(\varphi ) = \int_{\mathbb {R}^{m}} \varphi (t) d \mu (t), \varphi \in \mathcal{D}\left(\mathbb {R}^{m}\right), defines a generalized function αμ∈D′(Rm)\alpha_{\mu } \in \mathcal{D}^{\prime }\left(\mathbb {R}^{m}\right).

(2)

Let μ\mu be a measure on B(Rm)\mathcal{B}\left(\mathbb {R}^{m}\right), for which there exists k∈Nk \in \mathbb {N} such that (1+∥x∥)−k∈L1(Rm,μ)(1+ \left\| x\right\| )^{-k} \in L_{1}\left(\mathbb {R}^{m}, \mu \right) (such a measure is called a tempered measure). Prove that the formula from item (1) defines a tempered distribution αμ∈S′(Rm)\alpha_{\mu } \in S^{\prime }\left(\mathbb {R}^{m}\right).

(3)

Show that not every measure, which attains finite values on compacts, defines αμ∈S′(Rm)\alpha_{\mu } \in \mathcal{S}^{\prime }\left(\mathbb {R}^{m}\right).

Problem 14.87

Prove the following properties of the Fourier transform in S′(Rm)\mathcal{S}^{\prime }\left(\mathbb {R}^{m}\right) :

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(1)

F[f(x−a)](y)=ei(y,a)F[f](y),a∈RmF\left[f(x-a)\right](y) = e^{i(y, a)} F\left[f\right](y), a \in \mathbb {R}^{m};

(2)

F[f](y+b)=F[ei(b,x)f(x)](y),b∈RmF\left[f\right](y+b) = F\left[e^{i(b, x)} f(x)\right](y), b \in \mathbb {R}^{m}.

Problem 14.88

Prove in S′(Rm)S^{\prime }\left(\mathbb {R}^{m}\right) the equalities

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(1)

F[δ0]=1(2π)m2F\left[\delta_{0}\right] = \frac{1}{(2 \pi )^{\frac{m}{2}}};

(2)

F[δa]=ei(y,a)(2π)m2,a∈RmF\left[\delta_{a}\right] = \frac{e^{i(y, a)}}{(2 \pi )^{\frac{m}{2}}}, a \in \mathbb {R}^{m};

(3)

F[1]=(2π)m2δ0F\left[1\right] = (2 \pi )^{\frac{m}{2}} \delta_{0}.

Problem 14.89

Prove in S′(Rm)S^{\prime }\left(\mathbb {R}^{m}\right) the equalities

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(1)

F[Dαδ0]=(−iy)α(2π)m2F\left[D^{\alpha } \delta_{0}\right] = \frac{(-i y)^{\alpha }}{(2 \pi )^{\frac{m}{2}}};

(2)

F[xα]=(2π)m2(−1)∣α∣Dαδ0F\left[x^{\alpha }\right] = (2 \pi )^{\frac{m}{2}}(-1)^{\left|\alpha \right|} D^{\alpha } \delta_{0}, where xα=∏k=1mxkαkx^{\alpha } = \prod_{k = 1}^{m} x_{k}^{\alpha_{k}}.