Convolution of a distribution with a test function

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Problem 14.56 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)

Prove that

(1)

$\delta_{0} * \varphi = \varphi , \delta_{a} * \varphi = \varphi (x-a)$ for all $\varphi \in \mathcal{D}(\mathbb {R})$ and $a \in \mathbb {R}$ .

(2)

$f * \varphi \in C^{\infty }(\mathbb {R})$ and $(f * \varphi )^{(n)} = f * \varphi^{(n)} = f^{(n)} * \varphi$ for all $n \geq 1, f \in \mathcal{D}^{\prime }(\mathbb {R})$ and $\varphi \in \mathcal{D}(\mathbb {R})$ .

(3)

$\delta_{0}^{(n)} * \varphi = \varphi^{(n)}$ for all $n \geq 1$ and $\varphi \in \mathcal{D}(\mathbb {R})$ .

Problem 15.33 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Chapter 15 "Test (basic) function spaces"

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

(A \varphi )(t) = \int _{\mathbb {R}} \varphi (s) \alpha (t-s) d s

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.

Problem 18.21 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 18.3 "Convolution of generalized functions"

Prove that for a generalized function $G \in \mathcal{E}^{\prime }$ the map $\varphi \rightarrow G * \varphi$ defines a continuous operator $S_{G}: \mathcal{D} \rightarrow \mathcal{D}$ and the inclusion holds:

\operatorname {supp}(G * \varphi ) \subset \operatorname {supp} G+\operatorname {supp} \varphi = \left\{ x+y: x \in \operatorname {supp} G, y \in \operatorname {supp} \varphi \right\} .

Problem 18.22 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 18.3 "Convolution of generalized functions"

Prove that if a regular generalized function $G = G_{\varphi }$ is generated by a test function $\varphi$ , then for any generalized function $F$ the convolution $F * G$ is a regular function, generated by the convolution $F * \varphi$ (which is computed by Definition 18.2).

Definition 18.2: For any test function $\varphi \in \mathcal{D}$ (or $\mathcal{S}$ , or $\mathcal{E}$ ) its convolution with a generalized function $F \in \mathcal{D}^{\prime }$ (respectively, $\mathcal{S}^{\prime }$ , $\mathcal{E}^{\prime }$ ) as a function of the variable $x$ is defined as follows: $(F * \varphi )(x) := \left\langle F(t), \varphi (x-t) \right\rangle$ .

Problem 524 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.2 "Convolutions of Generalized Functions"

Prove the identity $f*1=\langle f,1\rangle \cdot 1$ for any generalized function $f\in \mathcal{E}'(\mathbb {R})$ .

Problem 526 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.2 "Convolutions of Generalized Functions"

(a)(b)(c)

Let $f\in S'(\mathbb {R}^{n})$ , $\varphi \in S(\mathbb {R}^{n})$ . The convolution $f*\varphi$ is defined by the formula $f*\varphi =S(\check\varphi )'f$ , i.e., $\langle f*\varphi ,\psi \rangle =\langle f,\varphi *\psi \rangle$ . Prove that:

(a)

$f*\varphi$ is a regular generalized function;

(b)

$(f*\varphi )(x)=\langle f,T(-x)\check\varphi \rangle$ ;

(c)

$(f*\varphi )(x)$ is not of greater than polynomial growth in $\left|x\right|$ at infinity.

Problem 527 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.2 "Convolutions of Generalized Functions"

(a)(b)(c)(d)

Let $f\in \mathcal{E}'(\mathbb {R}^{n})$ , $\varphi \in \mathcal{E}(\mathbb {R}^{n})$ . The convolution $f*\varphi$ is defined by the formula $f*\varphi =S(\check\varphi )f$ (cf. Problem 526). Prove that:

(a)

$f*\varphi$ is a regular generalized function;

(b)

$(f*\varphi )(x)=\langle f,T(-x)\check\varphi \rangle$ ;

(c)

$f*\varphi \in \mathcal{E}(\mathbb {R}^{n})$ ;

(d)

the operator $S(f):\mathcal{E}(\mathbb {R}^{n})\to \mathcal{E}(\mathbb {R}^{n})$ is continuous.

Problem 529 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.2 "Convolutions of Generalized Functions"

Let $\mathcal{E}(\mathrm{T}^n)$ be the space of infinitely differentiable functions on the torus $\mathrm{T}^n$ (with the topology of uniform convergence of all derivatives), and $\mathcal{E}'(\mathrm{T}^n)$ the dual space of generalized functions. Define a convolution operation in $\mathcal{E}'(\mathrm{T}^n)$ and prove that $\mathcal{E}'(\mathrm{T}^n)*\mathcal{E}(\mathrm{T}^n)\subset \mathcal{E}(\mathrm{T}^n)$ .

Problem 3.6.24 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.6 "Test Functions and Distributions"

(a)(b)(c)(d)

Let $H$ be the Heaviside function on $R$ , defined by

H(x) = \begin{cases} 1 & \text{ if } x > 0 \\ 0 & \text{ if } x \leq 0\end{cases}

and let $\delta$ be the Dirac measure.

(a)

Show that $(H * \phi )(x) = \int_{-\infty }^{x} \phi (s) d s$ , if $\phi \in \mathscr {D}(R)$ .

(b)

Show that $\delta^{\prime } * H = \delta$ .

(c)

Show that $1 * \delta^{\prime } = 0$ . (Here 1 denotes the locally integrable function whose value is 1 at every point and which is thought of as a distribution).

(d)

It follows that the associative law fails:

1 *\left(\delta ^{\prime } * H\right) = 1 * \delta = 1

but

\left(1 * \delta ^{\prime }\right) * H = 0 * H = 0

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.

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