Convolution operators

[12/67%]

Show

LaTeX

Show LaTeX

Problem 11B.10 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11B "Fourier Series and $L^{p}$ of Unit Circle"

Suppose $f \in L^{1}(\partial \mathbf{D})$ . Define $T: L^{2}(\partial \mathbf{D}) \rightarrow L^{2}(\partial \mathbf{D})$ by $T g = f * g$ .

(a)

Show that $T$ is a compact operator on $L^{2}(\partial \mathbf{D})$ .

(b)

Prove that $T$ is injective if and only if $\widehat{f}(n) \neq 0$ for every $n \in \mathbf{Z}$ .

(c)

Find a formula for $T^{*}$ .

(d)

Prove: $T$ is self-adjoint if and only if all Fourier coefficients of $f$ are real.

(e)

Show that $T$ is a normal operator.

Problem 11C.19 from Axler S. "Measure, Integration & Real Analysis" [Axl19]More problems in Section 11C "Fourier Transform"

(a)(b)

Suppose $y > 0$ . Define on operator $T$ on $L^{2}(\mathbb {R})$ by $T f = f * P_{y}$ .

(a)

Show that $T$ is a self-adjoint operator on $L^{2}(\mathbb {R})$ .

(b)

Show that $\operatorname {sp}(T) = [0,1]$ .

Remark.

Because the spectrum of each compact operator is a countable set (by 10.93), part (b) above implies that $T$ is not a compact operator. This conclusion differs from the situation on the unit circle-see Exercise 9 in Section 11B.

Problem 10.90 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 10.2 "Problems to Solve"

Let $K: \mathbb {R} \rightarrow \mathbb {C}$ be a $2 \pi$ -periodic Lebesgue measurable function and $K \in L_{1}(\left[0,2 \pi \right])$ . Prove that for $1 < p < +\infty$ the operator

(A x)(t) = \int _{0}^{2 \pi } K(t-s) x(s) d s, t \in [0,2 \pi ], x \in L_{p}(\left[0,2 \pi \right])

is compact.

Problem 10.92 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 10.2 "Problems to Solve"

Let $u \in L_{1}(\mathbb {R})$ . Prove that the operator

(A x)(t) = \int _{\mathbb {R}} u(t-s) x(s) d s, t \in \mathbb {R}, x \in L_{2}(\mathbb {R})

is linear and continuous in $L_{2}(\mathbb {R})$ . Under what conditions on the function $u$ is the operator $A$ compact?

Problem 10.93 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 10.2 "Problems to Solve"

Let $u \in L_{1}([0,+\infty ))$ and $(A x)(t) = \int_{0}^{t} u(t-s) x(s) d s, t \in [0,+\infty ), x \in L_{2}([0,+\infty ))$ . Prove that $A \in \mathcal{L}\left(L_{2}([0,+\infty ))\right)$ . Under what conditions on the function $u$ is the operator $A$ compact?

Problem 13.30 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 13.2 "Problems to Solve"

(1)(2)

(1)

Find all $\lambda \in \mathbb {R}$ , for which the homogeneous integral equation $x(t) = \lambda \int e^{-\left|t-s\right|} x(s) d s, t \in \mathbb {R}$ , has a nonzero solution $x \in L_{\infty }(\mathbb {R})$ .

(2)

Prove that for all $\lambda < \frac{1}{2}$ and $y \in L_{\infty }(\mathbb {R})$ the integral equation $x(t) = \lambda \int e^{-\left|t-s\right|} x(s) d s+y(t), t \in \mathbb {R}$ , has a unique solution in the space $L_{\infty }(\mathbb {R})$ , which is expressed by the formula

x(t) = \frac{\lambda }{\sqrt{1-2 \lambda }} \int _{\mathbb {R}} e^{-\sqrt{1-2 \lambda }\left|t-s\right|} y(s) d s+y(t), t \in \mathbb {R}

Problem 14.82 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"

Let $f \in L_{1}(\mathbb {R})$ . The operator $A \in \mathcal{L}\left(L_{2}(\mathbb {R})\right)$ is defined by the formula $A g = f * g$ (the convolution operator with function $f$ ). Find $\sigma (A)$ .

Problem II.4.12 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section II.4 "Compact Operators"

Verify the statement in Example 4.12: Let $h \in \mathcal{H} = L_{\mathbb {C}}^{2}(-\pi , \pi )$ and define $K: \mathcal{H} \rightarrow \mathcal{H}$ by $(Kf)(x) = (2\pi )^{-1/2} \int_{-\pi }^{\pi } h(x-y) f(y) \, dy$ . If $\lambda_{n} = (2\pi )^{-1/2} \int_{-\pi }^{\pi } h(x) \exp (-inx) \, dx = \hat{h}(n)$ , the $n$ th Fourier coefficient of $h$ , then $Ke_{n} = \lambda_{n} e_{n}$ , where $e_{n}(x) = (2\pi )^{-1/2} \exp (-inx)$ . (Note that the operator $K$ in this example is diagonalizable.)

Problem 4.3.21 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 4.3 "Space of Integrable Functions"

(a)(b)

For functions $f, g \in L^2(\mathbb {R})$ , define the convolution $f*g$ by

f*g(x) = \int _{-\infty }^{+\infty } f(y) g(x-y) \, dy,

where the integral is with respect to Lebesgue measure.

(a)

Show that $f*g \in L^{\infty }(\mathbb {R})$ . Do not neglect to check the measurability of $f*g$ .

(b)

Suppose that $f$ has the property that for all $g \in L^2(\mathbb {R})$ the convolution $f*g$ is also an element of $L^2(\mathbb {R})$ . Define $T_f : L^2(\mathbb {R}) \to L^2(\mathbb {R})$ by $T_f(g) = f*g$ . It is evident that $T_f$ is a linear operator; you need not check this. Show that $T_f$ is a bounded linear operator.

Problem 509 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.1 "Convolutions of Test Functions"

Suppose that $L$ is a Banach space of functions on a locally compact abelian group $G$ with invariant measure $\mu$ , with a norm $\left\| \cdot \right\|_{L}$ having the properties:

$\left\| T(a)f\right\|_{L}=\left\| f\right\|_{L}$ , where $T(a)$ is the translation operator $T(a)f(x)=f(x+a)$ ;
$\left\| T(a)f-f\right\|_{L}\to 0$ as $a\to 0$ in $G$ . Prove that the operator $S(\varphi )$ , $\varphi \in L^{1}\left({G,\mu }\right)$ , carries $L$ into itself and has a norm in this space not exceeding $\left\| \varphi \right\|_{L}$ .

Problem 514 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.1 "Convolutions of Test Functions"

Let $f\in L^{1}\left({G,\mu }\right)$ . Prove that $S(f)$ is a bounded operator in the Hilbert space $L^{2}\left({G,\mu }\right)$ . Compute the adjoint operator $S(f)^*$ .

Problem 694 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection V.2.1 "Reduction of an Operator to Multiplication by a Function"

(a)(b)(c)

Let $G$ be a locally compact abelian group with invariant measure $\mu$ . Determine conditions on a function $f\in L^{1}\left({G,\mu }\right)$ under which the convolution operator $S(f)$ is:

(a)

selfadjoint;

(b)

unitary;

(c)

compact.

References

[Axl19] Axler S. Measure, Integration & Real Analysis. 3rd ed., Springer, 2019.

[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.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

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