Explicit computation of convolutions

[9/67%]

Show

LaTeX

Show LaTeX

Problem 18.4 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 18.1 "Convolution of functions in $L^1(\mathbb{R})$"

Compute the convolution of the characteristic functions $\chi_{[1,2]}(x) * \chi_{[3,4]}(x)$ .

Problem 18.5 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 18.1 "Convolution of functions in $L^1(\mathbb{R})$"

Compute the convolution $f(x) * \chi_{[2,4]}(x)$ , where

f(x) = \begin{cases} x+1, & -1 \leqslant x \leqslant 0 \\ 1-x, & 0 < x \leqslant 1 \end{cases}

Problem 18.6 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 18.1 "Convolution of functions in $L^1(\mathbb{R})$"

Prove that the convolution $\omega_{a, b} * \chi_{[c, d]}$ , where $\omega_{a, b}$ is the "bump" from problem 15.1, is a function of the "hat" type from problem 15.2.

Problem 3837 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.3 "Change of variable in improper integrals. Differentiation and integration under the integral sign for improper integrals"

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

Find the Weierstrass transform

F(x)=\dfrac {1}{\sqrt{\pi }}\int _{-\infty }^{+\infty }e^{-(x-y)^{2}}f(y)\, dy .

Let:

(a)

$f(y)=1$ ;

(b)

$f(y)=y^{2}$ ;

(c)

$f(y)=e^{2ay}$ ;

(d)

$f(y)=\cos ay$ .

Problem 3839 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 7.3 "Change of variable in improper integrals. Differentiation and integration under the integral sign for improper integrals"

Compute the integral, which is of significant importance in probability theory,

\varphi (x)=\dfrac {1}{2\pi \sigma _{1}\sigma _{2}}\int _{-\infty }^{+\infty }e^{-\frac{\xi ^{2}}{2\sigma _{1}^{2}}}e^{-\frac{(x-\xi )^{2}}{2\sigma _{2}^{2}}}\, d\xi \quad (\sigma _{1}>0,\ \sigma _{2}>0) .

Problem 521 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.1 "Convolutions of Test Functions"

(a)(b)

Let $f_1$ and $f_2$ be locally integrable functions on $\mathbb {R}$ whose supports are bounded to the left. Then the convolution $f_1*f_2$ can be defined, and it has the same property. Compute explicitly the convolutions:

(a)

$[\theta (x)x^\alpha ]*[\theta (x)x^\beta ]$ ;

(b)

$[e^{ax}\theta (x)]*[e^{bx}\theta (x)]$ .

Problem 2.9.2 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.9 "Fourier Transforms"

Compute the Fourier transform of the characteristic function of an interval. For $n = 1,2,3, \ldots$ , let $g_{n}$ be the characteristic function of $[-n, n]$ , let $h$ be the characteristic function of $[-1,1]$ , and compute $g_{n} * h$ explicitly. (The graph is piecewise linear). Show that $g_{n} * h$ is the Fourier transform of a function $f_{n} \in L^{\mathbf{1}}$ ; except for a multiplicative constant,

f_{n}(x) = \frac{\sin x \sin n x}{x^{2}}

Show that $\left\| f_{n}\right\|_{1} \rightarrow \infty$ and conclude that the mapping $f \rightarrow \hat{f}$ maps $L^{1}$ into a proper subset of $C_{0}$ . Show, however, that the range of this mapping is dense in $C_{0}$ .

Problem 10.5.8 from Vestrup E.M. "The Theory of Measures and Integration" [Ves03]More problems in Section 10.5 "Convolutions"

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

The following parts give some concrete examples of convolutions courtesy of Jones, pages 278-279.

(a)

Let $0 < b \leq a$ . Let $f = \; \mathbb {1}_{(-a, a)}$ and $g = \; \mathbb {1}_{(-b, b)}$ . Then for each $x \in \mathbb {R}$

f * g(x) = \begin{cases} 2b & \left|x\right| \leq a - b \\ a + b - \left|x\right| & a - b \leq \left|x\right| \leq a + b \\ 0 & a + b \leq \left|x\right| . \end{cases}

(b)

Let $a$ and $b$ denote distinct real numbers. Let $f(x) = e^{-ax} \; \mathbb {1}_{[0, +\infty )}(x)$ and $g(x) = e^{-bx} \; \mathbb {1}_{[0, +\infty )}(x)$ for each $x \in \mathbb {R}$ . Then

f * g(x) = \begin{cases} \dfrac {e^{-ax} - e^{-bx}}{b - a} & x \geq 0 \\ 0 & x < 0 . \end{cases}

(c)

If in (b) we have $a = b$ , then $f * g(x) = x e^{-ax}$ for each $x \in \mathbb {R}$ .

(d)

If $a + b > 0$ , $f(x) = e^{-ax^{2}}$ , and $g(x) = e^{-bx^{2}}$ [for all $x \in \mathbb {R}$ ], then

f * g(x) = \sqrt{\frac{\pi }{a + b}} \, e^{-\frac{ab}{a + b} x^{2}} \quad \text{for all } x \in \mathbb {R} .

Problem 3.4 from Gallager R.G. "Stochastic Processes: Theory for Applications" [Gal13]More problems in Chapter 3 "Gaussian random vectors and processes"

(a)(b)(c)

(a)

Let $X_1 \sim N(0,\sigma_1^2)$ and let $X_2 \sim N(0,\sigma_2^2)$ be independent of $X_1$ . Convolve the density of $X_1$ with that of $X_2$ to show that $X_1+X_2$ is Gaussian, $N(0,\sigma_1^2+\sigma_2^2)$ .

(b)

Let $W_1, W_2$ be IID normalized Gaussian rv s. Show that $a_1W_1+a_2W_2$ is Gaussian, $N(0,a_1^2+a_2^2)$ .

(c)

Combine (b) with induction to show that all linear combinations of IID normalized Gaussian rv s are Gaussian.

References

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

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

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

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.

[Ves03] Vestrup E.M. The Theory of Measures and Integration. Wiley-Interscience, 2003.

[Gal13] Gallager R.G. Stochastic Processes: Theory for Applications. Cambridge University Press, 2013.