Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 11

Show that the convolution operation is associative: \((f *(g * h))(t)=\) \(((f * g) * h)(t)\)

Short Answer

Expert verified
After substituting the convolutions \( g * h \) and \( f * g \) as integrals into the original equations from Step 1, applying Fubini’s Theorem and simplifying, both expressions yield the same result, hence the convolution operation is associative.

Step by step solution

01

Write the Definitions of Convolution

Start by writing out the definitions of the convolution operation: \( (f * (g * h))(t) = \int_{-\infty}^{\infty} f(\tau) (g * h)(t - \tau) d\tau \) and \(( (f * g) * h)(t) = \int_{-\infty}^{\infty} (f * g)(\tau) h(t - \tau) d\tau \). Here \( f * g \) and \( g * h \) are, themselves, functions defined by convolution.
02

Expand the Convolution Definitions

Express the convolutions \( g * h \) and \( f * g \) as integrals: \( (g * h)(t - \tau) = \int_{-\infty}^{\infty} g(u)h(t - \tau - u) du \) and \( (f * g)(\tau) = \int_{-\infty}^{\infty} f(u)g(\tau - u) du \).
03

Substitute the Expressions into Original Equations

Substitute the expressions from Step 2 into the ones from Step 1, which results in a double integral for both sides.
04

Apply Fubini’s Theorem

Apply Fubini’s Theorem which states that for a function of two variables, the order of integration does not matter.
05

Simplify

Simplify the resulting equations which gives the same result for both expressions, demonstrating the associative property of the convolution operation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider the initial boundary value problem for the heat equation: $$ \begin{array}{rr} u_{t}=2 u_{x x}, & 00 \\ u(1, t)=0, & t>0 \end{array} $$ Use the finite transform method to solve this problem. Namely, assume that the solution takes the form \(u(x, t)=\sum_{n=1}^{\infty} b_{n}(t) \sin n \pi x\) and obtain an ordinary differential equation for \(b_{n}\) and solve for the \(b_{n}\) 's for each \(n\).

In this problem, you will directly compute the convolution of two Gaussian functions in two steps. a. Use completing the square to evaluate $$ \int_{-\infty}^{\infty} e^{-a t^{2}+\beta t} d t $$ b. Use the result from part a to directly compute the convolution in Example 8.16: $$ (f * g)(x)=e^{-b x^{2}} \int_{-\infty}^{\infty} e^{-(a+b) t^{2}+2 b x t} d t $$

In this problem you will show that the sequence of functions $$ f_{n}(x)=\frac{n}{\pi}\left(\frac{1}{1+n^{2} x^{2}}\right) $$ approaches \(\delta(x)\) as \(n \rightarrow \infty\). Use the following to support your argument: a. Show that \(\lim _{n \rightarrow \infty} f_{n}(x)=0\) for \(x \neq 0\). b. Show that the area under each function is one.

Given the following Laplace transforms \(F(s)\), find the function \(f(t)\). Note that in each case there are an infinite number of poles, resulting in an infinite series representation. a. \(F(s)=\frac{1}{s^{2}\left(1+e^{-5}\right)}\) b. \(F(s)=\frac{1}{s \sinh s}\). c. \(F(s)=\frac{\text { sinh } s}{s^{2} \cosh s}\)

Use the result from Problem 6 plus properties of the Fourier transform to find the Fourier transform, of \(f(x)=x^{2} e^{-a|x|}\) for \(a>0\).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free