To prove the above, we may use the fact that the space of polynomials with nite order is dense in the space of
continuous functions on a compact set under the sup norm. One example is the Bernstein polynomials, dened
as follows: Let for some f C([0, 1]) a
Substitution leads to:
A
(f )ei2f t df
lim
A
f =A
A
( )ei2f d ei2f t df
= lim
A
f =A
A
= lim
ei2f ei2f t d df
( )
A
f =A
A
= lim
ei2f ( t) df d
( )
A
(4.3)
f =A
A
1
By Lemma 4.5.1, the fact that f =A ei2f ( t) df = (t ) sin(2A(t ) represents a distributio
Chapter 1
Banach and Hilbert Signal Spaces and
Representation of Signals
In this chapter, some of the discussions we had in class are presented in a compactied form, with some additional
topics included, although many of these were already covered in Math
Let X be a normed space of signals and let us dene the dual space of X as the set of linear and continuous
functions on X to R or C, and let us denote this space by X . This space is a linear space, under pointwise
addition and scalar multiplication of fu
Theorem 2.2.1 For any singular distribution, there exists a sequence of regular distributions which converges
to the singular distribution.
The proof of this argument follows from the following. Let fn be an approximate identity sequence so that
fn S. The
and call this value the frequency response of the system for frequency f , whenever it exists.
A similar discussion applies for a discrete-time system.
Let u l (Z; R). If u(n) = ei2f n , then
y(n) =
h(m)ei2f m ei2f m
m=
with the frequency response functio
Queens University
Mathematics and Engineering and Mathematics and Statistics
MTHE/MATH 335
Mathematics of Engineering Systems
Supplemental Course Notes - Winter 2013
Serdar Y ksel
u
March 18, 2013
This document is generated to supplement the textbook (by
MTHE/MATH 335
Winter 2013
Sample Midterm II
There are four questions.
Be as neat as possible, clearly state your results
Student Number:
Problem 1 [40 Points]
Let x, y f (Z; R); that is, x, y map Z to R, such that x = cfw_. . . , x2 , x1 , x0 , x1 , x2 ,
MTHE/MATH 335
Winter 2013
Sample Midterm I
There are four questions.
Be as neat as possible, clearly state your results
Student Number:
Problem 1 [40 Points]
Let X be a Hilbert space and x, y X. Prove the following:
a) [15 Points]
|x + y|2 + |x y|2 = 2|x|
MTHE/MATH 335
Winter 2013
Sample Midterm III
There are four questions.
Be as neat as possible, clearly state your results
Student Number:
Problem 1 [30 Points]
Consider a discrete-time system described by the equation:
y(n + 1) = a1 y(n) + a2 y(n 1) + u(n