Math 5467, Spring 2000: HW 4: solutions of # 2 and # 4
2: Verify that the space V0 generated by the integer translates of the triangle scaling function T (t) shown
in gure 2.2(b) generates a multiresolution analysis, even though the translates of the tria
Math 5467, Spring 2000
Homework 1, numbers 3(iii) and 4.
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3: For x R, dene H (x) := 1 if 0 < x < 1/2, H (x) := 1 if 1/2 < x < 1, and let H (x) := 0 for all other x.
(i) Find j, k so that Hjk = H.
(ii) Shetch the graphs of H2,1 , H2,1 , and H3,1 .
(i
Math 5467, Spring 2000: Sample Questions: Test 2
What are the formulas for the Fourier transforms of f (t h) and f (t), > 0, in terms of f ( )?
What are the formulas for the Fourier transforms of f (t) and f (t), in terms of f ( )?
State Plancherels Formu
Math 5467
Spring 00
Fourier Transform Facts
3/2/00
p1
A list of Fourier Transform formulas and properties
The list that follows is derived or obtained from references (in the same order) following the list.
f (x)eix dx, where
f ( ) :=
(1)
(1 )
|f (x)| dx
Math 5467, Spring 2000
Homework 2, numbers 2(b) and 3.
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2(b): Suppose (for this part) f (t) dt = 0. Explain why it is impossible for cjk = 0 to be true for all j suciently
negative. The question behind this: what is the rle of the very-low-frequency
Math 5467, Spring 2000
Homework 3, numbers 1 and 2.
Page 1
1:
Find the element of V0 that is closest to H00 + H11 . Is there a simple recipe for solving this sort of
minimization problem? If so, what is it?
The solution that I had in mind (there are many
Math 5467, Spring 2000: Assignments
Assignments are due at the start of class on the due date.
Give credit for help received, including books and hints from me and others; mention discussions.
If a problem is dicult, please include a narrative that tells
Math 5467
Spring 98
Checking a dierentiability hypothesis
We have
i 2
F (, x) = e(x+ 2 ) = ex eix e
2
2
4
and we want to show that, for each , there is an integrable function g (x) such that, for all suciently close to
zero (well assume that | | 1),
F ( +
Math 5467, Spring 2000
Homework 5, number3. Remarks on 1, 2, 4.
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3: The Shannon scaling function is (t) = sinc(t) = sin t . Find ( ) and the coecients h(n) for its dilation
t
equation. Find the formula (not a series) for the Shannon wavelet.
A solut
Math 5467
Spring 00
Lebesgue theory - an overview
Page 1
Introduction
The theory of the Lebesgue integral is more complicated to learn that is the theory of the Riemann integral. But it
has distinct advantages, both theoretical and practical. First, a fun