Math 5467 meets 3 times weekly:
at 12:20 - 1:10pm MWF, VinH 20.
Office Hours: 11:15am-12:05pm MWF and 1:20-1:55 MWF
or by appointment-- or call 5-3855 to see if I'm in and available, especially before class.
You can ask questions by email too: jodeit@math.umn.edu
Please, send me messages in TEXT mode only! Also, here is a Web link to
a list of
special codes for math symbols (most of them start with the \ character).
Announcements, hints, "news" will be here.
Links to PDF documents
News May 16: At 2:24pm I got this note at the Registrar grade site:
Grades have been submitted to the registrar.
Grades will be available to students online in 24 hours.
I wish you all well! Thank you for listening to the math.
I'll be back June 13. If you want to look at your Final Exam, come by
at a Math 4606 Office Hour (not set yet) or call or send email to arrange
a time to meet.
News May 15:Here is the link to
John Hickey's audio denoising project 5/15/05
News May 12b:
Hints for two Exercises in the new "phi to psi" notes were requested:
(S7.5) *** There are two things to do: verify that each set in the union
is in V_1, and verify that the union is orthogonal.
(S10.01) *** Use the "orthonormal basis if the only thing orthogonal
to everything in the proposed basis is zero." By (S7.5) this is all that's
left to do.
Solutions for Test 1 #3, Test 2, #4 and #6 were requested.
Solution of Assgt 8 #2 was requested. These are on the Web,
added to "Some Solutions."
One person asked for solutions of unspecified Exercises in the new version
of the "phi to psi" notes. Be sure to specify which problem, and say why
you're stuck: time to answer grows short!
Here are some questions and answers, or
pointers to answers.
### "Can you clarify the proof of Theorem 6 from Hilbert Space Notes (An
inner product space, with ||v|| as norm, is indeed a normed space)?
In particular, I'm a little confused as to how the Schwarz equality is used
to go from Re <v,w> to ||v||*||w||."
*** Follow the link to "Complex Numbers;" look at (6a) and (6b).
### "In the definition of an IP space, the second condition is set up
logically as <v,v> = 0 => v = 0. Is there a reason why this is not an if
and only if statement?"
*** "v = 0 => <v,v> = 0" follows using "0 times (v=zero vector)=v"
and the scalar multiplication part of the definition. Thus, it need not be
part of the definition.
Solutions for Test 1 #3, Test 2, #4 and #6 were requested.
Solution fro Assgt 8 #2 was requested. These are being prepared and will
be placed on the Web later.
News May 9: The Final Exam is 8am-10am in our class-
room. No books, note, calculators or other electronic devices
are allowed, not even cell phones.
The new version of the Phi-to-Psi notes contain 5 Exercises that
you may want to use as part of your review of recent stuff.
You may ask questions by email. At this point, the answer will
be put on the Web so all can see the question and the answer.
So check this page again.
There is a link for Mr. DeSalvo's use of wavelets to produce
his title page: "DeSalvo Title Page."
News Apr 30: Assignment page: updated.
News Apr 29: Since my "original" for "Assignments" has
disappeared, I'll update it late tonight or tomorrow. The next
Assignment, the last one, is due Wednesday May 4, and consists of
one problem: prepare an Abstract of your project, together with
some sample output (if output is not ready, describe some).
News Apr 28: A major revision of the note
"From a scaling function to a wavelet" is posted.
Changes in the MRA notes were made too, to reflect
the importance of the Low-pass Filter Equations (h1) - (h4)
derived there. That's posted too.
News Apr 26: Homework typo!! Where it says \psi(2^j-t),
it should say \psi(2^j t - k).
The Assignment sheet has been corrected. Thanks to Mr. DeSalvo!
News Apr 24: Some asked that Assignment 11 #1 and #3
be solved for the Web. They are in "Some Solutions."
News Apr 21: The next Assignment is on the Web.
News Apr 19: Someone asked that Test 2 #4
be solved for the Web; that's now in "Some Solutions."
News Apr 15: Someone asked that Assignment 7 #1
be solved for the Web; that's new in "Some Solutions."
An alternate solution is included, that was used in the
computer presentation made Monday, 4/11.
News Apr 14: The new Assignment is on the Web,
as well as a solution for Assgt 10.
News Apr 8: Couldn't get online last night!
News Apr 7:
Someone asked that Assignment 8 #1 be solved for the Web;
that's in "Some Solutions."
News Apr 6a: The new Assignment is on the Web.
Someone asked that Assignment 8 #3 be solved for the Web;
that's in "Some Solutions." A few typos in various notes
have been corrected. A new Assignment will be posted later,
this evening I hope. Good luck preparing for Test 2!
News Apr 4: TEST 2: Friday. A newer version of
"Compact support" is up. Problem #2 on Wednesday's Homework
should refer to V_{j+1} instead of V_j.
The Assignment sheet has been corrected.
"Lebesgue Facts" now contains the Fact (33a) that an interval of
positive length is not a null set.
News Mar 30: Assignments are updated.
TEST 2 has been moved from Monday April 4 to Friday April 8.
News Mar 27: The main part of the put-off argument from Friday's
class is now on the Web as "Compact support." The part not yet done will be
somewhat longer, but it's technical, being a proof that our "good" functions
are dense amongst all the square-integrable functions that vanish in [M_1, M_2].
News Mar 24: Assignments are updated. A new Note,
From a scaling function to a wavelet,
has been added, and some typos in
From "low-pass filter" to scaling function
have been corrected. Test 2: April 4.
News Mar 12: A solution of Special Problem 3 has been added to
"Some Solutions."
News Mar 11a: Assignment 7, #1 had a typo (the first h_3 should be h_2).
A very important typo in the MRA notes has been corrected: on page 6,
in the last displayed "formula" the {n-M\over 2} should say {n+M\over 2}.
There is a new note too, MRA remarks, a discussion of the Definition
of an MRA, that I hope will usefully spell out some details for you.
News Mar 9a: Some typos in the MRA notes have been corrected, all
on page 3: after "periodization argument" the dt's should be d\xi's.
Later on the page, the \pi + 2\pi should be \pi + 2\pi k.
Some more (I hope) clarifying changes have been made to:
From a Lowpass Filter to a Scaling Function.
News Mar 8: Some revisions have been made to
From a Lowpass Filter to a Scaling Function.
News Mar 7: Assignments are updated already, so you can
get your questions in before the break. Test 2 will occur April 4.
News Mar 4a: Thanks to Mr. Jamey for reminding me to post new
notes about getting from a lowpass filer to a scaling function: posted!
The new version of the MRA notes is posted.
News Mar 3: Assignments are updated. You may need to ask questions!
The new scaling function there is the Shannon scaling function.
The new version of the MRA notes will be posted tomorrow.
News Mar 1: A note on Uniform Convergence (and much more!)
is now available.
News Feb 28a: "Some Solutions," link below, is new.
Several typos in From a "Multiresolution Analysis"
to a "Scaling Function" have been corrected. The after-class office
hours have been adjusted. If we are engrossed in a discussion, the
time can easily expand beyond the stated one!!
News Feb 24: The notes on L^1 Fourier Facts are
newly revised. They consist of a list of formulas followed by a
list of their derivations or notes thereon.
News Feb 23: The Assignment page has been updated.
Test 1 had a very high average! That's very unusual.
News Feb 21: The missing part of Exercise (16.1) has been restored.
That is, the Hilbert space notes have been changed to reflect that and other
corrections, primarily on pages one and six.
News Feb 16a: The Assignment page has finally been updated! You can ask
questions by email! The note on "little L^2" has been changed slightly.
(16.1), an Exercise, has been added to the Hilbert space notes. Also,
the observation that every Hilbert space has an orthonormal basis
has been added to the Remarks after (17.1).
Special Problem 2 (the Problem at (9) in Fourier series, v1)
is still due Friday. I've been told that math software works fine, so
it should be quick to do. Your solution will consist of (copy and paste)
the instructions you gave your software and a copy of the graph it produced.
Test 1 is Monday Feb. 21!
Test 1 will cover what has been done thru today, Feb. 16, including the
Homework assignment due today.
News Feb 15: The note on "little L^2" is on the Web now.
News Feb 13: A new version of a note on MRAs has been posted.
We will begin it Monday 2/14.
News Feb 11: More revisions of the Hilbert Space notes and
the Fourier series notes have been posted.
The revisions are at the end of Fourier series but are
scattered in Hilbert spaces: (1) has been slightly reworded;
(1.2) is new; the span of a set is defined on page 4.0; the proof
of (12) has been redone; (13.2), p 6.05, is new; it is a much shorter version
of a proof in class 2/9; (17.1) you know about; (22.1) is new.
News Feb 9: The Assignment page has been updated and posted,
with Assignment 3 added. Revisions of the Hilbert Space notes and
the Fourier series notes have been posted too.
News Feb 7: A new handout, on Fourier series, is posted.
News Feb 4: The Assignment page has been corrected,
with Special Problem 1 restored.
News Feb 3: The Assignment page has been updated and posted,
with Assignment 2 added. The Alternate Proofs of the Schwarz Inequality has
been corrected (in the second alternate proof).
News Feb 1: The newest version of the Inner Product and Hilbert Spaces
note has been posted.
News Jan 28a: An even newer version of the Inner Product and Hilbert Spaces
note has been posted. The Assignment page now has a Special Problem, due 2/4.
News Jan 27: The Assignment page has been updated and posted.
News Jan 26: A newer version of the Inner Product and Hilbert Spaces
note has been posted. Note: The typo-finding rule: it holds only for
handouts, not for solutions/hints or for the assignment page or this
Web page or the Old News page! And the typo has to be in the current
version of the handout!
Assignment 1: Exercises in the Inner Product Spaces and Hilbert Spaces
note marked (1.1), (12.1) and (13.1). The first counts as one problem, the
other two as two problems each. This will be added to the "official" assignment
page soon, but maybe tomorrow.
News Jan 25: The note on equivalence relations has been
updated. A note giving two alternate proofs of the Schwarz Inequality
has also been posted. This page has been re-arranged.
News Jan 24: A note on equivalence relations has been
added.
News Jan 21a: Office Hours (above) have been changed!
We're still working on pages 1 - 5 of
Monday we'll begin with the Schwarz Inequality.
The note on complex numbers has been posted.
News Jan 20: Forgotten yesterday: scheduling Office Hours (the ones posted
were for 2003). We'll decide tomorrow.
Tomorrow I'll talk about inner product and Hilbert spaces. I'll revise the
document
on Inner Product Spaces and Hilbert Spaces (link below)
as we go, influenced by you. The first 5 pages will probably stay
mostly the same, but with more examples, if you think they are needed.
News Jan 16: To begin reading the Bultheel paper, read the Abstract,
skip the History for now. The Motivation can be skimmed. The part thru the
middle of page 3 should not be taken seriously. In particular, we will not
assume familiarity with Fourier analysis. You may find it useful to refer again and
again to pages 3.5 - 5.2.
Section 3 should be rapidly skimmed for now, or put off.
We will begin with a discussion of your backgrounds. The aim is to lead up to
the introduction of the Haar system. A new version of the Introduction has been
posted. Please read it carefully!
This course, Introduction to the Mathematics of Wavelets,
covers Wavelets (via Multiresolution Analysis) and background,
including but not limited to Hilbert Space, Fourier transforms
and (lightly!) Fourier series, and the Fast Fourier Transform.
See the Syllabus for other information (link below, to a PDF document).
To get an early start, you may want to read the Introduction (link below),
try working the exercises there, and glance at the other documents as well.
The most basic of these is the one on Inner Product Spaces and Hilbert Spaces.
The Notes posted below will be updated as we go. The ones there now are
from Spring 2003, and they were posted "top down," so the earliest one is
at the bottom.
There is no "official" Text yet. My intention is to let the Notes,
PDF documents linked to this page below, serve as our Text, along with a
survey paper by Adhemar Bultheel. The Syllabus has a short list of books.
Some of these (and others) I've tried and found too advanced or not advanced
enough. The one that tempted me this time was
An Introduction to Wavelets Through Linear Algebra, by Michael W. Frazier.
But thanks to John Hickey, who found that a survey article that I like,
Learning to swim in a sea of wavelets, by Adhemar Bultheel,
Bull. Belg. Math. Soc. 2 (1995) 1-44, can be downloaded, I decided to see
whether the class would like to use that article and the other Notes as the
text this semester.
Links, if any, to other Web pages pages related to this course
will go here.
The links below are to PDF files (you need
Adobe Acrobat Reader to view them).
Follow this link to download the latest Acrobat Reader
from Adobe at no charge.
This is the "Express" URL
Math 5467, S'05, Syllabus. 1/2/05
Assignments 4/30/05
"Learning to swim in a sea of wavelets," by Adhemar Bultheel, Bull. Belg. Math. Soc. 2 (1995) 1-44 Our text?
"DeSalvo Title Page" 5/9/05
Compact Support 4/6/05
From a scaling function to a wavelet 4/28/05
MRA remarks 3/11/05
From a Lowpass Filter to a Scaling Function 3/24/05
From a "Multiresolution Analysis" to a "Scaling Function" 4/28/05
On Uniform Convergence (and more!) 3/1/05
Some Solutions 5/12/05
Fourier Transform Facts: L^1 2/24/05
On "little L^2" 2/16/05
Fourier series, v1 2/11/05
more Schwarz Inequality proofs 2/3/05
Equivalence relations 1/25/05
Complex numbers 1/21/05
Introduction 1/16/05
The Gram-Schmidt "orthogonalization" process
From a "Scaling Function" to a "Wavelet:" 4/22/03
From the Low-pass filter to a "Scaling Function" 4/3/03
The Lebesgue facts 4/4/05
The L^2 Fourier Transform Exists!
The document below gives a mathematical introduction to the kinds of vector spaces that wavelets "live" in.
A Brief Introduction to Inner Product Spaces and Hilbert Spaces. 2/21/05
