Thanks for mentioning the Icosahedral Snyder Equal Area projection — it’s news to me.

I have to ask: drupal pages?

(And no, this is ordinary, free WordPress.)

Rip

Btw: regarding 12 pentagons, you might find it interesting to look up/find out about the ISEA DGGs (Icosahedral Snyder Equal Area projection, for Discrete Global Grids).

I’ll be reading your stuff in the next little while…
(ps: have you somehow hacked wordpress so it serves up drupal pages?!!)

You’re welcome. I try to put out a new post every weekend — usually Sunday, but sometimes Monday evening.

Feel free to ask questions about what’s on the blog. (But I can’t promise to answer questions about _other_ material.)

I don’t know why it took me so long to remember my manners.

You’re quite welcome.

Thanks for the comment — and please feel free to ask questions.

Rip

Thanks again for asking; as I said, I learned something, too: that Daubechies’ 6 assumptions are redundant.

Rip

Thanks again for your help,

Andrei

As I understand it, you have asked one question: why are the phi_0,k in V0? And you have raised another: why is the invariance of V0 sometimes omitted?

First question. You said…

“…. in the definition of an MRA, the condition of invariance under integer translation for the space V0 is not listed, but after it is said that the functions phi_0,k=phi(x-k) (defined just like in daubechies) are all in V0 and they say that this follows from the fact that the integer translates of phi are an orthonormal basis for V0 (in your text, condition (5.1.6) ). this is not at all clear, because it’s not specified that a basis for a subspace contains elements from the subspace itself, not from L2(R).”

Well… they must be. The vectors that make up a basis for a space S are themselves in the space S — even if we didn’t assume that, each one of them is trivially in the span of the basis (since each is 1 times itself plus zeroes times the other basis vectors).

“in many books it’s also required that V0 is invariant under integer translation, but in others this condition is missing. can you give me an argument that the functions phi_0,k are in V0 for all integers k without assuming that V0 is invariant under integer translations?”

Sure; I just did. That the phi_0,k are a basis for V0 implies the phi_0,k are in V_0. I need not even think about whether V0 invariant. (You’re asking for such an argument because they didn’t give us invariance; don’t worry about it.)

The real insight is this: that the phi_0,k are a basis also implies that V_0 is invariant under integer translation: if f(t) is any function in V_0, then f(t-k) is also in V_0 for all integer k.

Say it loud: condition 5.1.5 is redundant; it follows from 5.1.6. That’s why it’s often omitted!

I had not realized that. Hernandez and Weiss did not say it was redundant — they simply didn’t list it! — but they provide a reference (Madych, “Some Elementary Properties of Multiresolution Analysis of L2(Rn)” in Chui, “Wavelets: A Tutorial in Theory and Applications” ISBN 0-12-174590-2) that does say it explicitly. (And, as we would expect, Daubechies herself did not say that 5.1.5 was redundant, not in the “Ten Lectures”.)

Madych doesn’t prove that 5.1.5 is redundant (“It is clear…”) but once said, it does seem clear to me. If f(t) is in V_0, then it has an expansion in terms of the basis. We have basis functions and coefficients. Now slide the basis functions by k (replace a_j phi_j(t) by a_j phi_j(t-k) ), and we’re looking at the expansion of f(t-k). Since f(t-k) is in the span of the basis, it’s in V_0.

If this isn’t clear — or if I’ve made a mistake — please let me know.

Thanks again for asking.

Rip

i found by chance your website when i was looking for something that’s bothering me lately. i’m reading hernandez & weiss’ “first course in wavelets” for my ms degree final exam. in the definition of an MRA, the condition of invariance under integer translation for the space V0 is not listed, but after it is said that the functions phi_0,k=phi(x-k) (defined just like in daubechies) are all in V0 and they say that this follows from the fact that the integer translates of phi are an orthonormal basis for V0 (in your text, condition (5.1.6) ). this is not at all clear, because it’s not specified that a basis for a subspace contains elements from the subspace itself, not from L2(R). in many books it’s also required that V0 is invariant under integer translation, but in others this condition is missing.
can you give me an argument that the functions phi_0,k are in V0 for all integers k without assuming that V0 is invariant under integer translations?
i would appreciate very much your answer.
thank you.

andrei