Orthogonality Problem (From Fourier Analysis Text)

[GridironCPJ]

Hello all,

I'm working through a majority of the problems in "A First Course in Wavelets with Fourier Analysis" and have stumbled upon a problem I'm having difficulty with. Please view the PDF attachment, it shows the problem and what I have done with it so far.

Once you have seen the problem and my work so far, what do you suggest I do for the next step? I thought about using the definition of the space D and setting everything less than or equal to the double integral of 1^(j+k_/2, which is just the double integral of 1 and you get xy, then I'm at a dead end, so perhaps that's not the correct route. Your ideas would be greatly appreciated.

 

[AlephZero]

Before you dive into the math, think what these functions look like geometrically. The region is the unit circle...

You should be able to see "why" it is true geometrically, and therefore see how to prove it before you start writing down integrals.

(Hint: the word "circle" is in bold)

 

[Bacle2]

Another approach: You may also want to consider using the Cauchy-Goursat theorem, if you're familiar with it.

Think also of the exponential rep. e^iθ. Maybe this is what Aleph was suggesting.

BTW, your factorization is incorrect.

And, sorry to nitpick,but you're not really letting your θ_n's be in L²(D); they are there already; you're letting your θ_n's be given by (x+iy)ⁿ

EDIT: Cauchy-Goursat does not apply here, since the product function is not analytic.

 

[GridironCPJ]

Yes, I realized earlier that my expansion was incorrect. I used polar coordinates, so we start with

∫∫[(rcosθ+irsinθ)^j(rcosθ-rsinθ)^k]rdrdθ

From here I integrated with respect to r first and then I pulled in out and was left with

[r^(j+k+2)/j+k+2]∫(cosθ+isinθ)^j(cosθ-isinθ)^k dθ

where j≠k. I am not sure exactly how to proceed. Is there an expansion of (x+iy)^j(x-iy)^k that I am not thinking of? The value in the integrand will have the form cosθ^j+k+" "+/-isinθ^j+k, where the " " represents combinations of sinθcosθ's, which integrate to zero. So I'm left with the first and last terms, which should be zero if θ=∏, correct?

If my method or logic is not optimal, please point me in the right direction.

 

[Bacle2]

Hi, Grid, sorry for the delay:

I think De Moivre's Formula :

http://en.wikipedia.org/wiki/De_Moivre's_formula

Should help, if you work with polar coords.