Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 5 / b / Solution

Fourier transforming the refinement equation and changing variables gives

f (2 t) = m (t) f (t), m (t) = \frac{1}{2} n \sum a_{n} e^{- in t} .

By Parseval identity, orthonormality of the translates is equivalent to

\frac{1}{2 π} \int_{R} ∣ f (t) ∣^{2} e^{in t} d t = δ_{n 0} .

These are precisely the Fourier coefficients of the periodization

P (t) = \sum_{k} ∣ f (t + 2 πk) ∣^{2}

. Therefore

{ϕ (\cdot - n)} is orthonormal ⟺ k \sum ∣ f (t + 2 πk) ∣^{2} = 1 a.e.

 Ancestors (11)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version