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I am searching for a "Fourier transform" $T$ on formal series with coefficients in dyadic numbers, such that:

$$T(fg)=T(f)*T(g)$$

Where $*$ is the "coefficient by coefficient " multiplication $(\sum a_i x^i ) *(\sum b_i x^i ) = \sum a_i b_i x^i $. Possibly, $T$ should be invertible.

Note: "dyadic" means $2$-adic as in the Wikipedia article P-adic number.

Note: "coefficient by coefficient" multiplication is also known as the Hadamard product.

Somos
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frame95
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  • Did you mean to put a $*$ in that multiplication definition? – aschepler Feb 26 '18 at 14:13
  • Yep,edited! Thank you :) – frame95 Feb 26 '18 at 15:22
  • What is the significance of "dyadic" numbers. How would the situation be different using rational numbers? Also, do you mean formal power series? – Somos Feb 26 '18 at 15:28
  • Actually, I don't know if it has some significance, but it is the setting in which I am working. If you have any hints of why it does not depend on the setting they are welcomed. The meaning of dyadic numbers, if this is what you mant to ask, is the set of numbers of the form ${ a_0 + 2a_1 +..+2^k a_k + \ldots : \ \ a_i \in {0,1} }$, with addition and multiplication as you can imagine. It is like you have "added" numbers with an infinite base-2 expansion! – frame95 Feb 27 '18 at 11:34
  • A quick search for "dyadic number" doesn't bring up that definition, but mostly hits for "dyadic rationals", which are something entirely different. You should probably specify the definition of dyadic number in the question. – aschepler Mar 02 '18 at 00:19
  • My answer is turning out to be really long, but basically $T(f) = \sum f(2^{k+1}) x^k$ works. – aschepler Mar 08 '18 at 12:09
  • Please see MSE question 8787 on Hadamard product of power series. – Somos Mar 21 '18 at 00:05
  • @aschpler: is it invertible? Or at least surjective? Thank you for the aid! – frame95 Mar 27 '18 at 20:43
  • In particular I need a function such that T(f) equals \sum (x/3)^k. This is not only what I need but would be a good step. Thank you again for the aid. – frame95 Mar 27 '18 at 20:50

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