Fourier Analysis on Local Fields. (MN-15)

Polyphase matrix characterization of framelets on local fields of positive characteristic

“…The properties of the Fourier transform on the local field K are quite similar to those of the Fourier analysis on the real line (See Taibleson [16]). In…”

“…Let χ be a fixed character on K + that is trivial on D but is nontrivial on B −1 . Therefore, χ is constant on cosets of D so if y ∈ B k , then χ y (x) = χ(yx), x ∈ K. Suppose that χ u is any character on K + , then clearly the restriction χ u |D is also a character on D. Therefore, if {u(n) : n ∈ N 0 } is a complete list of distinct coset representative of D in K + , then, as it was proved in [16], the set χ u(n) : n ∈ N 0 of distinct characters on D is a complete orthonormal system on D.…”

“…Such an element is called a prime element of K. Therefore, for such an ideal B in D, we have B = p = pD. As it was proved in [16], the set D is compact and open. Hence, B is compact and open.…”

“…The p-adic fields are examples of local fields. We use the notation of the book by Taibleson [16]. In the rest of this paper, we use the symbols N, N 0 and Z to denote the sets of natural, nonnegative integers and integers, respectively.…”

“…Then, it can be proved that D * is a group of units in K * and if x = 0, then we may write x = p k x , x ∈ D * . For a proof of this fact we refer to [16]. Moreover, each B k = p k D = x ∈ K : |x| < q −k is a compact subgroup of K + and usually known as the fractional ideals of…”