tailieunhanh - On orthogonal systems of shifts of scaling function on local fields of positive characteristic

We present a new method for constructing an orthogonal step scaling function on local fields of positive characteristic, which generates multiresolution analysis. Chinese mathematicians Jiang et al. in the article introduced the notion of multiresolution analysis (MRA) on local fields. | Turk J Math (2017) 41: 244 – 253 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On orthogonal systems of shifts of scaling function on local fields of positive characteristic Gleb Sergeevich BERDNIKOV, Iuliia Sergeevna KRUSS∗, Sergey Fedorovich LUKOMSKII Department of Mathematical Analysis, Saratov State University, Saratov, Russia Received: • Accepted/Published Online: • Final Version: Abstract: We present a new method for constructing an orthogonal step scaling function on local fields of positive characteristic, which generates multiresolution analysis. Key words: Local field, scaling function, multiresolution analysis 1. Introduction Chinese mathematicians Jiang et al. in the article [10] introduced the notion of multiresolution analysis (MRA) on local fields. For the fields F (s) of positive characteristic p they proved some properties and gave an algorithm for constructing wavelets for a known scaling function. Using these results they constructed ”Haar MRA” and corresponding ”Haar wavelets”. The problem of constructing orthogonal MRA on the field F (1) was studied in detail in the works [6–8, 12, 14, 15]. In [11] a necessary condition and sufficient conditions for wavelet frame on local fields were given. Behera and Jahan [2] constructed the wavelet packets associated with MRA on local fields of positive characteristic. In the article [1] necessary and sufficient conditions for a function φ ∈ L2 (F (s) ) under which it is a scaling function for MRA were obtained. These conditions are as follows: ∑ |φ(ξ ˆ + u(k))|2 = 1 (1) k∈N0 for . ξ in unit ball D , lim |φ(p ˆ j ξ)| = 1 f or . ξ ∈ F (s) , j→∞ (2) and there exists an integral periodic function m0 ∈ L2 (D) such that φ(ξ) ˆ = m0 (pξ)φ(pξ) ˆ f or . ξ ∈ F (s) , (3) where {u(k)} is the set of shifts and p is a prime element. Behera and Jahan [3] proved also that if the .