Abstract
The aim of this work is to study of the q^2 -Fourier multiplier operators on R_q and we give for them Calderon's reproducing formulas and best approximation on the q^2-analogue Sobolev type space H_q using the theory of q^2-Fourier transform and reproducing kernels.
Highlights
The q2-analogue differential-difference operator ∂q, called q-Rubin’s operator defined on Rq in [11,12]by f (q−1z) + f (−q−1z) − f + f (−qz) − 2f (−z)2(1 − q)z if z = 0 ∂qf (z) =lim ∂qf (z) in Rq z→0 if z = 0.This operator has correct eigenvalue relationships for analogue exponential Fourier analysis using the functions and orthogonalities of [9].The q2-analogue Fourier transform we employ to make our constructions and results in this paper is based on analogue trigonometric functions and orthogonality results from [9] which have important applications to Received January 13th, 2020; accepted January 30th, 2020; published May 1st, 2020.2010 Mathematics Subject Classification. 46E35; 43A32
The aim of this work is to study of the q2-Fourier multiplier operators on Rq and we give for them Calderon’s reproducing formulas and best approximation on the q2-analogue Sobolev type space Hq using the theory of q2-Fourier transform and reproducing kernels
L2-Multiplier operators for the q-Rubin-Fourier transform we study the q2-Fourier-multiplier operators and we establish theirs Calderon’s reproducing formulas in L2-case
Summary
Let a ∈ R+q , m ∈ L2q and f a smooth function on Rq. We define the q2-Fourier L2-multiplier operators Tm for a regular function f on Rq as follow From the definition of the q2-Fourier L2-multiplier operators (3.1) and relations (2.5) and (2.8) we get that the function Tmf belongs to L2q, and we have Iii) Let m ∈ L2q, and f ∈ L2q, from inversion formula we get Tmf ∈ L∞ q , and by relation (2.5) we obtain
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