Calderón’s reproducing formulas for the Weinstein L2-multiplier operators

Abstract

The aim of this work is the study of the Weinstein L 2 -multiplier operators on R d+1 + and we give for them Calderón's reproducing formulas and best approximation using the theory of Weinstein transform and reproducing kernels. (2010). Primary 43A32; Secondary 44A15. Mathematics Subject Classification

L2 -Uncertainty Principle for the Weinstein-Multiplier Operators

“…Hence, according to last inequality and (4.3)T w,m,σ (ϕ) 2,α ≤ m 1,α ϕ 2,α (µ α (Ω)) 1 2α+d+2) dΘ α (σ, x) 1 ( + ν) ϕ 2,α .Applying Plancherel formula[14, Theorem 2.3], we obtainm α,1 (µ α (Ω)) 1 2α+d+2) dΘ α (σ, x) 1 ( + ν),which completes the proof of the theorem.…”

“…Let m be a function in L 2 α (R d+1 + ) and let σ be a positive real number. The Weinstein L 2 -multiplier operators is defined for smooth functions ϕ on R d+1 + , in [14] as T w,m,σ ϕ(x) := F −1 W,α (m σ F W,α (ϕ)) (x), x ∈ R d+1 + , (1.1) where the function m σ is given by m σ (x) = m(σx).…”

“…These operators are a generalization of the multiplier operators T m associated with a bounded function m and given by T m (ϕ) = F −1 (mF(ϕ)), where F(ϕ) denotes the ordinary Fourier transform on R n . These operators gained the interest of several Mathematicians and they were generalized in many settings in [1,3,6,13,14,[16][17][18].…”

Two-wavelet theory in Weinstein setting

“…Very recently, many authors have been investigating the behaviour of the Weinstein transform (2.5) with respect to several problems already studied for the classical Fourier transform. For instance, Heisenberg-type inequalities [17], Littlewood-Paley g-function [19], Shapiro and HardyâĂŞLittlewoodâĂŞSobolev type inequalities [16,18], Paley-Wiener theorem [9], Uncertainty principles [12,21,24], multiplier Weinstein operator [20], wavelet and continuous wavelet transform [4,11], Wigner transform and localization operators [22,23], and so forth...…”

Time-frequency Analysis of two-wavelet theory in Weinstein setting

“…Very recently, many authors have been investigating the behaviour of the Weinstein transform (2.5) with respect to several problems already studied for the classical Fourier transform. For instance, Heisenberg-type inequalities [29], Littlewood-Paley g-function [31], Shapiro and HardyâĂŞLittlewoodâĂŞSobolev type inequalities [28,30], Paley-Wiener theorem [21], Uncertainty principles [24,32,37], multiplier Weinstein operator [33], wavelet and continuous wavelet transform [15,23], Wigner transform and localization operators [34,36], and so forth...…”