TY - JOUR

T1 - On the Rate of Decay of a Meyer Scaling Function

AU - Vinogradov, O. L.

N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/4/6

Y1 - 2022/4/6

N2 - A function with the following properties is called a Meyer scaling function: φ: ℝ → ℝ, its integral shifts φ(· + n), n ∈ ℤ, are orthonormal in L2(ℝ), and its Fourier transform φ̂(y)=12π∫ℝφ(t)e−iytdt has the following properties: φ̂ is even, φ̂ = 0 outside [−π −ε, π +ε], φ̂=12π on [−π + ε, π − ε], where ε∈(0π3]. Here is the main result of the paper. Assume that ω:[0+∞)→[0+∞) and the function ω(x)x decreases. Then the following assertions are equivalent. 1. For every (or, equivalently, for some) ε∈(0π3], there exists x0 > 0 and a Meyer scaling function φ such that φ̂ = 0 outside [−π − ε, π + ε] and |φ(x)| ≤ e−ω(|x|) for all |x| > x0. 2. ∫1+∞ω(x)x2dx<+∞.

AB - A function with the following properties is called a Meyer scaling function: φ: ℝ → ℝ, its integral shifts φ(· + n), n ∈ ℤ, are orthonormal in L2(ℝ), and its Fourier transform φ̂(y)=12π∫ℝφ(t)e−iytdt has the following properties: φ̂ is even, φ̂ = 0 outside [−π −ε, π +ε], φ̂=12π on [−π + ε, π − ε], where ε∈(0π3]. Here is the main result of the paper. Assume that ω:[0+∞)→[0+∞) and the function ω(x)x decreases. Then the following assertions are equivalent. 1. For every (or, equivalently, for some) ε∈(0π3], there exists x0 > 0 and a Meyer scaling function φ such that φ̂ = 0 outside [−π − ε, π + ε] and |φ(x)| ≤ e−ω(|x|) for all |x| > x0. 2. ∫1+∞ω(x)x2dx<+∞.

UR - http://www.scopus.com/inward/record.url?scp=85127670971&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/bf39242f-823f-34e6-b56a-5af0f9d59c4c/

U2 - 10.1007/s10958-022-05787-y

DO - 10.1007/s10958-022-05787-y

M3 - Article

AN - SCOPUS:85127670971

VL - 261

SP - 763

EP - 772

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -