Geodesic
On the rate of decay of a Meyer scaling function
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 48, Tome 491 (2020), pp. 52-65 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A function with the following properties is called a Meyer scaling function: $\varphi\colon\Bbb R\to\Bbb R$, its integral shifts $\varphi(\cdot+n)$, $n\in\Bbb Z$, are orthonormal in $L_2(\Bbb R)$, and its Fourier transform $\widehat{\varphi}(y)=\frac{1}{\sqrt{2\pi}}\int\limits_{\Bbb R}\varphi(t)e^{-iyt} dt$ has the following properties: $\widehat{\varphi}$ is even, $\widehat{\varphi}=0$ outside $[-\pi-\varepsilon,\pi+\varepsilon]$, $\widehat{\varphi}=\frac{1}{\sqrt{2\pi}}$ on $[-\pi+\varepsilon,\pi-\varepsilon]$, where $\varepsilon\in\bigl(0,\frac{\pi}{3}\bigr]$. Here is the main result of the paper. Assume that $$ \omega\colon[ 0,+\infty)\to [ 0,+\infty) $$ and the function $\frac{\omega(x)}{x}$ decreases. Then the following assertions are equivalent. 1. For every (or, equivalently, for some) $\varepsilon\in(0,\frac{\pi}{3}]$ there exists $x_0>0$ and a Meyer scaling function $\varphi$ such that $\widehat{\varphi}=0$ outside $[-\pi-\varepsilon,\pi+\varepsilon]$ and $|\varphi(x)|\leqslant e^{-\omega(|x|)}$ for all $|x|>x_0$. 2. $\int\limits_1^{+\infty}\frac{\omega(x)}{x^2} dx<+\infty$. 
@article{ZNSL_2020_491_a3,
     author = {O. L. Vinogradov},
     title = {On the rate of decay of a {Meyer} scaling function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {52--65},
     year = {2020},
     volume = {491},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a3/}
}
TY  - JOUR
AU  - O. L. Vinogradov
TI  - On the rate of decay of a Meyer scaling function
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2020
SP  - 52
EP  - 65
VL  - 491
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a3/
LA  - ru
ID  - ZNSL_2020_491_a3
ER  - 
%0 Journal Article
%A O. L. Vinogradov
%T On the rate of decay of a Meyer scaling function
%J Zapiski Nauchnykh Seminarov POMI
%D 2020
%P 52-65
%V 491
%U http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a3/
%G ru
%F ZNSL_2020_491_a3
O. L. Vinogradov. On the rate of decay of a Meyer scaling function. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 48, Tome 491 (2020), pp. 52-65. http://geodesic.mathdoc.fr/item/ZNSL_2020_491_a3/

[1] Y. Meyer, “Principe d'incertitude, bases hilbertiennes et algèbres d'opérateurs”, Séminaire Bourbaki, 662, 1985–1986, 209–223 | MR

[2] I. Dobeshi, Desyat lektsii po veivletam, RKhD, Izhevsk, 2001

[3] I. Ya. Novikov, V. Yu. Protasov, M. A. Skopina, Teoriya vspleskov, Fizmatlit, M., 2005 | MR

[4] A. E. Ingham, “A note on Fourier transforms”, J. London Math. Soc., 9 (1934), 29–32 | DOI | MR

[5] N. Levinson, Gap and density theorems, AM, New York, 1940 | MR

[6] W. A. J. Luxemburg, J. Korevaar, “Entire functions and Müntz – Szász type approximation”, Trans. Amer. Math. Soc., 157 (1971), 23–37 | MR | Zbl

[7] V. Khavin, B. Jöricke, The uncertainty principle in harmonic analysis, Springer-Verlag, Berlin –Heidelberg, 1994 | MR | Zbl

[8] G. M. Fikhtengolts, Kurs differentsialnogo i integralnogo ischisleniya, v. 2, Fizmatgiz, M., 1959

[9] E. Laeng, “Uncertainty inequalities for Fourier series of pairs of reciprocal positive functions”, Bull. London Math. Soc., 31 (1999), 314–322 | DOI | MR

[10] P. L. Ulyanov, “O klassakh beskonechno differentsiruemykh funktsii”, Mat. sb., 181:5 (1990), 589–609

[11] Y. Taguchi, “Fourier coefficients of periodic functions of Gevrey classes and ultradistributions”, Yokohama Math. J., 35 (1987), 51–60 | MR | Zbl