Trigonometric approximation and a general form of the Erdős Turán inequality
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- by Leonardo Colzani, Giacomo Gigante and Giancarlo Travaglini PDF
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Abstract:
There exists a positive function $\psi (t)$ on $t\geq 0$, with fast decay at infinity, such that for every measurable set $\Omega$ in the Euclidean space and $R>0$, there exist entire functions $A\left ( x\right )$ and $B\left ( x\right )$ of exponential type $R$, satisfying $A(x)\leq \chi _{\Omega }(x)\leq B(x)$ and $\left \vert B(x)-A(x)\right \vert \leqslant \psi \left ( R\operatorname *{dist} \left ( x,\partial \Omega \right ) \right )$. This leads to Erdős Turán estimates for discrepancy of point set distributions in the multi-dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.References
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Additional Information
- Leonardo Colzani
- Affiliation: Dipartimento di Matematica e Applicazioni, Edificio U5, Università di Milano - Bicocca, via R. Cozzi 53, 20125 Milano, Italia
- MR Author ID: 50785
- Email: leonardo.colzani@unimib.it
- Giacomo Gigante
- Affiliation: Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università di Bergamo, viale Marconi 5, 24044 Dalmine, Italia
- MR Author ID: 666574
- Email: giacomo.gigante@unibg.it
- Giancarlo Travaglini
- Affiliation: Dipartimento di Statistica, Edificio U7, Università di Milano - Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italia
- MR Author ID: 199040
- ORCID: 0000-0002-7405-0233
- Email: giancarlo.travaglini@unimib.it
- Received by editor(s): February 24, 2009
- Received by editor(s) in revised form: December 10, 2009
- Published electronically: September 20, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 1101-1123
- MSC (2000): Primary 11K38, 42C15
- DOI: https://doi.org/10.1090/S0002-9947-2010-05287-0
- MathSciNet review: 2728598