Behaviour of $L_{q}$ norms of the $\operatorname {sinc}_{p}$ function
HTML articles powered by AMS MathViewer
- by David E. Edmunds and Houry Melkonian
- Proc. Amer. Math. Soc. 147 (2019), 229-238
- DOI: https://doi.org/10.1090/proc/14264
- Published electronically: October 12, 2018
- PDF | Request permission
Abstract:
An integral inequality due to Ball involves the $L_{q}$ norm of the function $\textrm {{sinc}} x:=\frac {\sin x}{x}$; the dependence of this norm on $q$ as $q\rightarrow \infty$ is now understood. By use of recent inequalities involving $p-$trigonometric functions $(1<p<\infty )$ we obtain asymptotic information about the analogue of Ball’s integral when $\sin$ is replaced by $\sin _{p}.$References
- Keith Ball, Cube slicing in $\textbf {R}^n$, Proc. Amer. Math. Soc. 97 (1986), no. 3, 465–473. MR 840631, DOI 10.1090/S0002-9939-1986-0840631-0
- Barkat Ali Bhayo and Matti Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory 164 (2012), no. 10, 1415–1426. MR 2961189, DOI 10.1016/j.jat.2012.06.003
- David Borwein, Jonathan M. Borwein, and Isaac E. Leonard, $L_p$ norms and the sinc function, Amer. Math. Monthly 117 (2010), no. 6, 528–539. MR 2662705, DOI 10.4169/000298910X492817
- P. J. Bushell and D. E. Edmunds, Remarks on generalized trigonometric functions, Rocky Mountain J. Math. 42 (2012), no. 1, 25–57. MR 2876267, DOI 10.1216/RMJ-2012-42-1-25
- Jan Lang and David Edmunds, Eigenvalues, embeddings and generalised trigonometric functions, Lecture Notes in Mathematics, vol. 2016, Springer, Heidelberg, 2011. MR 2796520, DOI 10.1007/978-3-642-18429-1
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
- Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR 0453984
- Ron Kerman, Rastislav Oľhava, and Susanna Spektor, An asymptotically sharp form of Ball’s integral inequality, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3839–3846. MR 3359575, DOI 10.1090/proc/12505
- F. Olver, Asimptotika i spetsial′nye funktsii, “Nauka”, Moscow, 1990 (Russian). Translated from the English by Yu. A. Brychkov; Translation edited and with a preface by A. P. Prudnikov. MR 1108396
Bibliographic Information
- David E. Edmunds
- Affiliation: Department of Mathematics, University of Sussex, Brighton BN1 9QH, United Kingdom
- MR Author ID: 61855
- Email: davideedmunds@aol.com
- Houry Melkonian
- Affiliation: Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom – and – Department of Mathematics, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Penryn, TR10 9FE, United Kingdom
- MR Author ID: 1169384
- Email: hm189@hw.ac.uk, h.melkonian@exeter.ac.uk
- Received by editor(s): April 13, 2018
- Published electronically: October 12, 2018
- Communicated by: Mourad Ismail
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 229-238
- MSC (2010): Primary 33F05; Secondary 42A99
- DOI: https://doi.org/10.1090/proc/14264
- MathSciNet review: 3876745