Abstract
The problem of understanding the Fourier-analytic structure of the cone of positive functions on a group has a long history. In this article, we develop the first quantitative spectral concentration results for such functions over arbitrary compact groups. Specifically, we describe a family of finite, positive quadrature rules for the Fourier coefficients of band-limited functions on compact groups. We apply these quadrature rules to establish a spectral concentration result for positive functions: given appropriately nested band limits \(\mathcal {A}\subset \mathcal {B} \subset\widehat{G}\), we prove a lower bound on the fraction of L 2-mass that any \(\mathcal {B}\)-band-limited positive function has in \(\mathcal {A}\). Our bounds are explicit and depend only on elementary properties of \(\mathcal {A}\) and \(\mathcal {B}\); they are the first such bounds that apply to arbitrary compact groups. They apply to finite groups as a special case, where the quadrature rule is given by the Fourier transform on the smallest quotient whose dual contains the Fourier support of the function.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alagic, G., Russell, A.: Uncertainty principles for compact groups. Ill. J. Math. 52(4), 1315–1324 (2008)
Davis, P., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, San Diego (1984)
Erdős, P., Fuchs, W.: On a problem in additive number theory. J. Lond. Math. Soc. 31, 67–73 (1956)
Filbir, F., Mhaskar, H.N.: A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel. J. Fourier Anal. Appl. 16(5), 629–657 (2010)
Folland, G.: A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)
Fulton, W., Harris, J.: Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129. Springer, Berlin (1991)
Goldoni, G.: A visual proof for the sum of the first n squares and for the sum of the first n factorials of order two. Math. Intell. 24, 67–69 (2002)
Gräf, M., Potts, D.: Sampling sets and quadrature formulae on the rotation group. Numer. Funct. Anal. Optim. 30, 665–688 (2009)
Kueh, K., Olson, T., Rockmore, D., Tan, K.: Nonlinear approximation theory on compact groups. J. Fourier Anal. Appl. 7(3), 257–281 (2001)
Logan, B.: An interference problem for exponentials. Mich. Math. J. 35, 369–401 (1988)
Maslen, D.: Efficient computation of Fourier transforms on compact groups. J. Fourier Anal. Appl. 4(1), 19–52 (1988)
Maslen, D.: Sampling of functions and sections for compact groups. Tech. Report PMA-TR99-193, Dartmouth College Department of Mathematics (1999)
Reed, M., Simon, B.: Methods of Mathematical Physics I, Functional Analysis. Academic Press, San Diego (1980)
Schmid, D.: Scattered data approximation on the rotation group and generalizations. Ph.D. thesis, Technische Universität München (2009)
Serre, J.-P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42. Springer, Berlin (1977)
Shapiro, H.: Majorant problems for Fourier coefficients. Q. J. Math. 26, 9–18 (1975)
Simon, B.: Representations of Finite and Compact Groups. Graduate Studies in Mathematics, vol. 10. Am. Math. Soc., Providence (1996)
Vilenkin, N., Klimyk, A.: Representation of Lie Groups and Special Functions. Volume 1: Simplest Lie Groups, Special Functions and Integral Transforms. Mathematics and Its Applications. Soviet Series. Kluwer Academic, Dordrecht (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hans G. Feichtinger.
Gorjan Alagic would like to thank the University of Connecticut, NSERC, MITACS and the US ARO for their support. Alexander Russell gratefully acknowledges support from the NSF under grants 0829917 and 0835735 and the US ARO.
Rights and permissions
About this article
Cite this article
Alagic, G., Russell, A. Spectral Concentration of Positive Functions on Compact Groups. J Fourier Anal Appl 17, 355–373 (2011). https://doi.org/10.1007/s00041-011-9174-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-011-9174-5