Abstract
The purpose of this survey paper is to examine some specific applications of the process of averaging over finite groups, compact topological groups, and compact homogeneous manifolds. These applications occur in a variety of different disciplines of pure and applied mathematics, such as (i) classical representation theory of finite groups, (ii) practical Fourier analysis, (iii) classical harmonic analysis, (iv) approximation theory, (v) complex analysis in several variables, and (vi) combinatorics. In each case, the symmetrization technique reveals itself to be a simple and efficient tool to produce far-reaching identities and inequalities in a unifying manner.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
L. Auslander and R. Tolimieri, Is computing with the finite Fourier transform pure of applied mathematics? Bull. Amer. Math. Soc. 1 (1979), 847–897.
D.L. Berman, On the impossibility of constructing a linear polynomial operator giving an approximation of best order. Uspehi Mat. Nauk 14 (1959), 141–142.
E.W. Cheney, Introduction to Approximation Theory. McGraw-Hill Book Company, New York, St. Louis, San Francisco, Toronto, London, Sydney, 1966.
R.R. Coifman and G. Weiss, Representations of compact groups and spherical harmonics. Enseignement Math. 14 (1960), 121–173.
C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers, New York, London, Sydney, 1962.
P. Delsarte, J.M. Goethals, and J.J. Seidel, Spherical codes and designs. Geometriae Dedicata 6 (1977), 363–388.
B. Dreseler and W. Schempp, On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces. Manuscripta Math. 13 (1974), 321–337.
J.M. Goethals and J.J. Seidel, Spherical designs. In: Relations between Combinatorics and Other Parts of Mathematics. (Proc. Symp. Pure Math., Vol. 34). Amer. Math. Soc, Providence, R.I., 1979.
A. Hurwitz, Über die Erzeugung der Invarianten durch Integration. Nachr. k. Ges. Göttingen, Math.-phys. Klasse 1897, 71–90. Also in: Mathematische Werke, Band II, pp. 546-564. Birkhäuser, Basel, 1933.
N. Jacobson, Basic Algebra II. W.H. Freeman and Company, San Francisco, 1980.
F. Lorenz and A. Schönhage, Über ein zahlentheoretisches Problem aus der Fourieranalysis. Math. Z. 120 (1971), 369–372.
S. Lozinski, On a class of linear operators. Doklady Akad. Nauk SSSR 61 (1948), 193–196.
J. Marcinkiewicz, Quelques remarques sur l’interpolation. Acta Litt. Sci. Szeged 8 (1936/37), 127–130. Also in: Collected papers, pp. 200-203. Państwowe Wydawnictwo Naukowe, Warszawa, 1964.
H. Maschke, Über den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen. Math. Ann. 50 (1898), 482–498.
D.J. Newman, The nonexistence of projections from L1 to H1. Proc. Amer. Math. Soc. 12 (1961), 98–99.
W. Rudin, Projections on invariant subspaces. Proc. Amer. Math. Soc. 13 (1962), 429–432.
W. Rudin, Function Theory in the Unit Ball of C n (Grundlehren der math. Wissenschaften 241). Springer-Verlag, New York, Heidelberg, Berlin, 1980.
W. Schempp, Approximation on compact Riemannian globally symmetric manifolds of rank one. J. Approx. Theory 21 (1977), 236–245.
W. Schempp, Approxmation in G-homogeneous Banach spaces. In: Constructive Theory of Functions of Several Variables. (Proc. Conf. Oberwolfach 1976, W. Schempp and K. Zeller, Eds., Lecture Notes in in Math., Vol. 571, pp. 220-225). Springer-Verlag, Berlin, Heidelberg, New York, 1977.
W. Schempp, A note on the Newman-Schoenberg phenomenon. Math. Z. 167 (1979), 1–6.
W. Schempp, Periodic splines and nilpotent harmonic analysis. C.R. Math. ReP. Acad. Sci. Canada 3 (1981), 69–74.
W. Schempp, Approximation und Transformationsmethoden III. In: Approximation and Functional Analysis (Proc. Conf. Oberwolfach 1980, P.L. Butzer, E. Görlich, and B. Sz.-Nagy, Eds., ISNM 60). Birkhäuser Verlag, Basel, Boston, Stuttgart, 1981.
W. Schempp, Attenuation factors and nilpotent harmonic analysis. Resultate Math, (in press).
W. Schempp, Complex Contour Integral Representation of Cardinal Spline Functions. With a preface by I.J. Schoenberg (Contemporary Mathematics). Amer. Math. Soc, Providence, R.I. (in press).
W. Schempp, Cardinal splines and nilpotent harmonic analysis (to appear).
W. Schempp, Harmonic Analysis on Heisenberg Groups with Applications to Spline Functions (Research Notes in Mathematics). Pitman, San Francisco, London, Melbourne (in preparation).
W. Schempp, Gruppentheoretische Aspekte der digitalen Signalübertragung und der kardinalen Splines (to appear).
W. Schempp and B. Dreseler, Einführung in die harmonische Analyse (Mathematische Leitfäden). Teubner, Stuttgart, 1980.
V.M. Sidelnikov, New bounds for the density of sphere packings in an n-dimensional Euclidean space. Math. USSR Sbornik 24 (1974), 147–157.
H. Weyl, The Classical Groups, their Invariants and Representations. Princeton University Press, Princeton, N.J. 1939.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer Basel AG
About this chapter
Cite this chapter
Schempp, W. (1983). Identities and Inequalities via Symmetrization. In: Beckenbach, E.F., Walter, W. (eds) General Inequalities 3. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 64. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6290-5_16
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6290-5_16
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-6292-9
Online ISBN: 978-3-0348-6290-5
eBook Packages: Springer Book Archive