Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T05:32:34.438Z Has data issue: false hasContentIssue false
  • English
  • Français

Composite Cosine Transforms

Part of: General convexity Harmonic analysis in several variables

Published online by Cambridge University Press:  21 December 2009

E. Ournycheva  and
B. Rubin
E. Ournycheva
Affiliation:
Department of Mathematical Sciences, Kent State University, Mathematics and Computer Science Building, Summit Street, Kent OH 44242, U.S.A. E-mail: [email protected]
B. Rubin
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, U.S.A. and Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel. E-mail: [email protected]
Get access

Abstract

The cosine transforms of functions on the unit sphere play an important role in convex geometry, Banach space theory, stochastic geometry and other areas. Their higher-rank generalization to Grassmann manifolds represents an interesting mathematical object useful for applications. More general integral transforms are introduced that reveal distinctive features of higher-rank objects in full generality. These new transforms are called the composite cosine transforms, by taking into account that their kernels agree with the composite power function of the cone of positive definite symmetric matrices. It is shown that injectivity of the composite cosine transforms can be studied using standard tools of the Fourier analysis on matrix spaces. In the framework of this approach, associated generalized zeta integrals are introduced and new simple proofs given to the relevant functional relations. The technique is based on application of the higher-rank Radon transform on matrix spaces.

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 53 - 68
Copyright
Copyright © University College London 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[A]Alesker, S., The α-cosine transform and intertwining integrals, Preprint, 2003.Google Scholar
[AB]Alesker, S., and Bernstein, J., Range characterization of the cosine transform on higher Grassmannians, Advances Math., 184 (2004), 367–379.CrossRefGoogle Scholar
[BSZ]Barchini, L., Sepanski, M., and Zierau, R., Positivity of zeta distributions and small representations, Preprint (2004).Google Scholar
[C1]Clerc, J.-L., Zeta distributions associated to a representation of a Jordan algebra, Math. Z. 239 (2002), 263–276.CrossRefGoogle Scholar
[Es]Eskin, G. I., Boundary Value Problems for Elliptic Pseudodifferential Equations, Amer. Math. Soc. (Providence, R.I., 1981).Google Scholar
[FK]Faraut, J., and Korányi, A., Analysis on Symmetric Cones, Clarendon Press (Oxford, 1994).CrossRefGoogle Scholar
[Ga]Gardner, R. J., Geometric Tomography, Cambridge University Press (New York, 1995).Google Scholar
[GŠ]Gel'fand, I. M., and Šapiro, Z. Ja., Homogeneous functions and their applications, Uspekhi Mat. Nauk, 10 (1955), 3–70 (in Russian).Google Scholar
[GSh]Gel'fand, I. M., and Shilov, G. E., Generalized Functions, Vol. 1: Properties and Operations, Academic Press (New York-London, 1964).Google Scholar
[Gi]Gindikin, S. G., Analysis on homogeneous domains, Russian Math. Surveys, 19 (1964), 1–89.CrossRefGoogle Scholar
[GH]Gonzalez, F., and Helgason, S., Invariant differential operators on Grassmann manifolds, Adv. Math., 60 (1986), 81–91.Google Scholar
[Goo]Goodey, P., Applications of representation theory to convex bodies, II International Conference in “Stochastic Geometry, Convex Bodies and Empirical Measures” (Agrigento, 1996), Rend. Circ. Mat. Palermo (2) Suppl. No. 50 (1997), 179–187.Google Scholar
[GH1]Goodey, P., and Howard, R., Processes of flats induced by higher-dimensional processes, Adv. Math., 80 (1) (1990), 92–109.Google Scholar
[GH2]Goodey, P., and Howard, R., Processes of flats induced by higher-dimensional processes. II, Integral Geometry and Tomography (Arcata, CA, 1989), Contemp. Math., 113, Amer. Math. Soc. (Providence, RI, 1990), 111–119.Google Scholar
[GZ]Goodey, P., and Zhang, G., Inequalities between projection functions of convex bodies, Amer. J. Math., 120 (1998), 345–367.Google Scholar
[Gr]Grinberg, E. L., Radon transforms on higher Grassmannians, J. Differential Geom., 24 (1986), 53–68.Google Scholar
[GR]Grinberg, E., and Rubin, B., Radon inversion on Grassmannians via Gårding-Gindikin fractional integrals, Annals Math., 159 (2004), 809–843.CrossRefGoogle Scholar
[Herz]Herz, C., Bessel functions of matrix argument, Ann. Math., 61 (1955), 474–523.Google Scholar
[Kh]Khekalo, S. P., The Igusa zeta function associated with a composite power function on the space of rectangular matrices, Preprint POMI RAN, 10 (2004), 1–20.Google Scholar
[Ko]Koldobsky, A., Inverse formula for the Blaschke-Levy representation: Houston J. Math., 23 (1997), 95–107.Google Scholar
[Le]Lemoine, C., Fourier transforms of homogeneous distributions, Ann. Scuola Norm. Super. Pisa Sci. Fis. e Mat., 26 (1972), No. 1, 117–149.Google Scholar
[Ma]Mathai, A. M., Jacobians of Matrix Transformations and Functions of Matrix Argument, World Sci. Publ. Co. Pte. Ltd (Singapore, 1997).Google Scholar
[Mat1]Matheron, G., Un théoréme d'unicité pour les hyperplans poissoniens, J. Appl. Probability, 11 (1974), 184–189.Google Scholar
[Mat2]Matheron, G., Random Sets and Integral Geometry, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons (New York-London-Sydney, 1975).Google Scholar
[Mu]Muirhead, R. J., Aspects of Multivariate Statistical Theory, John Wiley & Sons. Inc. (New York, 1982).CrossRefGoogle Scholar
[OR1]Ournycheva, E., and Rubin, B., Radon transform of functions of matrix argument, Preprint, 2004 (math.FA/0406573).Google Scholar
[OR2]Ournycheva, E., and Rubin, B., The composite cosine transform on the Stiefel manifold and generalized zeta integrals, Contemporary Math. (to appear).Google Scholar
[OR3]Ournycheva, E., and Rubin, B., Higher-rank Radon transforms, Preprint 2005.Google Scholar
[OR4]Ournycheva, E., and Rubin, B., An analogue of the Fuglede formula in integral geometry on matrix spaces, Contemporary Math., 382 (2005), 305–320.Google Scholar
[P]Petrov, E. E., The Radon transform in spaces of matrices, Trudy seminara po vektornomu i tenzornomu analizu, M.G.U., Moscow, 15 (1970), 279–315 (Russian).Google Scholar
[Ra]Raïs, M., Distributions homogènes sur des espaces de matrices, Bull. Soc. Math. France, Mem., 30 (1972), 3–109.Google Scholar
[Ru]Rubin, B., Inversion of fractional integrals related to the spherical Radon transform, J. Functional Analysis, 157 (1998), 470–487.CrossRefGoogle Scholar
[Ru4]Rubin, B., Riesz potentials and integral geometry in the space of rectangular matrices, Advances Math. (to appear).Google Scholar
[Sa]Samko, S. G., Generalized Riesz potentials and hypersingular integrals with homogeneous characteristics, their symbols and inversion, Proc. Steklov Inst. Math., 2 (1983), 173–243.Google Scholar
[Schn]Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, (1993).Google Scholar
[Se]Semyanistyi, V. I., Some integral transformations and integral geometry in an elliptic space, Trudy Sem. Vektor. Tenzor. Anal., 12, (1963), 397–441 (in Russian).Google Scholar
[Sh1]Shibasov, L. P., Integral problems in a matrix space that are connected with the functional , Izv. Vysš. Učebn. Zaved. Matematika (1973), (135), 101–112 (Russian).Google Scholar
[Sh2]Shibasov, L. P., Integral geometry on planes of a matrix space, (Russian) Harmonic Analysis on Groups. Moskov. Gos. Zaočn. Ped. Inst. Sb. Naučn. Trudov Vyp. 39 (1974), 68–76.Google Scholar
[Str1]Strichartz, R. S., The explicit Fourier decomposition of L 2(SO(n)/SO(n – m)), Canad. J. Math., 27 (1975), 294–310.Google Scholar
[Str2]Strichartz, R. S., Lp estimates for Radon transforms in Euclidean and non-Euclidean spaces, Duke Math. J. 48 (1981), 699–727.Google Scholar
[T]Terras, A, Harmonic Analysis on Symmetric Spaces and Applications, Vol. II, Springer (Berlin, 1988).Google Scholar
[TT]Ton-That, T., Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc., 216 (1976), 1–46.Google Scholar