A functional from geometry with applications to discrepancy estimates and the Radon transform

Rogers, Allen D.

doi:10.1090/S0002-9947-1994-1169082-8

A functional from geometry with applications to discrepancy estimates and the Radon transform
HTML articles powered by AMS MathViewer

by Allen D. Rogers

Trans. Amer. Math. Soc. 341 (1994), 275-313

DOI: https://doi.org/10.1090/S0002-9947-1994-1169082-8

PDF | Request permission

Abstract:

Estimates of discrepancy, or irregularities of distribution, are obtained for measures without atoms. Two estimators are used, the half-space, or separation, discrepancy ${D_S}$ and a geometric functional ${I^\alpha }$. A representation formula for the generalized energy integral ${I^\alpha }$ is developed. Norm inequalities for the Radon transform are obtained as an application of the continuous discrepancy results. Integral geometric notions play a prominent role.

References

Ralph Alexander, Generalized sums of distances, Pacific J. Math. 56 (1975), no. 2, 297–304. MR 513964
R. Alexander, Geometric methods in the study of irregularities of distribution, Combinatorica 10 (1990), no. 2, 115–136. MR 1082645, DOI 10.1007/BF02123006
Ralph Alexander, Principles of a new method in the study of irregularities of distribution, Invent. Math. 103 (1991), no. 2, 279–296. MR 1085108, DOI 10.1007/BF01239514
Ralph Alexander and Kenneth B. Stolarsky, Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc. 193 (1974), 1–31. MR 350629, DOI 10.1090/S0002-9947-1974-0350629-3
J. Beck, On a problem of K. F. Roth concerning irregularities of point distribution, Invent. Math. 74 (1983), no. 3, 477–487. MR 724016, DOI 10.1007/BF01394247
József Beck, Sums of distances between points on a sphere—an application of the theory of irregularities of distribution to discrete geometry, Mathematika 31 (1984), no. 1, 33–41. MR 762175, DOI 10.1112/S0025579300010639
József Beck and William W. L. Chen, Irregularities of distribution, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, Cambridge, 1987. MR 903025, DOI 10.1017/CBO9780511565984
Göran Björck, Distributions of positive mass, which maximize a certain generalized energy integral, Ark. Mat. 3 (1956), 255–269. MR 78470, DOI 10.1007/BF02589412
Salomon Bochner, Harmonic analysis and the theory of probability, University of California Press, Berkeley-Los Angeles, Calif., 1955. MR 0072370
Stanley R. Deans, The Radon transform and some of its applications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. MR 709591
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
P. Erdős, Problems and results on diophantine approximations, Compositio Math. 16 (1964), 52–65 (1964). MR 179131
Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633

Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions

3

Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Alexander Hertle, Continuity of the Radon transform and its inverse on Euclidean space, Math. Z. 184 (1983), no. 2, 165–192. MR 716270, DOI 10.1007/BF01252856
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0419394
D. M. Oberlin and E. M. Stein, Mapping properties of the Radon transform, Indiana Univ. Math. J. 31 (1982), no. 5, 641–650. MR 667786, DOI 10.1512/iumj.1982.31.31046

Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen

165

Johann Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Computed tomography (Cincinnati, Ohio, 1982) Proc. Sympos. Appl. Math., vol. 27, Amer. Math. Soc., Providence, R.I., 1982, pp. 71–86 (German). MR 692055

Theory and applications of a functional from metric geometry

K. F. Roth, On irregularities of distribution, Mathematika 1 (1954), 73–79. MR 66435, DOI 10.1112/S0025579300000541
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
Wolfgang M. Schmidt, Irregularities of distribution. IV, Invent. Math. 7 (1969), 55–82. MR 245532, DOI 10.1007/BF01418774
Wolfgang M. Schmidt, Lectures on irregularities of distribution, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 56, Tata Institute of Fundamental Research, Bombay, 1977. Notes taken by T. N. Shorey. MR 554923
I. J. Schoenberg, On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space, Ann. of Math. (2) 38 (1937), no. 4, 787–793. MR 1503370, DOI 10.2307/1968835
I. J. Schoenberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), no. 3, 522–536. MR 1501980, DOI 10.1090/S0002-9947-1938-1501980-0
Kenneth B. Stolarsky, Sums of distances between points on a sphere. II, Proc. Amer. Math. Soc. 41 (1973), 575–582. MR 333995, DOI 10.1090/S0002-9939-1973-0333995-9
Gerold Wagner, On means of distances on the surface of a sphere (lower bounds), Pacific J. Math. 144 (1990), no. 2, 389–398. MR 1061328
Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864