Approximation, Probability, and Related Fields

440 Pages·1994·14.53 MB·English

by  José A. Adell

Approximation, Probability, and
Related Fields
Approximation, Probability, and
Related Fields
Edited by
George Anastassiou
Memphis State University
Memphis, Tennessee
and
Svetlozar T. Rachev
University of California
Santa Barbara, California
Springer Science+Business Media, LLC
Librar y o f Congress Cataloging-in-Publicatio n Data
Approximation, probability , and relate d field s / edite d b y George
Anastassiou and Svetloza r T. Rachev.
p. cm .
"Proceedings o f a Conference o n Approximation , Probability , an d
Related Fields , hel d May 20-22 , 1993, i n Santa Barbara , California" -
-T.p. verso .
Includes bibliographica l reference s and index .
ISBN 978-1-4613-6063-6 ISBN 978-1-4615-2494-6 (eBook)
DOI 10.1007/978-1-4615-2494-6
1. Approximatio n theory—Congresses . 2 . Probabilities -
-Congresses. 3 . Stochasti c processes—Congresses . I . Anastassiou,
George A. , 1952 - . II . Rachev, S. T. (Svetloza r Todorov)
III . Conference o n Approximation , Probability , and Relate d Field s
(1993 : Santa Barbara , Calif. )
QA221.A634 199 4
511.4—dc20 94-2769 3
CIP
Proceedings of a conference on Approximation, Probability, and Related Fields, held May 20-22,
1993, in Santa Barbara, California
ISBN 978-1-4613-6063-6
© 1994 Springer Springer Science+Business Media New York
Originally published by Plenum Press, New York in 1994
Softcover reprint of the hardcover 1st edition 1994
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form
or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise,
without written permission from the Publisher
DEDICATED TO
LEOPOLDO NACHBIN
1922-1993
v
DEDICATION
Leopoldo Nachbin was born on January 7, 1922 in the city of Recife, Brazil, and died on
April 3, 1993. It is not easy to summarize in a few paragraphs Nachbin's manifold
contributions to mathematics. Fortunately, the interested reader can consult the analysis
that Professor John Horvath made for the volume honoring Nachbin on the occasion of his
sixtieth birthday, published by North-Holland under the title Aspects of Mathematics and
its Applications (Editor: J.A. Barroso). This volume appeared in 1986 and Professor
Horvath's paper is entitled "The Life and Works of Leopoldo Nachbin," and occupies
pages 1 to 75.
Nachbin moved from Recife to Rio de Janeiro in 1938 and received an engineering degree
in 1943. But already in 1941, while still an undergraduate student, he had published his
first paper in mathematics, in Volume 13 of the Proceedings of the Brazilian Academy of
Sciences, and in 1942 he published two more papers, one in Italy and the other in
Argentina. All three papers were reviewed in Mathematical Reviews. In 1946, his eighth
paper appeared in the Comptes Rendus de I'Academie des Sciences Paris. In that same
year, two other papers were published, one in the Transactions of the American
Mathematical Society and the other in Summa Brasiliensis Mathematicae. In 1948 he
published his fIrst book. The subject was topological vector spaces. Horvath remarks that
this was the fIrst text ever written on the theory of general topological vector spaces. But
the mathematical activity of Nachbin covered a wide range of subjects and in each of
them he left his imprint: Boolean algebras, general topology, ordered topological spaces,
topological vector spaces, an extension of the Hahn-Banach theorem for linear
transfonnations, ordered topological vector spaces, approximation theory, hannonic
analysis, infInite-dimensional holomorphy.
Nachbin's work on approximation theory started with a 1949 paper establishing the
conditions for density of subalgebras of r-times continuously differentiable functions on
open subsets of n-dimensional Euclidean space, or more generally on n-dimensional
differentiable manifolds, for the topology of unifonn convergence of order r on compact
subsets, extending to this case the so-called Stone-Weierstrass theorem. In 1948 H.
Whitney had described the closure in this topology of an ideal, and Nachbin solved the
problem of density of subalgebras, but the problem of describing the closure of a
subalgebra remains open.
The Stone-Weierstrass theorem for modules and the theorem of Whitney on the closure of
an ideal led Nachbin to his work on algebras of fInite differentiable order and the
operational calculus, which appeared in the Annals of Mathematics of 1959. His next
interest was the study of the Bernstein weighted approximation problem. His fIrst paper in
this area appeared in 1961. At this time, more exactly during the two academic years
vii
1961/62 and 1962/63, Nachbin was a visiting professor at the University of Paris. He was
an invited lecturer at the 1962 International Congress of Mathematicians held in
Stockholm, and his lecture dealt with algebraic problems in approximation theory. What
he did with the Bernstein weighted approximation problem is similar to what Stone had
done with the Wierstrass approximation problem. His work in this area can be found in
his book Elements of Approximation Theory, published by Van Nostrand in 1967, and
reprinted by Krieger in 1976. Since the academic year 1963/64, Nachbin worked in
Rochester, NY, and since 1967 he had the title of George Eastman Professor at the
University of Rochester. Previously, he had spent the years 1948/49 and 1949/50 at the
University of Chicago, and 1956/58 again in Chicago and at the Institute for Advanced
Study in Princeton.
Already in 1967 his first paper in infinite-dimensional holomorphy introduced a new
topology for the space of all holomorphic functions on a given open subset. From then on,
most of his efforts were in infinite-dimensional holomorphy, with occasional looks at
approximation theory. See, for example, his contribution to the volume Approximation
Theory and Functional Analysis, Proceedings of an International Symposium on
Approximation Theory, held in Campinas in 1977, North-Holland Math. Studies 35.
Nachbin developed during his entire life and intensive work as teacher and advisor of
many students from several countries. His editorial work is impressive. The well-known
Notas de Matematica are the best testimonial for his dedication to mathetimatics, and will
for a long time recall his name to those perusing those volumes. For a long period
Nachbin dedicated most of his time to make IMPA at Rio de Janeiro one of the most
active and prestigious mathematical centers of Latin America.
Leopoldo married Maria de Gra~a Mousinho in 1956. They had three children: Andre,
Lea, and Luis. Gra~a died in 1982 after a long chronic illness.
Joao B. Prolla
viii
PREFACE
These proceedings contain selected papers presented at the Conference on Approximation,
Probability and Related Fields held in Santa Barbara, California, on May 20-22, 1993.
The main topics of the conference were:
1) approximation of functions by polynomials, splines, and operators, and applications to
stochastics
2) numerical methods for approximation of deterministic and stochastic integrals
3) orthogonal polynomials and stochastic processes
4) positive linear operators and related deterministic and stochastic inequalities
5) multivariate approximation and interpolation
6) rate of convergence in probability theory
7) approximations and martingales
8) deterministic and stochastic inequalities
9) stability of deterministic and stochastic models
10) signal analysis
11) prediction theory
12) wavelets and approximations based on wavelets
The Conference was very successful and received many compliments. We quote some of
the letters sent by the participants:
- "Many thanks for the wonderful conference and the exemplary organization!"
- "Many thanks for your good and effective work"
We would like to thank the international organizing committee consisting of Paul Butzer,
Stamatis Cambanis and Zuhair Nashed as well as the local organizing committee consisting
of Bessy Athanasopoulos, Raisa Feldman, and Gleb Haynatzki for their superb work and
for their contribution to the success of the Conference.
A
The chairmer
~ C·p,.~lflIl",
George Anastassiou Svetlozar Rachev
ix
CONTENTS
1. Preservation of Moduli of Continuity for Bernstein-type Operators ....... .
Jose A. Adell and Jesus de la Cal
2. Lp-Korovkin Type Inequalities for Positive Linear Operators ............ 19
G.A. Anastassiou
3. On Some Shift-Invariant Integral Operators, Multivariate Case .......... 41
G.A. Anastassiou and Heinz H. Gonska
4. Multivariate Probabilistic Wavelet Approximation ...................... 65
G.A. Anastassiou, S.T. Rachev, and X.M. Yu
5. Probabilistic Approach to the Rounding Problem with
Applications to Fair Representation ............................... 75
Bessy Athanasopoulos
6. Limit Theorems for Random Multinomial Forms. . . . . . . . . . . . .. . . . . . .. .. 101
Alfredas Basalykas
7. Multivariate Boolean Trapezoidal Rules ................................ 109
Gunter Baszenski and Franz-Jurgen Delvos
8. Convergence Results for an Extension of the Fourier Transform ......... 119
Carlo Belingeri and Paolo Emilio Ricci
9. Euler Functions Eo(z) with Complex a and Applications ............... 127
Paul L. Butzer, Stefan Flocke, and Michael Hauss
10. On the Role of loo in Approximation Theory ........................... 151
B.L. Chalmers and B. Shekhtman
11. The Action Constants. . . . .. .. . .. . . ... . . . . ... .. .... .. . . . . .. . . . .. . .. . . .. 161
B.L. Chalmers and B. Shekhtman
12. Bivariate Probability Distributions Similar to Exponential ............. 167
B. Dimitrov, S. Chukova, and Z. Khalil
13. Probability Waiting-Time Results for Pattern and Frequency Quotas
in the Same Inverse Sampling Problem Via the Dirichlet .......... 179
M. Ebneshahrashoob and Milton Sobel
xi
14. Asymptotic Normality of the Collision Resolution Interval
for a Multiple Access Protocol ................................... 187
Phillip M. Feldman
15. An Example of Non-Gaussian Density Process ......................... 201
Raisa Epstein Feldman
16. Limit Theorems for Functionals of Markov Processes and
Renormalizable Stable Fields ............. . . . . . . . . . . . . . . . . . . . . . . .. 207
Raisa E. Fieldman and Nagamani Krishnakumar
17. Stable Models in Testable Asset Pricing ............................... 223
Bertrand Gamrowski and Svetlozar T. Rachev
18. A Note on the Death Process with a Random Parameter ............... 237
J.M. Gani and G.R. Haynatzki
19. Stochastic Approximation under Dependent Noises,
Detecting Signals and Adaptive Control .......................... 247
Oleg N. Granichin
20. Uniform Norm of Polynomials Constrained with Many Zeros
in Two Disjoint Intervals ........................................ 273
Matthew He
21. Admissible Location of Sample Points of Interpolation
by Bivariate Cl Quadratic Splines ............................... 283
Tian-Xiao He
22. On a Bernstein-Type Operator of Bleimann, Butzer and Hahn III ...... 297
C. Jayasri and Y. Sitaraman
23. A Problem of Analytic Monotone Extension in Three Dimensions ...... 303
Shelby Kilmer, Xingping Sun, and Xiang Ming Yu
24. On the Joint Estimation of Stable Law Parameters .................... , 315
L.B. Klebanov, J.A. Melamed, and S.T. Rachev
25. Representations and Properties of Geometric Stable Laws .............. 321
Thomasz J. Kozubowski
26. Rearrangements of Conditionally Integrable Functions ................. 339
Thomas Kunkle
27. Wavelet Transforms and Wavelet Approximations ...................... 357
E.B. Lin
28. Approximations and Derivatives of Probability Functions .............. 367
K. Marti
xii
29. Rate of Convergence in the Central Limit Theorem for
Generalized Convolutions ........................................ 379
Anna K. Panorska.
30. Order Completeness of L1 with Applications to Stochastics ............. 395
D. Plachky
31. On Reliability Analysis of Consecutive-k-Out-Of-n-F Systems
and their Generalizations - A Survey ............................ 401
Wolfgang W. Preuss and Thomas K. Boehme
32. The Uniform Closure of Convex Semi-Lattices 413
Joao B. Prolla
33. Another Pairwise Independent, Stationary Chain ...................... 423
James B. Robertson
34. The Lebesgue Integral as an Improper Riemann Integral ............... 435
Oved Shisha
35. Tests of Existence of Generalized Riemann Integrals ................... 439
Oved Shisha
36. Remark on a Theorem About Polynomials ............................. 443
P. Yertesi
Participants .............................................................. 447
Index ............................................... :..................... 453
xiii