Asymptotic Expansions for Pseudodifferential Operators on Bounded Domains
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1152 Harold Widom
Asymptotic Expansions for Pseudodifferential Operators on Bounded Domains
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Harold Widom Department of Mathematics, University of California Santa Cruz, California 95064, USA
Mathematics Subject Classification (1980): 35S05, 45C05, 45E 10, 45P05, 47B35,47G05
ISBN 3-540-15701-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15701-8 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Vervvertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
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,
The author wishes to thank the Institut des Hautes Etudes Scientifiques. Bures-surYvette. for their hospitality durin, the preparation of this memoir and also the US. National Science Foundation for their support.
Tahle of CeateD..
Chapter I. Introduction
.
..
a
a
..
1
Chapter U. Derivation of the Series.
. . . . . . . . . . . . . . . ..
16
Chapter W. The Szego and Heat Expansions
. . . . . . . . . . . . . . . ..
36
Chapter IV. t I, it suffices that U be any open set containing the closure of
4
{p(ll,-.t): (ll,U E 0 XR It } .
We leave this last as a puzzle for the reader. The e:&pansion (1.1) will be established under the assumptions described above. Formulas for lOme of the coeffICient. will be Jiven in fJ.6 where also the method of proof will be described. 1.3. Here is a little bit of history. The oriJinal theorem of SzegCi (26] i. that if • is nonne.ative (and satisfies + E £1,10'+ E £1) then
lim N.-
{+ J
(de' TN(. » 1/N - exp
lo••(ele)de}.
(1.9)
This relation is important in the study of the asymptotic. of polynomials orthogonal on the unit circle. He generalized this by showinS (27] that if • is real and bounded then for any Riemann integrable function f lim N.-
where
N- 1 I f().I,N)'"
).1,11
lim N.-
+ J
f(.(e"»de
are the eigenvalues of TN(.). Equivalently
N- 1 tr f(TN
1 (.»'" :;1
2.
0
(1.10)
f(+(e"»de.
Clearly (1.9) is the special case of this where fee) 10' e. Later, in answer to a question of L. Onsager which arose in statistical mechanics,SzegCi [28] sharpened (1.9) to 10gdetTN(.)-N(log.r(0) +
I•
11
k (log.nk)(log.r(-k) +0(1)
(1.11)
assuming. is positive and sufficiently smooth. Continuous analogues of these results were soon discovered [14, 15]. The function .(e 18) on the unit circle was replaced by a function a(t) on R and the discrete convolution given by the Toeplitz matrix replaced by convolution by a (z) on an interval [O,a] where a is a large parameter. (The switch from " to" is a result of notati
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