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Introduction to the Theory of Singular Integral Operators with Shift : Mathematics and Its Applications - Viktor G. Kravchenko

Introduction to the Theory of Singular Integral Operators with Shift

Mathematics and Its Applications

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Published: 31st May 1994
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problem (0. 2) was the same u that of problem (0. 1). Incidentally, later on Mandzhavidze and Khvedclidze (I) and Simonenko (I) achieved a direct reduction of problem (0. 2) to problem (0. 1) with the help of conformal mappings. Apparenlly, the first paper in which SIES were considered was the paper by Vekua (2) published in 1948. Vekua verified that the equation (0. 3) where (1; C(f), 5 is the operator of 'ingular integration with a Cauchy kernel (Srp)(!) "" (". i)-I fr(T - t)-lrp(T)dT, W is the shift operator (WrpHt) = rp{a(tĀ», in the case 01 = - (13,0, = 0. , could be reduced to problem (0. 2). We note thai, in problem (0. 2), the shift ott) need not be a Carlemao shift, . ei. , it is oot necessary that a . . (t) :::: t for some integer 11 ~ 2, where ai(l) "" o(ok_dt)), 0(1(1) ::::!. For the first time, the condition 0,(1) == 1 appeared in BPAFS theory in connection with the study of the problem (0. 4) by Carle man (2) who, in particular, showed that problem (0. 4) Wall a natural generalization of the problem on the existence of an a. utomorphic function belonging to a certain group of Fucs. Thus, the paper by Vckua (2) is also the fint paper in which a singular integral equation with a nonĀ·Carieman 5hifl is on c sidered.

Introduction
Background informationp. 1
On Noetherian operatorsp. 1
On the operator of singular integrationp. 15
On the shift function and shift operatorp. 26
On C*-algebrasp. 33
Noetherity criterion and a formula for the index of a singular integral functional operator of first order in the continuous casep. 37
Criterion of Noetherity for singular integral functional operators of first order with orientation-preserving shiftp. 40
The calculation of the index of a singular integral functional operator of the first order with a shift preserving the orientationp. 60
The Noetherity criterion and the index formula for a singular integral functional operator of the first order with a shift changing the orientationp. 81
References and a survey of similar or closed resultsp. 89
The Noether theory of a singular integral functional operator of finite order in the continuous casep. 101
The Noetherity criterion and the index formula for a system of singular integral equations with a Cauchy kernel and continuous coefficients on a closed contourp. 104
Theorems concerning decreasing the order of functional and singular integral operatorsp. 107
Noetherity criterion and a formula for the index for systems of singular integral equation with a Carleman shiftp. 113
An invertibility criterion for a matrix functional operator with a non-Carleman shiftp. 118
Noether theory for singular integral functional operators of superior ordersp. 148
References and a survey of similar resultsp. 154
The Noether theory of singular integral functional operators with continuous coefficients on a non-closed contourp. 161
The Noetherity criterion and the index formula for a singular integral operator with continuous coefficients on a non-closed contourp. 164
The Noetherity criterion and the index formula for singular integral functional operators with continuous coefficients on a non-closed contourp. 186
Systems of singular integral operators with shift on a non-closed contourp. 202
References and a survey of closely related resultsp. 208
The Noether theory in algebras of singular integral functional operatorsp. 211
C*-algebras of singular integral operatorsp. 214
C*-algebras of singular integral operators with a Carleman shiftp. 221
C*-algebras of singular integral operators with non-Carleman shift which has periodic pointsp. 228
Further development of a local method for studying the Noetherity of bounded linear operators of non-local type and its applications. Commentaries to the literaturep. 238
Referencesp. 265
Subject Indexp. 287
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792328643
ISBN-10: 0792328647
Series: Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 288
Published: 31st May 1994
Publisher: SPRINGER VERLAG GMBH
Country of Publication: NL
Dimensions (cm): 23.39 x 15.6  x 1.91
Weight (kg): 0.61