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Mathematics and Its Applications

By: Victor G. Kravchenko, V. G. Kravchenko, Georgii S. Litvinchuk

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Hardcover | 31 May 1994

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308 Pages

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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; â¬ 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(t, 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 non·Carieman 5hifl is on c sidered.

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Functional Analysis & Transforms Differential Calculus & Equations Integral Calculus & Equations Complex Analysis

Introduction
Background information	p. 1
On Noetherian operators	p. 1
On the operator of singular integration	p. 15
On the shift function and shift operator	p. 26
On C*-algebras	p. 33
Noetherity criterion and a formula for the index of a singular integral functional operator of first order in the continuous case	p. 37
Criterion of Noetherity for singular integral functional operators of first order with orientation-preserving shift	p. 40
The calculation of the index of a singular integral functional operator of the first order with a shift preserving the orientation	p. 60
The Noetherity criterion and the index formula for a singular integral functional operator of the first order with a shift changing the orientation	p. 81
References and a survey of similar or closed results	p. 89
The Noether theory of a singular integral functional operator of finite order in the continuous case	p. 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 contour	p. 104
Theorems concerning decreasing the order of functional and singular integral operators	p. 107
Noetherity criterion and a formula for the index for systems of singular integral equation with a Carleman shift	p. 113
An invertibility criterion for a matrix functional operator with a non-Carleman shift	p. 118
Noether theory for singular integral functional operators of superior orders	p. 148
References and a survey of similar results	p. 154
The Noether theory of singular integral functional operators with continuous coefficients on a non-closed contour	p. 161
The Noetherity criterion and the index formula for a singular integral operator with continuous coefficients on a non-closed contour	p. 164
The Noetherity criterion and the index formula for singular integral functional operators with continuous coefficients on a non-closed contour	p. 186
Systems of singular integral operators with shift on a non-closed contour	p. 202
References and a survey of closely related results	p. 208
The Noether theory in algebras of singular integral functional operators	p. 211
C*-algebras of singular integral operators	p. 214
C*-algebras of singular integral operators with a Carleman shift	p. 221
C*-algebras of singular integral operators with non-Carleman shift which has periodic points	p. 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 literature	p. 238
References	p. 265
Subject Index	p. 287
Table of Contents provided by Blackwell. All Rights Reserved.

Mathematics and Its Applications

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