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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Integral equations
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One-dimensional Linear Singular Integral Equations, v. 2 - General Theory and Applications (Hardcover)
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One-dimensional Linear Singular Integral Equations, v. 2 - General Theory and Applications (Hardcover)
Series: Operator Theory: Advances and Applications, 54
Expected to ship within 12 - 17 working days
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6 Preliminaries.- 6.1 The operator of singular integration.- 6.2
The space Lp(?, ?).- 6.3 Singular integral operators.- 6.4 The
spaces $$L_{p}^{ + }(\Gamma, \rho ), L_{p}^{ - }(\Gamma, \rho ) and
\mathop{{L_{p}^{ - }}}\limits^{^\circ } (\Gamma, \rho )$$.-
6.5 Factorization.- 6.6 One-sided invertibility of singular
integral operators.- 6.7 Fredholm operators.- 6.8 The local
principle for singular integral operators.- 6.9 The interpolation
theorem.- 7 General theorems.- 7.1 Change of the curve.- 7.2 The
quotient norm of singular integral operators.- 7.3 The principle of
separation of singularities.- 7.4 A necessary condition.- 7.5
Theorems on kernel and cokernel of singular integral operators.-
7.6 Two theorems on connections between singular integral
operators.- 7.7 Index cancellation and approximative inversion of
singular integral operators.- 7.8 Exercises.- Comments and
references.- 8 The generalized factorization of bounded measurable
functions and its applications.- 8.1 Sketch of the problem.- 8.2
Functions admitting a generalized factorization with respect to a
curve in Lp(?, ?).- 8.3 Factorization in the spaces Lp(?, ?).- 8.4
Application of the factorization to the inversion of singular
integral operators.- 8.5 Exercises.- Comments and references.- 9
Singular integral operators with piecewise continuous coefficients
and their applications.- 9.1 Non-singular functions and their
index.- 9.2 Criteria for the generalized factorizability of power
functions.- 9.3 The inversion of singular integral operators on a
closed curve.- 9.4 Composed curves.- 9.5 Singular integral
operators with continuous coefficients on a composed curve.- 9.6
The case of the real axis.- 9.7 Another method of inversion.- 9.8
Singular integral operators with regel functions coefficients.- 9.9
Estimates for the norms of the operators P?, Q? and S?.- 9.10
Singular operators on spaces H?o(?, ?).- 9.11 Singular operators on
symmetric spaces.- 9.12 Fredholm conditions in the case of
arbitrary weights.- 9.13 Technical lemmas.- 9.14 Toeplitz and
paired operators with piecewise continuous coefficients on the
spaces lp and ?p.- 9.15 Some applications.- 9.16 Exercises.-
Comments and references.- 10 Singular integral operators on
non-simple curves.- 10.1 Technical lemmas.- 10.2 A preliminary
theorem.- 10.3 The main theorem.- 10.4 Exercises.- Comments and
references.- 11 Singular integral operators with coefficients
having discontinuities of almost periodic type.- 11.1 Almost
periodic functions and their factorization.- 11.2 Lemmas on
functions with discontinuities of almost periodic type.- 11.3 The
main theorem.- 11.4 Operators with continuous coefficients - the
degenerate case.- 11.5 Exercises.- Comments and references.- 12
Singular integral operators with bounded measurable coefficients.-
12.1 Singular operators with measurable coefficients in the space
L2(?).- 12.2 Necessary conditions in the space L2(?).- 12.3
Lemmas.- 12.4 Singular operators with coefficients in ?p(?).
Sufficient conditions.- 12.5 The Helson-Szegoe theorem and its
generalization.- 12.6 On the necessity of the condition a ? Sp.-
12.7 Extension of the class of coefficients.- 12.8 Exercises.-
Comments and references.- 13 Exact constants in theorems on the
boundedness of singular operators.- 13.1 Norm and quotient norm of
the operator of singular integration.- 13.2 A second proof of
Theorem 4.1 of Chapter 12.- 13.3 Norm and quotient norm of the
operator S? on weighted spaces.- 13.4 Conditions for Fredholmness
in spaces Lp(?, ?).- 13.5 Norms and quotient norm of the operator
aI + bS?.- 13.6 Exercises.- Comments and references.- References.
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