Articoli correlati a Principal Functions
Sinossi
During the decade and a half that has elapsed since the intro- duction of principal functions (Sario [8 J), they have become impor- tant tools in an increasing number of branches of modern mathe- matics. The purpose of the present research monograph is to systematically develop the theory of these functions and their ap- plications on Riemann surfaces and Riemannian spaces. Apart from brief background information (see below), nothing contained in this monograph has previously appeared in any other book. The basic idea of principal functions is simple: Given a Riemann surface or a Riemannian space R, a neighborhood A of its ideal boundary, and a harmonic function s on A, the principal function problem consists in constructing a harmonic function p on all of R which imitates the behavior of s in A. Here A need not be connected, but may include neighborhoods of isolated points deleted from R. Thus we are dealing with the general problem of constructing harmonic functions with given singularities and a prescribed behavior near the ideal boundary. The function p is called the principal function corresponding to the given A, s, and the mode of imitation of s by p. The significance of principal functions is in their versatility.
Le informazioni nella sezione "Riassunto" possono far riferimento a edizioni diverse di questo titolo.
Contenuti
Introduction: What are Principal Functions?.- 0 Prerequisite Riemann Surface Theory.- §1. Topology of Riemann Surfaces.- 1. Definition of a Riemann surface.- Conformal structure.- Holomorphic functions.- Holo-morphic mappings.- Basis for conformal structure.- Examples.- Bordered Riemann surfaces.- Arcs.- Orientation.- The double.- Open and closed surfaces.- 2. Compactifications.- One-point compactification.- Topological representatives.- Boundary components.- 3. Homology.- Chains.- Boundary operator.- Intersection numbers.- Canonical bases.- §2. Analysis on Riemann Surfaces.- 1. Harmonic functions.- Harmonic functions.- Subharmonic functions.- The Dirichlet problem.- Regular subregions and partitions.- Directed limits.- Countability.- 2. Differential forms.- Notational conventions.- Differentials.- A second order differential.- Exterior algebra.- 3. Integration.- Line integrals.- Area integrals.- Stokes’ theorem.- Integration over open sets.- The Dirichlet integral.- Convergence in Dirichlet norm.- A normal family criterion.- I The Normal Operator Method.- §1. The Main Existence Theorem.- 1. A lemma on harmonic functions.- The q-lemma.- 2. The main theorem.- Normal operators.- The main theorem.- Reduction of the problem.- An invertible operator.- Existence of X.- The main theorem with estimates.- §2. Normal Operators.- 1. Operators on compact bordered surfaces.- The operator L0.- The operator L1.- Partitions of ?.- 2. Operators on open surfaces.- Limits of normal operators.- Convergence of L0?.- Consistent partitions.- Convergence of L1?.- Direct sum operators.- 3. Convergence of principal functions.- Convergence of operators.- Convergence of p?.- §3. The Principal Functions p0 and p1.- 1. Integral representations.- Auxiliary functions.- Integral representations.- Convergence of auxiliary functions.- 2. Convergence of p0 p1.- Proof of Li?? Li.- Principal functions with singularities.- §4. Special Topics.- 1. An integral equation.- 2. Estimates of q.- The Poincaré metric.- Poincaré diameter.- Harmonic metric.- Harmonic diameter.- Comparison.- 3. The space of normal operators. Ahlfors’ problem.- The space N(A).- The space N(?, ?).- The space M(?, ?).- Ahlfors’ problem.- A counterexample.- 4. Principal functions and orthogonal projection.- Weyl’s lemma.- Poincaré type inequality.- Existence proof by orthogonal projection.- II Principal Functions.- §1. Main Extremal Theorem.- 1. An extremal function.- Principal functions.- The extremal functional.- Main Extremal Theorem.- 2. Special cases.- The functions p0+p1.- Meromorphic and logarithmic poles.- Regular principal functions.- §2. Conformal Mapping.- 1. Parallel slit mappings.- Principal meromorphic functions.- Horizontal slit mapping.- Extremal property.- 2. Mapping by P0+P1.- Compact bordered surfaces.- Boundary behavior of P0?.- Convexity of the boundary.- Extremal property of P0+P1.- Extremal property of P0—P1.- 3. Circular and radial slit planes.- Principal meromorphic functions F0, F1.- Mapping and extremal properties.- §3. Reproducing Differentials.- 1. The Hilbert space ?h.- Harmonic differentials.- Subspaces of ?h.- 2. Reproducing differentials.- Basic properties.- Construction of ??.- Proof of the theorem.- 3. Orthogonal projections.- The space ?a.- The spaces ?hm and ?h*se??ho.- The space ?ho.- The spaces ?he and ?h*se??he.- The space ?hse.- The space ?ase.- §4. Interpolation Problems.- 1. Generalities.- 2. Bordered surfaces.- Reflection of differentials.- Interpolation on bordered surfaces.- 3. Open regions.- §5. The Theorems of Riemann-Roch and Abel.- 1. Riemann-Roch type theorems.- The classical case.- An application to conformal mapping.- 2. Abel’s theorem.- §6. Extremal Length.- 1. Fundamentals.- Linear densities.- Extremal length.- Extremal and conjugate extremal distance.- Level curves.- An application.- A geometric inequality.- 2. Infinite extremal length.- Basic properties.- Examples.- Level curves.- Relative homology.- Integration.- Principal functions.- 3. Extremal length of homology classes.- Generalized homology.- Compact surfaces.- Open surfaces.- ?ho-reproducers.- III Capacity Stability and Extremal Length.- §1. Generalized Capacity Functions.- 1. Construction of u.- Induced partitions.- Convergence of uni.- Dependence of u(?0, ?1, ?0, ?1) on ?0.- Boundary behavior of du.- 2. Construction of p.- Construction of p(?, ?1, ?0, ?1).- Capacity.- 3. A maximum problem.- Boundary components of maximal capacity.- §2. Extremal Length.- 1. Continuity lemma.- Definition of F F*.- Continuity lemma.- Restatement.- Notation and terminology.- Definition of F’.- Proof of (a) and (b’’).- Proof of (c).- 2. Extremal and conjugate extremal distance.- Proof of ?(F*) = || du ||2.- 3. Properties of u and p.- Uniqueness of du.- Monotone properties.- Properties of p.- Uniqueness of dp.- 4. Capacities.- Notation.- Capacities.- Extremal properties.- Green’s function.- §3. Exponential Mappings of Plane Regions.- 1. Extremal slit annuli and disks.- The mappings U, P.- A reduction to annuli.- Area of slits.- Characterization of R.- 2. Special cases.- Circular slit annulus.- Minimal circular slit annulus.- Extremal property of P?.- Extremal radial slit annulus.- Minimal radial slit annulus.- 3. Generalizations.- Removable boundary components.- A dual problem.- §4. Stability.- 1. Parallel slit mappings.- The mappings P?.- Extremal length properties.- 2. The mapping P0+P1.- A property of P?.- Convexity theorem.- Proof of (b’).- 3. Stability.- Strong weak and unstable components.- A condition for weakness.- A condition for strength.- Extendability.- 4. Vanishing capacity.- Nonuniqueness.- Evans potential.- IV Classification Theory.- §1. Inclusion Relations.- 1. Properties of principal functions.- Reproducing differentials.- Spans.- Univalent functions.- Capacities and extremal length.- 2. Classes of Riemann surfaces.- Notation.- Classes OHY.- Classes OAY for planar surfaces.- Capacities on planar surfaces.- Classes OAY for nonplanar surfaces.- Parabolic surfaces.- Summary.- §2. Other Properties of the O-Classes.- 1. Normal operators and ideal boundary properties.- Degeneracies of normal operators.- Removable singularities.- Properties of the ideal boundary.- An example.- 2. Extremal distances.- Plane regions.- V Analytic Mappings.- §1. The Proximity Function.- 1. Use of principal functions.- Construction of p(?, ?1).- The proximity function s(?,a).- Proof of symmetry.- Boundedness from below.- Bounds for p(?,a).- Proof of lemma.- 2. The conformal metric.- Area of S.- Curvature.- §2. Analytic Mappings.- 1. First Main Theorem.- Notation.- The fundamental functions.- 2. Second Main Theorem.- Estimate of F2(h).- Evaluation of G2.- 3. Defects and ramifications.- Admissible functions.- Defect-ramification relation.- Consequences.- §3. Meromorphic Functions.- 1. The classical case.- Proximity function.- Specializing R.- Classical defect-ramification relations.- Admissible functions.- 2. Rp-surfaces.- VI Principal Forms and Fields on Riemannian Spaces.- §1. Principal Functions on Riemannian Spaces.- 1. Fundamentals of Riemannian spaces.- Riemannian spaces.- Differential forms.- 0-forms.- Green’s functions.- Harmonic functions.- 2. The main theorem.- Normal operators.- The main theorem.- Operators L0 and L1.- 3. Functions with singularities.- Construction on regular subregions.- Extremum property of Pµ?.- The span of ?.- A convergence theorem.- Non-compact regions.- The span for R.- 4. Classification of Riemannian spaces.- The class HD.- Green’s function.- Harmonic measures.- Capacity functions.- The capacity.- Completeness and degeneracy.- Other null classes.- List of problems.- 5. Interpolation problem.- 6. Principal functions in physics.- §2. Principal Forms on Locally Flat spaces.- 1. p-forms on regular regions.- Tangential and normal parts.- The point norm.- 2. Bounded principal forms.- Locally flat spaces.- Problem.- Normal operators.- A q-lemma.- The Main Existence Theorem.- Proof of Theorem 2B for parallel V.- 3. Border reduction.- Generalized Dirichlet operator.- Border reduction theorem.- Solution to Problem 3B.- §3. Principal Forms on Riemannian Spaces.- 1. Classes of p-forms.- Weak derivatives.- Subclasses of harmonic forms.- Green’s formulas.- 2. Principal harmonic fields.- Problem.- Main theorem.- Specialization.- Spherelike components.- Point singularities.- Ahlfors’ method.- 3. Principal harmonic forms.- Problem.- Main theorem.- Specialization.- 4. Principal semifields.- Semifields.- Tensor potentials.- 5. Generalization.- LT-principal forms.- Existence.- System of operators.- Special cases.- VII Principal Functions on Harmonic Spaces.- §1. Harmonic Spaces.- 1. Harmonic structures.- Regularity of open sets.- Definition of harmonic space.- Basic properties.- Perron family.- 2. Dirichlet’s problem.- Regular points.- Outer-regular sets.- 3. Classification.- The operator B.- Parabolicity.- §2. Harmonic Functions with General Singularities.- 1. Problem and its reduction.- Problem.- Reformulation.- Reduction.- 2. Riesz-Schauder theory.- Dual operator T*.- Eigenvalues.- The eigenvalue 1.- Invariant measure.- 3. Solution of Problem 1C.- Result.- 4. Solution of Problem 1B.- Flux.- Result.- Solution of the original problem.- §3. General Principal Function Problem.- 1. Principal functions.- Quasinormal operators.- Associated operator.- L-flux.- 2. Generalized main existence theorem.- Result.- Appendix Sario Potentials on Riemann Surfaces.- §1. Continuity Principle.- 1. Joint continuity of s(? a).- Definition of s(?,a).- Continuity outside the diagonal set.- Decomposition of s(?,a).- 2. Sario potentials.- Potential-theoretic principles.- Local maximum principle.- Continuity principle.- 3. Unicity principle.- Uniqueness.- §2. Maximum Principle.- 1. Frostman’s maximum principle.- Global maximum.- 2. Fundamental theorem.- Capacity.- Capacitary measure.- Subadditivity.- 3. Energy principle.- Ninomiya’s theorem.- Unicity of capacitary measure.- Author Index.
Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.
(nessuna copia disponibile)
Cerca: Inserisci un desiderataNon riesci a trovare il libro che stai cercando? Continueremo a cercarlo per te. Se uno dei nostri librai lo aggiunge ad AbeBooks, ti invieremo una notifica!
Inserisci un desiderata
