Hilbert space

About: Hilbert space is a research topic. Over the lifetime, 29705 publications have been published within this topic receiving 637043 citations.

A Generalized Representer Theorem

Abstract: Wahba's classical representer theorem states that the solutions of certain risk minimization problems involving an empirical risk term and a quadratic regularizer can be written as expansions in terms of the training examples. We generalize the theorem to a larger class of regularizers and empirical risk terms, and give a self-contained proof utilizing the feature space associated with a kernel. The result shows that a wide range of problems have optimal solutions that live in the finite dimensional span of the training examples mapped into feature space, thus enabling us to carry out kernel algorithms independent of the (potentially infinite) dimensionality of the feature space.

Invariance theory, the heat equation, and the Atiyah-Singer index theorem

Abstract: Pseudo-Differential Operators Introduction Fourier Transform and Sobolev Spaces Pseudo-Differential Operators on Rm Pseudo-Differential Operators on Manifolds Index of Fredholm Operators Elliptic Complexes Spectral Theory The Heat Equation Local Index Formula Variational Formulas Lefschetz Fixed Point Theorems The Zeta Function The Eta Function Characteristic Classes Introduction Characteristic Classes of Complex Bundles Characteristic Classes of Real Bundles Complex Projective Space Invariance Theory The Gauss-Bonnet Theorem Invariance Theory and Pontrjagin Classes Gauss-Bonnet for Manifolds with Boundary Boundary Characteristic Classes Singer's Question The Index Theorem Introduction Clifford Modules Hirzebruch Signature Formula Spinors The Spin Complex The Riemann-Roch Theorem K-Theory The Atiyah-Singer Index Theorem The Regularity at s = 0 of the Eta Function Lefschetz Fixed Point Formulas Index Theorem for Manifolds with Boundary The Eta Invariant of Locally Flat Bundles Spectral Geometry Introduction Operators of Laplace Type Isospectral Manifolds Non-Minimal Operators Operators of Dirac Type Manifolds with Boundary Other Asymptotic Formulas The Eta Invariant of Spherical Space Forms A Guide to the Literature Acknowledgment Introduction Bibliography Notation

Reproducing Kernel Hilbert Spaces in Probability and Statistics

Abstract: 1 Theory.- 2 RKHS AND STOCHASTIC PROCESSES.- 3 Nonparametric Curve Estimation.- 4 Measures And Random Measures.- 5 Miscellaneous Applications.- 6 Computational Aspects.- 7 A Collection of Examples.- to Sobolev spaces.- A.l Schwartz-distributions or generalized functions.- A.1.1 Spaces and their topology.- A.1.2 Weak-derivative or derivative in the sense of distributions.- A.1.3 Facts about Fourier transforms.- A.2 Sobolev spaces.- A.2.1 Absolute continuity of functions of one variable.- A.2.2 Sobolev space with non negative integer exponent.- A.2.3 Sobolev space with real exponent.- A.2.4 Periodic Sobolev space.- A.3 Beppo-Levi spaces.

A course in abstract harmonic analysis

Abstract: Banach Algebras and Spectral Theory Banach Algebras: Basic Concepts Gelfand Theory Nonunital Banach Algebras The Spectral Theorem Spectral Theory of *-Representations Von Neumann Algebras Notes and References Locally Compact Groups Topological Groups Haar Measure Interlude: Some Technicalities The Modular Function Convolutions Homogeneous Spaces Notes and References Basic Representation Theory Unitary Representations Representations of a Group and Its Group Algebra Functions of Positive Type Notes and References Analysis on Locally Compact Abelian Groups The Dual Group The Fourier Transform The Pontrjagin Duality Theorem Representations of Locally Compact Abelian Groups Closed Ideals in L1(G) Spectral Synthesis The Bohr Compactification Notes and References Analysis on Compact Groups Representations of Compact Groups The Peter-Weyl Theorem Fourier Analysis on Compact Groups Examples Notes and References Induced Representations The Inducing Construction The Frobenius Reciprocity Theorem Pseudomeasures and Induction in Stages Systems of Imprimitivity The Imprimitivity Theorem Introduction to the Mackey Machine Examples: The Classics More Examples, Good and Bad Notes and References Further Topics in Representation Theory The Group C* Algebra The Structure of the Dual Space Tensor Products of Representations Direct Integral Decompositions The Plancherel Theorem Examples Appendices A Hilbert Space Miscellany Trace-Class and Hilbert-Schmidt Operators Tensor Products of Hilbert Spaces Vector-Valued Integrals