Sobolev spaces of functions on a Hilbert space endowed with a translation-invariant measure and approximations of semigroups

2020 ◽  
Vol 84 (4) ◽  
pp. 694-721
Author(s):  
V. M. Busovikov ◽  
V. Zh. Sakbaev
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Sobolev Spaces ◽  
Translation Invariant

On Learning Vector-Valued Functions

2005 ◽  
Vol 17 (1) ◽  
pp. 177-204 ◽  
Author(s):  
Charles A. Micchelli ◽  
Massimiliano Pontil
Keyword(s):  
Hilbert Space ◽  
Learning Theory ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Support Vector ◽  
Translation Invariant ◽  
Minimal Norm ◽  
Finite Set ◽  
Vector Valued Functions ◽  
Vector Valued

In this letter, we provide a study of learning in a Hilbert space of vector-valued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a reproducing kernel Hilbert space of functions whose values lie in Y. In this setting, we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory. We consider specific examples of such functionals corresponding to multiple-output regularization networks and support vector machines, for both regression and classification. Finally, we provide classes of operator-valued kernels of the dot product and translation-invariant type.

scholarly journals HAMILTONIAN FORMALISM FOR BLACK HOLES AND QUANTIZATION II

1996 ◽  
Vol 05 (03) ◽  
pp. 227-250 ◽  
Author(s):  
MARCO CAVAGLIÀ ◽  
VITTORIO DE ALFARO ◽  
ALEXANDRE T. FILIPPOV
Keyword(s):  
Hilbert Space ◽  
Black Holes ◽  
Black Hole ◽  
Invariant Measure ◽  
Hamiltonian Formalism ◽  
Quantization Method ◽  
Schwarzschild Black Hole ◽  
Canonical Formulation ◽  
Gauge Fixing ◽  
Rigid Transformations

We quantize by the Dirac-Wheeler-DeWitt method the canonical formulation of the Schwarzschild black hole developed in a previous paper. We investigate the properties of the operators that generate rigid symmetries of the Hamiltonian, establish the form of the invariant measure under the rigid transformations, and determine the gauge fixed Hilbert space of states. We also prove that the reduced quantization method leads to the same Hilbert space for a suitable gauge fixing.

scholarly journals TOPOLOGICAL MEASURE AND GRAPH-DIFFERENTIAL GEOMETRY ON THE QUOTIENT SPACE OF CONNECTIONS

1994 ◽  
Vol 03 (01) ◽  
pp. 207-210 ◽  
Author(s):  
JERZY LEWANDOWSKI
Keyword(s):  
Differential Geometry ◽  
Hilbert Space ◽  
Invariant Measure ◽  
Quotient Space ◽  
Vector Spaces ◽  
Integral Calculus ◽  
Topological Vector Spaces ◽  
Topological Measure ◽  
Non Linear ◽  
Densely Defined

Integral calculus on the space [Formula: see text] of gauge equivalent connections is developed. By carring out a non-linear generalization of the theory of cylindrical measures on topological vector spaces, a faithfull, diffeomorphism invariant measure is introduced on a suitable completion of [Formula: see text]. The strip (i.e. momentum) operators are densely-defined in the resulting Hilbert space and interact with the measure correctly

2-D QUANTUM GRAVITY AND LIOUVILLE THEORY

1990 ◽  
Vol 05 (18) ◽  
pp. 1411-1421 ◽  
Author(s):  
ERIC D’HOKER ◽  
P.S. KURZEPA
Keyword(s):  
Invariant Measure ◽  
Quantum Gravity ◽  
First Principles ◽  
Liouville Theory ◽  
Translation Invariant ◽  
Anomalous Dimensions ◽  
Conformal Gauge ◽  
Liouville Action

We quantize the Liouville theory, or 2-D quantum gravity, and quantum supergravity in the conformal gauge. We explicitly calculate the Jacobian accompanying the change from the Weyl invariant measure to the translation invariant one. We show that it is of the same form as the original Liouville action, thus establishing a conjecture of David and Distler and Kawai. This calculation yields dressed gravitational central charges and anomalous dimensions from first principles.

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