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1
Content available Finitely additive functions in measure theory and applications
Alpay Daniel, Jorgensen Palle
Opuscula Mathematica
|
2024
|
Vol. 44, no. 3
323--339
EN
In this paper, we consider, and make precise, a certain extension of the Radon–Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of μ-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures μ, and to adjoints of composition operators.
2
Content available remote New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry
Alpay Daniel, Jorgensen Palle E.T.
Opuscula Mathematica
|
2021
|
Vol. 41, no. 3
283--300
EN
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.
3
Content available remote Reproducing Kernel Hilbert Space and Coalescence Hidden-variable Fractal Interpolation Functions
Prasad Srijanani Anurag
Demonstratio Mathematica
|
2019
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Vol. 52, nr 1
467--474
EN
Reproducing Kernel Hilbert Spaces (RKHS) and their kernel are important tools which have been found to be incredibly useful in many areas like machine learning, complex analysis, probability theory, group representation theory and the theory of integral operator. In the present paper, the space of Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is demonstrated to be an RKHS and its associated kernel is derived. This extends the possibility of using this new kernel function, which is partly self-affine and partly non-self-affine, in diverse fields wherein the structure is not always self-affine.
4
Content available Frames and factorization of graph Laplacians
Jorgensen P., Tian F.
Opuscula Mathematica
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2015
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Vol. 35, no. 3
293--332
EN
Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space [formula] of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in [formula] we characterize the Priedrichs extension of the [formula]-graph Laplacian. We consider infinite connected network-graphs G = (V, E), V for vertices, and E for edges. To every conductance function c on the edges E of G, there is an associated pair [formula] where [formula] in an energy Hilbert space, and Δ (=Δc) is the c-graph Laplacian; both depending on the choice of conductance function c. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in [formula] consisting of dipoles. Now Δ is a well-defined semibounded Hermitian operator in both of the Hilbert [formula] and [formula]. It is known to automatically be essentially selfadjoint as an [formula]-operator, but generally not as an [formula] operator. Hence as an [formula] operator it has a Friedrichs extension. In this paper we offer two results for the Priedrichs extension: a characterization and a factorization. The latter is via [formula].
5
Content available Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels
Castro L. P., Saitoh S., Tuan N. M.
Opuscula Mathematica
|
2012
|
Vol. 32, no. 4
633-646
EN
This paper introduces a general concept of convolutions by means of the theory of reproducing kernels which turns out to be useful for several concrete examples and applications. Consequent properties are exposed (including, in particular, associated norm inequalities).
6
Content available Matrices related to some Fock space operators
Rudol K.
Opuscula Mathematica
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2011
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Vol. 31, no. 2
289-296
EN
Matrices of operators with respect to frames are sometimes more natural and easier to compute than the ones related to bases. The present work investigates such operators on the Segal-Bargmann space, known also as the Fock space. We consider in particular some properties of matrices related to Toeplitz and Hankel operators. The underlying frame is provided by normalised reproducing kernel functions at some lattice points.
7
Content available remote Recent developments in stabilized Galerkin and collocation meshfree methods
Chen J-S., Chi S-W., Hu H-Y.
Computer Assisted Mechanics and Engineering Sciences
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2011
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Vol. 18, no. 1-2
3-21
EN
Meshfree methods have been developed based on Galerkin type weak formulation and strong formulationwith collocation. Galerkin type formulation in conjunction with the compactly supported approximation functions and polynomial reproducibility yields algebraic convergence, while strong form collocationmethod with nonlocal approximation such as radial basis functions offers exponential convergence. In thiswork, we discuss rank instability resulting from the nodal integration of Galerkin type meshfree methodas well as the ill-conditioning type instability in the radial basis collocation method. We present the recentadvances in resolving these diffculties in meshfree methods, and demonstrate how meshfree methods can be applied to problems di?cult to be modeled by the conventional ?nite element methods due to their intrinsic regularity constraints.
8
Content available Próbkowanie sygnałów diagnostycznych. Część V. Próbkowanie sygnałów o nieograniczonym paśmie za pomocą nieklasycznych jąder
Syroka Z.
Diagnostyka
|
2010
|
nr 2(54)
65-70
PL
W pracy przedstawiono matematyczny opis próbkowania sygnałów diagnostycznych posiadających nieograniczone pasmo przy pomocy nieklasycznych jąder oraz próbkowania sygnałów określonych na zbiorach mierzalnych.
EN
In this article is presented mathematical description of diagnostic signals sampling with infinite frequency band using non clasical kernels and sampling of signals defined on measurment sets.
9
Content available remote On weighted harmonic Bergman spaces
Petrosyan A.
Demonstratio Mathematica
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2008
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Vol. 41, nr 1
73-83
EN
This paper is devoted to the investigation of the weighted Bergman harmonic spaces bp/alpha(B] in the unit ball in Rn. The reproducing kernel Ralpha for the ball is constructed and the integral representation for functions in bp/alpha(B) by means of this kernel is obtained. Besides an linear mapping between the bp/alpha(B) spaces and the ordinary L2-space on the unit sphere, which has an explicit form of integral operator along with its inversion, is established.
10
Content available Próbkowanie sygnałów diagnostycznych. Część 3. Próbkowanie w przestrzeni Hilberta z bazami wielomianowymi za pomocą nieklasycznych jąder
Syroka Z.
Diagnostyka
|
2007
|
nr 2(42)
35-41
PL
W pracy wyprowadzono regułą Cristoffela - Darboux oraz dokonano analizy próbkowania sygnałów diagnostycznych przy wykorzystaniu jądra Legendr'a, Czebyszewa, Laguerre'a i Hermite'a. Podano metodykę wyprowadzania jąder reprodukujących w bazach opartych o klasyczne wielomiany ortogonalne.
EN
In this article removaled the Cristoffela - Darboux rule. The analysis of the sampling diagnostic signals using Legendr's, Czebyszew's, Laguerre's and Hermite's kernals was made. Methodology of derivation of reproducing kernels in basics, based on clasical ortogonal polynomial.
11
Content available Próbkowanie sygnałów diagnostycznych. Część 2. Próbkowanie w przestrzeni Hilberta z bazami harmonicznymi za pomocą nieklasycznych jąder
Syroka Z.
Diagnostyka
|
2007
|
nr 2(42)
27-34
PL
W pracy dokonano analizy próbkowania sygnałów diagnostycznych przy wykorzystaniu jądra Dirichleta, Fejere'a, de la Vallee Poussina i Poissona. Pokazano pełne matematyczne wyprowadzenie tych jąder; pierwsze trzy jądra są ze sobą powiązane. Podano zależności między nimi oraz ich przebiegi graficzne.
EN
In this article the analysis of the sampling diagnostic signals using Dirichlet's, Fejer's, Poisson's and de la Vallee Poussin's kernels was made. The full derivation of those kernels have given; first three off them one connected. Dependences between them and their graphical representation have also given in this article.
12
Content available Próbkowanie sygnałów diagnostycznych. Część 1. Próbkowanie w przestrzeni Hilberta z reprodukującym jądrem Shanona
Syroka Z.
Diagnostyka
|
2007
|
nr 2(42)
19-26
PL
W pracy przedstawiono matematyczny opis sygnałów diagnostycznych przestrzeni Hilberta oraz sposób konstrukcji tej przestrzeni. Podano teorię jąder reprodukujących w zastosowaniu do próbkowania sygnałów diagnostycznych oraz zapis klasycznego twierdzenia o próbkowaniu Shanona wykorzystującego teorię jąder reprodukujących.
EN
In this article is defined the diagnostic signals in the reproducing kernel Hilbert space and the way this space is constructed. The theory of the reproducing kernel Hilbert space and Shanon theorem in this space were given.
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