Related papers: A remark on the reproducing kernel thesis for Hank…
In 2012, we obtained Oppenheim's inequality for positive semidefinite matrices and its equality condition by the reproducing kernel method. In this paper, as a continuation, we give a reproducing kernel proof of the block matrix version of…
We consider conditions on a given system $\mathcal{F}$ of vectors in Hilbert space $\mathcal{H}$, forming a frame, which turn $\mathcal{H}$ into a reproducing kernel Hilbert space. It is assumed that the vectors in $\mathcal{F}$ are…
We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of…
This study presents a theoretical model for a self-replicating mechanical system inspired by biological processes within living cells and supported by computer simulations. The model decomposes self-replication into core components, each of…
In this paper we combine the theory of reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with special emphasis on reproducing property of kernels. From the reproducing property of…
Compressions of Toeplitz operators to coinvariant subspaces of $H^2$ are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies…
Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the…
A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this…
This paper reviews the functional aspects of statistical learning theory. The main point under consideration is the nature of the hypothesis set when no prior information is available but data. Within this framework we first discuss about…
We study both canonical reproducing kernels and constructive reproducing kernels for holomorphic functions in $\CC^1$ and $\CC^n$. We compare and contrast the two, and also develop important relations between the two types of kernels. We…
In the classical Hardy space $H^2(\mathbb{D})$, it is well-known that the kernel of the Hankel operator is invariant under the action of shift operator S and sometimes nearly invariant under the action of backward shift operator $S^{*}$. It…
This work is concerned with the kernel-based approximation of a complex-valued function from data, where the frequency response function of a partial differential equation in the frequency domain is of particular interest. In this setting,…
We discuss replica analytic continuation using several simple models in order to prove mathematically the validity of replica analysis, which is used in a wide range of fields related to large scale complex systems. While replica analysis…
We review the basic properties of paired operators and their adjoints, the transposed paired operators, with particular reference to commutation relations, and we study the properties of their kernels, bringing out their similarities and…
Hilbertian kernel methods and their positive semidefinite kernels have been extensively used in various fields of applied mathematics and machine learning, owing to their several equivalent characterizations. We here unveil an analogy with…
We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of…
Two variants of generalizations of Hankel operators to the case of linearly ordered abelian groups are considered, criteria of the boundedness and compactness of these operators are given, among them in terms of functions of bounded mean…
The partial success of the block renormalization group techniques is analysed in terms of a functional operator which formalizes the idea of self-replicability of a system in terms of smaller blocks which are similar to the original. The…
We give a short proof of the trace formula for Hecke operators on modular forms for the modular group, using the action of Hecke operators on the space of period polynomials.
We provide a simple proof of Kamp's theorem.