[Show abstract][Hide abstract] ABSTRACT: Given a function \(f\) on the positive half-line \({\mathbb R}_+\) and a sequence (finite or infinite) of points \(X=\{x_k\}_{k=1}^\omega \) in \({\mathbb R}^n\), we define and study matrices \({\mathcal S}_X(f)=[f(\Vert x_i-x_j\Vert )]_{i,j=1}^\omega \) called Schoenberg’s matrices. We are primarily interested in those matrices which generate bounded and invertible linear operators \(S_X(f)\) on \(\ell ^2({\mathbb N})\). We provide conditions on \(X\) and \(f\) for the latter to hold. If \(f\) is an \(\ell ^2\)-positive definite function, such conditions are given in terms of the Schoenberg measure \(\sigma _f\). Examples of Schoenberg’s operators with various spectral properties are presented. We also approach Schoenberg’s matrices from the viewpoint of harmonic analysis on \({\mathbb R}^n\), wherein the notion of the strong \(X\)-positive definiteness plays a key role. In particular, we prove that each radial
\(\ell ^2\)
-positive definite function is strongly
\(X\)
-positive definite whenever \(X\) is a separated set. We also implement a “grammization” procedure for certain positive definite Schoenberg’s matrices. This leads to Riesz–Fischer and Riesz sequences (Riesz bases in their linear span) of the form \({\mathcal F}_X(g)=\{g(\cdot -x_j)\}_{x_j\in X}\) for certain radial functions \(g\in L^2({\mathbb R}^n)\).
Journal of Fourier Analysis and Applications 03/2015; 21(5). DOI:10.1007/s00041-015-9391-4 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The main object under consideration is a class $\Phi_n\backslash\Phi_{n+1}$
of radial positive definite functions on $\R^n$ which do not admit \emph{radial
positive definite continuation} on $\R^{n+1}$. We find certain necessary and
sufficient conditions for the Schoenberg representation measure $\nu_n$ of
$f\in \Phi_n$ in order that the inclusion $f\in \Phi_{n+k}$, $k\in\N$, holds.
We show that the class $\Phi_n\backslash\Phi_{n+k}$ is rich enough by giving a
number of examples. In particular, we give a direct proof of
$\Omega_n\in\Phi_n\backslash\Phi_{n+1}$, which avoids Schoenberg's theorem,
$\Omega_n$ is the Schoenberg kernel. We show that
$\Omega_n(a\cdot)\Omega_n(b\cdot)\in\Phi_n\backslash\Phi_{n+1}$, for $a\not=b$.
Moreover, for the square of this function we prove surprisingly much stronger
result: $\Omega_n^2(a\cdot)\in\Phi_{2n-1}\backslash\Phi_{2n}$. We also show
that any $f\in\Phi_n\backslash\Phi_{n+1}$, $n\ge2$, has infinitely many
negative squares. The latter means that for an arbitrary positive integer $N$
there is a finite Schoenberg matrix $\kS_X(f) :=
\|f(|x_i-x_j|_{n+1})\|_{i,j=1}^{m}$, $X := \{x_j\}_{j=1}^m \subset\R^{n+1}$,
which has at least $N$ negative eigenvalues.
[Show abstract][Hide abstract] ABSTRACT: Given a function $f$ on the positive half-line $\R_+$ and a sequence (finite
or infinite) of points $X=\{x_k\}_{k=1}^\omega$ in $\R^n$, we define and study
matrices $\kS_X(f)=\|f(|x_i-x_j|)\|_{i,j=1}^\omega$ called Schoenberg's
matrices. We are primarily interested in those matrices which generate bounded
and invertible linear operators $S_X(f)$ on $\ell^2(\N)$. We provide conditions
on $X$ and $f$ for the latter to hold. If $f$ is an $\ell^2$-positive definite
function, such conditions are given in terms of the Schoenberg measure
$\sigma(f)$. We also approach Schoenberg's matrices from the viewpoint of
harmonic analysis on $\R^n$, wherein the notion of the strong $X$-positive
definiteness plays a key role. In particular, we prove that \emph{each radial
$\ell^2$-positive definite function is strongly $X$-positive definite} whenever
$X$ is separated. We also implement a "grammization" procedure for certain
positive definite Schoenberg's matrices. This leads to Riesz--Fischer and Riesz
sequences (Riesz bases in their linear span) of the form
$\kF_X(f)=\{f(x-x_j)\}_{x_j\in X}$ for certain radial functions $f\in
L^2(\R^n)$. Examples of Schoenberg's operators with various spectral properties
are presented.
[Show abstract][Hide abstract] ABSTRACT: The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. We introduce and investigate the class of weakly regular boundary conditions. We show that this class is much broader than the class of regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2×22×2 Dirac type equations subject to irregular boundary conditions. Emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions.
[Show abstract][Hide abstract] ABSTRACT: We consider the second-order linear differential equation y" + A(t)y = 0 on the semiaxis with complex-valued potential function. Sufficient conditions for the potential function assuring that all solutions of the equation converge to zero at infinity are obtained. It is shown that the conditions imposed on the potential function are close to the necessary ones. One of the results seems to be new even in the case of real-valued function A(·).
Journal of Mathematical Sciences 04/2012; 182(1). DOI:10.1007/s10958-012-0730-6
[Show abstract][Hide abstract] ABSTRACT: The paper is concerned with the completeness problem of root functions of
general boundary value problems for first order systems of ordinary
differential equations. Namely, we introduce and investigate the class of
\emph{weakly regular boundary conditions}. We show that this class is much
broader than the class of {\em regular boundary conditions} introduced by G.D.
Birkhoff and R.E. Langer. Our main result states that the system of root
functions of a boundary value problem is complete and minimal provided that the
boundary conditions are weakly regular. Moreover, we show that in some cases
\emph{the weak regularity} of boundary conditions \emph{is also necessary} for
the completeness. Also we investigate the completeness for $2\times 2$ Dirac
and Dirac type equations subject to irregular or even to degenerate boundary
conditions.
We emphasize that our results are the first results on the completeness
problem for general first order systems even in the case of regular boundary
conditions.
[Show abstract][Hide abstract] ABSTRACT: We investigate negative spectra of one-dimensional (1D) Schrödinger operators with δ- and δ′-interactions on a discrete set in the framework of a new approach. Namely, using the technique of boundary triplets and
the corresponding Weyl functions, we complete and generalize the results of Albeverio and Nizhnik (Lett Math Phys 65:27–35,
2003; Methods Funct Anal Topol 9(4):273–286, 2003). For instance, we propose an algorithm for determining the number of negative
squares of the operator with δ-interactions. We also show that the number of negative squares of the operator with δ′-interactions equals the number of negative strengths.
Mathematics Subject Classification (2000)Primary 47A10–Secondary 34L40
Integral Equations and Operator Theory 05/2010; 67(1):1-14. DOI:10.1007/s00020-010-1759-x · 0.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: . A completeness theorem is proved involving a system of integro-differential equations with some λ-depending boundary conditions.
Also some sufficient conditions for the root functions to form a Riesz basis are established.
Integral Equations and Operator Theory 07/2009; 64(3):357-379. DOI:10.1007/s00020-009-1698-6 · 0.70 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Consider the Sobolev space W
2
n
(ℝ+) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients.
We study the problem of exact constants A
n,k
in inequalities of Kolmogorov type for the values of intermediate derivatives |f
(k)(0)| ≤ A
n,k
‖f‖. In the general case, the expression for the constants A
n,k
is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants
A
n,k
in the case of the following norms:
$
\left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } .
$
\left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } .
In the case of the norm ‖ · ‖1, formulas for the constants A
n,k
were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants A
n,k
is also studied in the case of the norm ‖ · ‖2. In addition, we prove a symmetry property of the constants A
n,k
in the general case.
Key wordsSobolev space-Kolmogorov-type inequalities-intermediate derivative-linear functional in Hilbert space-Vandermonde matrix-Cramer’s rule
[Show abstract][Hide abstract] ABSTRACT: We investigate negative spectra of 1--D Schr\"odinger operators with $\delta$- and $\delta'$-interactions on a discrete set in the framework of a new approach. Namely, using technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of S. Albeverio and L. Nizhnik. For instance, we propose the algorithm for determining the number of negative squares of the operator with $\delta$-interactions. We also show that the number of negative squares of the operator with $\delta'$-interactions equals the number of negative strengths. Comment: 14 pages
[Show abstract][Hide abstract] ABSTRACT: A completeness theorem is proved involving a system of integro-differential
equations with some $\lambda$-depending boundary conditions. Also some
sufficient conditions for the root functions to form a Riesz basis are
established.