L. L. Oridoroga

Donetsk National Technical University, Yuzovo, Donetsk, Ukraine

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Publications (15)4.74 Total impact

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    L. Golinskii, M. Malamud, L. Oridoroga
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    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.
  • L. Golinskii, M. Malamud, L. Oridoroga
    Journal of Fourier Analysis and Applications 01/2015; DOI:10.1007/s00041-015-9391-4 · 1.08 Impact Factor
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    L. Golinskii, M. Malamud, L. Oridoroga
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    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.
  • Mark M. Malamud, Leonid L. Oridoroga
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    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.
    Journal of Functional Analysis 10/2012; 263(7):1939–1980. DOI:10.1016/j.jfa.2012.06.016 · 1.15 Impact Factor
  • Anton A. Lunyov, Leonid L. Oridoroga
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    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
  • A.V. Agibalova, M.M. Malamud, L.L. Oridoroga
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    ABSTRACT: The paper is devoted to general 2×2 first order systems of ordinary differential equations with general linear boundary conditions. While the problem of completeness of root vectors has been studied thoroughly for a single higher order equation, for systems of differential equations much less is known. However, in several recent papers by the authors, the class of systems and boundary conditions, for which the set of root vectors is known to be complete or incomplete is being gradually extended. See, in particular, M. M. Malamud and L. L. Oridoroga [Funct. Anal. Appl. 34, No. 4, 308-310 (2000); translation from Funkts. Anal. Prilozh. 34, No. 4, 88–90 (2000; Zbl 0979.34058)], M. M. Malamud and L. L. Oridoroga [Dokl. Math. 82, No. 3, 899–904 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 435, No. 3, 298–304 (2010; Zbl 1231.34152)]. In the paper under review, this process is continued. The authors study various classes of systems not covered by earlier results. Here, the completeness or incompleteness depend on lower order terms. In the case of a holomorphic potential matrix and some nondegeneracy assumptions, even necessary and sufficient conditions are obtained.
    01/2012; 18(1).
  • Source
    M. M. Malamud, L. L. Oridoroga
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    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.
    Doklady Mathematics 09/2011; 82(3). DOI:10.1134/S1064562410060165 · 0.31 Impact Factor
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    N. Goloschapova, L. Oridoroga
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    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 · 0.58 Impact Factor
  • Seppo Hassi, Leonid Oridoroga
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    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 · 0.58 Impact Factor
  • A. A. Lunev, L. L. Oridoroga
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    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
    Mathematical Notes 06/2009; 85(5):703-711. DOI:10.1134/S0001434609050101 · 0.26 Impact Factor
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    Nataly Goloschapova, Leonid Oridoroga
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    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
  • Source
    Seppo Hassi, Leonid Oridoroga
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    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.
  • N. I. Goloshchapova, L. L. Oridoroga
    Mathematical Notes 07/2008; 84(1):125-129. DOI:10.1134/S0001434608070110 · 0.26 Impact Factor
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    L. L. Oridoroga, S. Hassi
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    ABSTRACT: The authors look at the first order differential equation -iBy ' +Qy=λy, where y is the vector y=(y 1 ,⋯,y n ) T ,y i ∈L 2 (0,1), B=B * =diag(b 1 ,b 2 ,⋯,b n ) and Q =(q ij ) 1≤i,j≤n where q ij ∈L ∞ (0,1)· The boundary conditions depend on the eigenvalue parameter P 0 (λ)y(0)+P 1 (λ)y(1)=0, where P i is a matrix polynomial such that det(P 0 P 0 * +P 1 P 1 * )>0 for λ∈ℂ· Under the above conditions the authors show that the eigenfunctions form a complete system in ⨁ 1 n L 2 (0,1)· Furthermore, if we remove certain “extra” associated eigenfunctions, then the remaining set is a Riesz basis.
    Mathematical Notes 02/2006; 79(3):589-593. DOI:10.1007/s11006-006-0067-x · 0.26 Impact Factor
  • L. L. Oridoroga, S. Hassi
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    ABSTRACT: Pi0(λ)y 2 (0) + Pi1(λ)y � 2 (0) + Pi2(λ)y 2 1 + Pi3(λ)y � 2 1 + Pi4(λ)y(0)y � (0) + Pi5(λ)y(0)y 1 + Pi6(λ)y(0)y � 1 + Pi7(λ)y � (0)y 1
    Mathematical Notes 06/2003; 74(1):302-307. DOI:10.1023/A:1025072628519 · 0.26 Impact Factor

Publication Stats

57 Citations
4.74 Total Impact Points

Institutions

  • 2008–2012
    • Donetsk National Technical University
      Yuzovo, Donetsk, Ukraine
  • 2003–2012
    • Donetsk National Medical University
      Yuzovo, Donets’ka Oblast’, Ukraine
  • 2009
    • Donetsk University
      Yuzovo, Donets’ka Oblast’, Ukraine