Space Of Square Summable Functions Research Articles

Symmetric operator extensions of composites of higher order difference operators

In this paper we have considered two higher order difference operators generated by two higher order difference functions on the Hilbert space of square summable functions. By allowing the leading coefficients to be unbounded and the other coefficients as constant functions, we have shown that the composites of two symmetric difference operators are symmetric if the leading coefficients are scalar multiple of each other and the common divisor of their orders is 1. Using examples, we have shown that these conditions of symmetry cannot be weakened. Furthermore, We have shown that the deficiency indices of the composites is equal to the sum of the deficiency indices of the individual operators and that the spectra of the self-adjoint operator extensions is the whole of the real line.

On the Completeness of Eigenfunctions of One 5th-Order Differential Operator

In this paper, we fully solve the problem of the completeness of the eigenfunctions of an ordinary 5th-order differential operator in the space of square-summable functions on the segment [0, 1] generated by the simplest differential expression \( y^{(5)} \) and two-point two-term boundary conditions \(\alpha_v y^{(v−1)}(0) + \beta_v
 y^{(v−1)}(1) = 0\) and \(v =
 \overline{1, 5}\), under the main assumption \(\alpha_v \ne 0\), \(v = \overline{1, 5}\) or \(\beta_v \ne 0\), \(v = \overline{1, 5}\) (in this case, without loss of generality, we can assume that all \(\alpha_v\) or all \(\beta_v\) , respectively, are equal to one). The classical methods of studying completeness, which go back to well-known articles by M. V. Keldysh, A. P. Khromov, A. A. Shkalikov, and many others, are not applicable to the operator under consideration. These methods are based on “good” estimates for the spectral parameter of the used generating functions (“classical”) for the system of eigenfunctions and associated functions. In the case of a strong irregularity of the operator under consideration, these «classical» generating functions have too large rate of grows in the spectral parameter. To solve the problem of multiple completeness, we propose a new approach that uses a special parametric solution that generalizes «classical» generating functions. The main idea of this approach is to select the parameters of this special solution to construct generating functions that are no longer «classical» with suitable estimates in terms of the spectral parameter. Such a selection for the operator under consideration turned out to be possible, although rather nontrivial, which allowed us to follow the traditional scheme of proving the completeness of the system of eigenfunctions in the space of square-summable functions on the segment [0, 1].

Method of Similar Operators in the Study of Spectral Properties of Perturbed First-Order Differential Operators

In this paper, we consider first-order differential operators with periodic boundary conditions acting in a Hilbert space of square summable functions on the segment [0, ω] perturbed by Hilbert– Schmidt integral operators. A similarity transformation of the original operator to the operator of the block-diagonal structure is performed; this allows one to study spectral properties of the perturbed operator. The research method is the method of similar operators, which also presented in this work.

Symplectic Geometry of the Koopman Operator

We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.

The Dirichlet Problem for an Equation of Mixed Type with Two Internal Lines of Type Change

In a rectangular domain, for the equation of mixed elliptic-hyperbolic type with the Lavrent’ev–Bitsadze operator and two perpendicular lines of type change, we investigate the first boundary value problem. A criterion for the uniqueness of its solution is established. The solution to the problem is constructed as the sum of a series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. On the basis of the completeness of the biorthogonal system in the space of square-summable functions, we prove the uniqueness of a solution to the problem. When proving the existence of a solution, a problem of small denominators arises. Therefore, we obtain estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.

The homogenization of diffusion-convection equations in non-periodic structures

We consider the homogenization of diffusion-convective problems with given divergence-free velocities in nonperiodic structures defined by sequences of characteristic functions the first sequence . These quence of concentration the second sequence is uniformly bounded in the space of square-summable functions with square-summable derivatives with respect to spatial variables. At the same time, the sequence of time-derivative of product of these concentrations on the characteristic functions, that define a nonperiodic structure, is bounded in the space of square-summable functions from time interval into the conjugated space of functions depending on spatial variables, withsquare-summable derivatives. We prove the strong compactness of the second sequences in the space of quadratically summable functions and use this result to homogenize the corresponding boundary value problems that depend on a small parameter.

О сопряженных операторах для операторов дробного дифференцирования

On a linear manifold of the space of square summable functions on a finite segment vanishing at its ends, we consider the operator of left-sided Caputo fractional differentiation. We prove that the adjoint for it is the operator of right-sided Caputo fractional differentiation. Similar results are established for the Riemann–Liouville fractional differentiation operators. We also demonstrate that the operator, which is represented as the sum of the left-sided and the right-sided fractional differentiation operators is self adjoint. The known properties of the Caputo and Riemann–Liouville fractional derivatives are used to substantiate the results.

О РЕГУЛЯРИЗАЦИИ ОБРАТНОЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ ТЕПЛОПРОВОДНОСТИ

В работе содержатся новые результаты по спектральным методам решения некорректно поставленных задач на примере задачи Коши для параболического уравнения. В работе предложен метод регуляризации решения обратной задачи. Регуляризованное уравнение получается за счет введения в уравнение теплопроводности биквадратного лапласиана с коэффициентом, равным параметру регуляризации. Это позволяет получить регуляризованное решение данной задачи в виде спектрального ряда, который хорошо сходится. Показано, что если решение исходной задачи существует, то разность между спектральными разложениями исходного и регуляризованного решений стремится к нулю при стремлении параметра регуляризации к нулю в пространстве функций, суммируемых с квадратом. Используя результаты по спектральной теории В.А. Ильина [4], [5]. Получены некоторые оценки для разности точного и регуляризованного решений. The paper contains new results on spectral methods for solving ill-posed problems using the example of the Cauchy problem for a parabolic equation. A method for regularizing the solution of the inverse problem is proposed. We obtain regularized equation by introducing into the heat equation a biquadratic Laplacian with a coefficient equal to the regularization parameter. This allows us to obtain a regularized solution as a well converging spectral series. It is shown that if the solution of the original problem exists, then the difference between the spectral expansions of the initial and regularized solutions tends to zero when the regularization parameter tends to zero in the space of square summable functions. Using the results of the spectral theory of V.A. Ilyin [4], [5]. Some estimates are obtained for the difference between the exact and regularized solutions.

Invariant subspaces of [formula omitted] and [formula omitted] preserving compatibility

Operators of multiplication by independent variables on the space of square summable functions over the torus and its Hardy subspace are considered. Invariant subspaces where the operators are compatible are described.

On the equivalence of the electromagnetic problem of diffraction by an inhomogeneous bounded dielectric body to a volume singular integro-differential equation

The paper is concerned with the smoothness of the solutions to the volume singular integrodifferential equations for the electric field to which the problem of electromagnetic-wave diffraction by a local inhomogeneous bounded dielectric body is reduced. The basic tool of the study is the method of pseudo-differential operators in Sobolev spaces. The theory of elliptic boundary problems and field-matching problems is also applied. It is proven that, for smooth data of the problem, the solution from the space of square-summable functions is continuous up to the boundaries and smooth inside and outside of the body. The results on the smoothness of the solutions to the volume singular integro-differential equation for the electric field make it possible to resolve the issues on the equivalence of the boundary value problem and the equation.

Multiple Completeness of the Root Functions for a Certain Class of Pencils of Ordinary Differential Operators

A class of polynomial pencils of ordinary differential operators with constant coefficients is considered in the article. The pencils from this class are generated by the n-th order ordinary differential expression on a finite segment (main segment) and two-point boundary conditions (at the ends of the segments). Coefficients of the differential expression are supposed to be polynomials of the spectral parameter of a special form with constant coefficients. The boundary conditions are supposed to depend on the spectral parameter polynomially. l (\({1\leq l \leq n-1}\)) of the boundary conditions are considered only at one end of the main segment. It is assumed that the roots of the characteristic equation of the pencils from this class are simple, non-zero and lie on two rays (or one as a special case) emanating from the origin. The author investigates m-fold (\({1\leq m \leq n}\)) completeness of the root functions of the pencils from this class in the space of square summable functions on the main segment. Sufficient conditions of the m-fold completeness of the root functions are obtained. The main idea of the proof of the theorem is a new asymptotics of the characteristic determinant of the pencil. The presented results supplement previous results of the author.

Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation

Abstract In the present paper, we consider the approximation of the solution of an ill-posed spherical pseudo-differential equation at a given point. While the methods for approximating the whole solution are well-studied in Hilbert spaces, such as the space of square-summable functions, the computation of values of the solution at given points is much less studied. This can be explained, in particular, by the fact that for square-summable functions the functional of pointwise evaluation is, in general, not well defined. To overcome this limitation we adjust the regularized least-squares method of An, Chen, Sloan and Womersley [Siam J. Numer. Anal. 50 (2012), no. 3, 1513–1534] by using a special a posteriori parameter choice rule. We also illustrate our theoretical findings by numerical results for the reconstruction of the solution at a given point.

О двукратной полноте собственных функций сильно нерегулярного квадратичного пучка дифференциальных операторов второго порядка

Saratov State Law Academy, Russia, 410056, Saratov, Volskaya st., 1, Oksana_Parfilova@mail.ruA class of strongly irregular pencils of ordinary differential operators of second order with constant coefficients is considered. Theroots of the characteristic equation of the pencils from this class are supposed to lie on a straight line coming through the originand on the both side of the origin. Exact interval on which the system of eigenfunctions is 2-fold complete in the space of squaresummable functions is finded.Key words: quadratic pencil, second order pencil, pencil of ordinary differential operators, two-point boundary conditions,homogeneous differential expression with constant coefficients, completeness of the system of eigenfunctions, non-completenessof the system of eigenfunctions.

Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions

IIn this work we study the global approximate multiplicative controllability for a weakly degenerate parabolic Cauchy-Robin problem. The problem is weakly degenerate in the sense that the diffusion coefficient is positive in the interior of the domain and is allowed to vanish at the boundary, provided the reciprocal of the diffusion coefficient is summable. In this paper, we will show that the above system can be steered, in the space of square-summable functions, from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear static controls. Moreover, we extend the above result relaxing the sign constraint on the initial-state.

The Lagrange trigonometric interpolation polynomial with the minimal norm considered as an operator from C 2π to C 2π

In this paper we perform a comparative analysis of Lebesgue functions and constants of a family of Lagrange polynomials. We prove that if a polynomial from the family has the minimal norm in the space of square summable functions, then it also has the minimal norm as an operator which maps the space of continuous functions into itself.

An alternative approach to Privault's discrete-time chaotic calculus

In this paper, we present an alternative approach to Privault's discrete-time chaotic calculus. Let Z be an appropriate stochastic process indexed by N (the set of nonnegative integers) and l 2 ( Γ ) the space of square summable functions defined on Γ (the finite power set of N ). First we introduce a stochastic integral operator J with respect to Z, which, unlike discrete multiple Wiener integral operators, acts on l 2 ( Γ ) . And then we show how to define the gradient and divergence by means of the operator J and the combinatorial properties of l 2 ( Γ ) . We also prove in our setting the main results of the discrete-time chaotic calculus like the Clark formula, the integration by parts formula, etc. Finally we show an application of the gradient and divergence operators to quantum probability.

On the Basis Properties of One Spectral Problem with a Spectral Parameter in a Boundary Condition

We consider a second-order ordinary differential operator with the same spectral parameter in the equation and in one of the boundary conditions. We study the basis property of the system of eigenfunctions of this operator in the space of square summable functions.

ON THE INTEGRAL OPERATORS OF THE THIRD KIND

We establish some necessary and sufficient conditions for an operator in the space of square summable functions to be representable as a sum of multiplication by a bounded function and an integral operator.

Simple wavelet frames for finite functions on discrete tori

A new approach to numerical wavelet transforms is described where sampled n-dimensional signal data are treated as finite functions on discrete tori T/sup n/. This approach shows that the undecimated numerical wavelet transform leads to decomposition onto a frame in the space of square-summable functions on the discrete torus l/sup 2/ (T/sup n/).

Induced representations of the one dimensional quantum Galilei group

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on l2, i.e. on the space of square summable functions on a one-dimensional lattice.