System Of Orthogonal Functions Research Articles

Semiclassical Moser–Trudinger inequalities

We extend the Moser–Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schrödinger operators on bounded domains.

Periodic functions with variable period – basic concepts and certain investigation results

Investigation of real signals is one of the most important applied areas of mathematics. According to their properties, signals are quite diverse, and methods of their research are different as well. Among this diversity, periodic signals with variable period make up a significant proportion. Till present, no attention was paid to the theory of such signals. In this paper, we define periodic functions with variable period, which are the model of these signals. Some properties of the variable period are considered. Examples of the analytical formulation of functions with variable period in the form of trigonometric functions with variable period are given and their variable periods are recorded. It is pointed out that these functions can be used as basic ones for constructing orthogonal system of trigonometric functions with variable period and its use for constructing Fourier series of functions with variable period.

Transforming image descriptions as a set of descriptors to construct classification features

<span>The article develops methods to solve a fundamental problem in computer vision: image recognition of visual objects. The results of the research on the construction of modifications for the space of classification features based on the application of the transformation of the structural description through the decomposition in the orthogonal basis and the implementation of the distance matrix model between the components of the description are presented. The application of the system of orthogonal functions as an apparatus for the transformation of the description showed the possibility of a significant gain in the speed of processing while maintaining high indicators of classification accuracy and interference resistance. The synthesized feature systems’ effectiveness has been confirmed in terms of a significant increase in the rate of codes and a sufficient level of efficiency. An experimental example showed that the time spent calculating the relevance of descriptions according to their modified presentation is more than ten times shorter than for traditional metric approaches. The developed classification features can be used in applied tasks where the time of visual objects’ identification is critical.</span>

Method for determining eigen numbers in heat conduction problems for a cylinder

RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed.METHODS. The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased.RESULTS. The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function).CONCLUSION. Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.

Dense weakly lacunary subsystems of orthogonal systems and maximal partial sum operator

It is shown that any finite orthogonal system of functions whose norms in $L_p$ are bounded by 1, where $p>2$, has a sufficiently dense subsystem with lacunarity property in the Orlicz space. The norm of the maximal partial sum operator for this subsystem has a better estimate than it is guaranteed by the classical Menshov-Rademacher theorem for general orthogonal systems. Bibliography: 17 titles.

Discrete Time Sequence Reconstruction of a Signal Based on Local Approximation Using a Fourier Series by an Orthogonal System of Trigonometric Functions

The article considers the development of mathematical and algorithmic support for the sample’s reconstruction in problem sections of a discrete sequence of a continuous signal. The work aimed to ensure the reconstruction of lost samples or sections of samples with a non-constant distorted time grid when sampling a signal with a uniform step and at the same time to reduce the computational complexity of digital reconstruction algorithms. The solution to the stated problem is obtained based on the local approximation method. The specific of this method application was the use of two subsequences of samples located symmetrically concerning the reconstructed section of the sequence. The approximating model is a Fourier series on an orthogonal system of trigonometric functions. The optimal solution to the approximation problem is based on the minimum square error criterion. Mathematical equations are obtained for this type of error. They allow us to estimate its value depending on the model order and the samples number in the subsequences used in the reconstruction process. The peculiarity of the mathematical equations obtained in this paper for signal reconstruction is that they do not require the preliminary calculation of the Fourier series coefficients. They provide a direct calculation of the values of reconstructed samples. At the same time, when the number of samples in the subsequences used for reconstruction will be even, it is not necessary to perform multiplication operations. All this made it possible to reduce the computational complexity of the developed algorithm for signal reconstruction. Experimental studies of the algorithm were carried out based on simulation modeling using a signal model that is an additive sum of harmonic components with a random initial phase. Numerical experiments have shown that the developed algorithm provides the reconstruction result of signal samples with a sufficiently low error. The algorithm is implemented as a software module. The operation of the module is carried out on the basis of asynchronous control of the sampling reconstruction process. It can be used as part of metrologically significant software for digital signal processing systems.

Application of systems of orthogonal functions for formation of sign space in image classification methods

The subject of the article's research is the improvement of structural methods of image classification in computer vision systems. The goal is to reduce computational costs for classification by implementing a device for decomposing image description components using a system of orthogonal functions and implementing feature space compression models. Applied methods: ORB key point detector, set theory apparatus and vector spaces, metric models for determining relevance to sets of multidimensional vectors, theory of orthogonal decomposition of vectors, elements of probability theory, software modeling. Obtained results: modifications of the image classification method based on the introduction of orthogonal data decomposition in vector space were developed, models were proposed for data compression in the transformed feature space, Tanimoto metric was introduced for image comparison, a threshold selection method was established for determining equivalent description components. The effectiveness of the developed modifications of the classifier depends on the selection of a subset of functions for decomposition, the metric for comparing descriptions, and the method of determining the equivalence threshold. The implementation of the apparatus of orthogonal functions not only reduced computational costs tenfold, but also ensured sufficiently high indicators of classification performance and interference resistance. The practical significance of the work is the construction of new models of the image classifier in the transformed space of features, confirmation of the functionality, speed and immunity of the proposed modifications on examples of images, the creation of a software application for the implementation of the developed classification methods in computer vision systems.

Systems of Discrete Walsh-Like Sequential Functions

In this paper the algorithm for constructing the discrete (0,1)-sequent functions constituting the whole symmetric systems of orthogonal equidistant functions on the example of the eighth-order systems developed. Discrete sequential functions form by replacing their piecewise constant values +1 or -1 in the time domain (from the original space) with numerical values 0 and 1 in the image space. We refer to Walsh-like functions as (0,1)-sequent functions in which the number of zeros and ones in each half of the definition interval is not necessarily the same as in classical Walsh functions. By the directed search method, each of the 30 formed whole groups of equidistant sequent functions unfolds, like the group of classical Walsh functions of the eighth order, into 28 symmetric systems of sequent functions. The main result achieved in this work should consider an expansion of the set of Walsh-like systems of the eighth order by more than an order of magnitude. The algorithm's simplicity for synthesizing such systems of sequential functions and the high speed of spectral processing of discrete signals provided by the proposed bases open the Walsh-like systems for broad prospects of application in various fields of science and technology.

ICHV: A New Compression Approach for Industrial Images

The rapid development of the Industrial Internet of Things and artificial intelligence has brought new pressures to the compression and storage of image information. Thus, efficient image compression approaches are important in the fourth industrial revolution (Industry 4.0), which can save storage resources, accelerate transmission speed, and improve imaging fidelity. In this article, a new approach of image compression based on Haar-V functions (ICHV) is investigated, which has good performance in industrial image compression. The Haar-V functions form a discontinuous orthogonal function system composed of piecewise polynomials. The construction and properties of Haar-V functions are introduced at first. Then, 2-D discrete orthogonal matrices of Haar-V functions and the corresponding image transform method are presented successfully. Finally, a fast computation of ICHV is proposed to enhance the computation speed of image compression. The experimental results indicate that compared with related algorithms, the new approach can effectively improve the quality of compressed images and consume less computational cost.

Computational method for solving boundary value problems of mechanics deformable body using non-orthogonal functions

In this paper a complete system of non-orthogonal functions was built on the basis of orthogonal sines and cosines. It is shown that the known orthogonal systems of functions are a degenerate case of non-orthogonal systems of functions. It has been proven that the continuous function can be approximated non-orthogonal functions in such a way that one selected nonorthogonal function will not included in this amount. The boundary value problem of the elasticity theory has been considered for an inhomogeneous plate. A new method for solving the boundary value problem is developed for the fourth-order equation with variable coefficients. The proposed method is based on separating of the stress state of the plate, use of complete systems of nonorthogonal functions and a generalized quadratic form. Criteria have been established under which the constructed approximate solution coincides with the exact solution. This method has been adapted to solving a boundary value problem for high-order differential equations. The high accuracy of the method has been confirmed by numerical calculations.

The Evaluation of Electric Power by Separate Analysis of Fourier Transform Components

This paper presents a substantiation of an approach for the evaluation of components of apparent power and intended to simplify the computational procedures which usually should be implemented in order to process the preliminary sampled waveform of instantaneous power. The results of carried out studies have shown that both active and reactive power can be calculated by the analysis of calculated components of sine and cosine Fourier transforms. This paper also presents the discussion of restrictions, which should be imposed on the duration of the analyzed signal and on frequencies of the auxiliary trigonometric functions, which are applied in order to calculate components of Fourier transform which are used for the evaluation of active and reactive power. The compliance with these restrictions allows us to eliminate the undesirable bias of active and reactive power estimation caused by the refusal from the decomposition of the analyzed waveform of the instantaneous power by applying the complete system of orthogonal trigonometric functions, as the evaluation of components of the apparent power is attained based on separate analysis of sine and cosine Fourier transforms calculated for the analyzed signal. The results of carried out simulations illustrate the frequency dependencies of sine Fourier transform calculated for the case of compliance with the restrictions, which allow to attain the highest accuracy of estimation and for the case when the duration of analyzed signal does not fit these restrictions.

Norm of iegnfunction of one-dimension photonic crystal

Relevance. In recent decades (about the 90-s ХХ century) there has been rapid development of photonic. Thus, to arise scientific interest to optic range of electromagnetic radiation. Currently, the diffraction problem about scattering electromagnetic waves on such object as photonic crystal is impotent problem. As well known, this problem can be reduced to a solution of wave equation. The need to calculate the norm iegnfunction spectral iegnfunction Sturm-Liouville problem, however, to arise in the transition from one complete orthogonal system to another complete orthogonal system of functions by separating variables method, correspondingly, for a wave equation solving. The purpose of the work. We indicate a direct approach to calculating of the norm of iegnfunction of spectral Sturm-Liouville problem for the tow-layer infinite one-dimension photonic crystal (a direct approach to calculating of the norm that is presuppose a direct integration); and propose a methodologically different approach, which is based on the marginal transition in the scalar product, which accordingly sets this norm. Materials and methods. Taking the limit in calculation the norm of the iegnfunction of spectral Sturm-Liouville problem for the tow-layer infinite one-dimension photonic crystal encounters difficulties, associated with the emergence of species uncertainty . Such infinitive investigates by the Lopital's rule. In turn, Lopital's rule entails the need to find a derivative of solution of spectral equation by a spectral param. In this way we have to face the solution a linear inhomogeneous differential equation 2-nd order. Results. We propose a methodic of calculating of norm of iegnfunction of spectral Sturm-Liouville problem for the tow-layer infinite one-dimension photonic crystal. Conclusion. Unlike the direct approach, proposed methodic to make it possible to understand the character of dependencies the required norm of iegnfunction itself (ending expression containing the iegnfunction itself). Further work in this direction of development of this approach may be aimed at simplifying the final expression for the norm.

Dirichlet-Neumann problem for the partial differential equations with deviation over the space argument

Dirichlet-Neumann problem for the typeless high order partial differential equation with deviating over the space argument is studied in the domain, which is the Cartesian product of the segment $(0,T)$ and the unit circle $\Omega=\mathbb R/(2\pi \mathbb Z)$. Dirichlet-Neumann problem for hyperbolic equations and their systems in case with absent argument deviation $h$ has been studied by the authors before. Correct solvability conditions have been established for these problems for almost all (with respect to Lebesgue measure) numbers $T>0$ and for almost all (with respect to Lebesgue measure) vectors, constructed by coefficients of the equation.
 In this paper, the solvability conditions of the problem for $h\ne0$ are described and the influence of the deviation $h$ on the solvability of the problem is studied. The solution of the problem is constructed in the form of the series with respect to the systems of orthogonal functions. Metric estimations (of exponential type) are proved for small denominators appearing during construction of the problem solution. These estimations guarantee the correctness of the problem in Sobolev spaces for almost all (with respect to Lebesgue measure) values $ T> 0 $ and for almost all (with respect to Lebesgue measure) values $ h \in [0,2\pi) $. The obtained results are based on the fact that the corresponding characteristic determinant permits factorization in the form of the product of hyperbolic functions with integer parameters.

On some Sobolev spaces with matrix weights and classical type Sobolev orthogonal polynomials

For every system of OPRL or OPUC, we construct Sobolev orthogonal polynomials , with explicit integral representations involving . Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in n) and generalized eigenvalues of a differential operator (in z) are given. We define suitable Sobolev spaces with matrix weights and consider measurable factorizations of weights. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar spaces are discussed.

New recurrence relations for Wilson polynomials via a system of Jacobi type orthogonal functions

In this paper, we will find an orthogonal basis in term of explicit Jacobi polynomials, of the corresponding weighted L2-space on the whole of the real numbers, which is mapped onto another orthogonal basis involving Wilson polynomials by the Jacobi-Dunkl transform. Next we give a tridiagonalization of the Jacobi-Dunkl Laplacian for the Jacobi type functions. This tridiagonalization leads to a new result, which can be viewed as contiguous relations for Wilson polynomials.

Оптимизация двоичных групповых последовательностей, полученных в результате нелинейного кодового уплотнения

This paper presents the challenge of optimization of binary group sequences obtained from the nonlinear code multiplexing. It shows that at optimal element wise methodology for group sequences in a full system of orthogonal Walsh functions, a maximum criterion for the minimum distance of group sequences is equal to a maximum criterion for the minimum correlation response for the information orthogonal Walsh functions which contain group sequences. After a nonlinear code multiplexing group sequences are often contained errors, that makes using this method of multiplexing more difficult. To eliminate this source of errors the algorithm of optimization of group sequences was suggested. In this algorithm some elements of group sequences can be replaced with the opposite elements. Additionally, the algorithm of receiving of the entire group sequence among the nearest group sequences was developed. This algorithm provides considerable increasing of immunity while group sequence receiving unlike the other algorithm of element-to-element receiving in a full system of orthogonal Walsh functions.

Risk of estimators for Sobol’ sensitivity indices based on metamodels

Sobol' sensitivity indices allow to quantify the respective effects of random input variables and their combinations on the variance of mathematical model output. We focus on the problem of Sobol' indices estimation via a metamodeling approach where we replace the true mathematical model with a sample-based approximation to compute sensitivity indices. We propose a new method for indices quality control and obtain asymptotic and non-asymptotic risk bounds for Sobol' indices estimates based on a general class of metamodels. Our analysis is closely connected with the problem of nonparametric function fitting using the orthogonal system of functions in the random design setting. It considers the relation between the metamodel quality and the error of the corresponding estimator for Sobol' indices and shows the possibility of fast convergence rates in the case of noiseless observations. The theoretical results are complemented with numerical experiments for the approximations based on multivariate Legendre and Trigonometric polynomials.

Two-Dimensional Distribution Law of Random Variables Having S. N. Kritskii and M. F. Menkel Three-Parameter Gamma Distributions: A Symmetrical Case

A method for constructing two-dimensional distribution law in a symmetrical case for three-parameter Kritskii and Menkel distribution is considered, and some results of the application of the model in applied hydrological studies are discussed. A system of orthogonal functions with a weight function in the form of a three-parameter gamma distribution is proposed to obtain linear correlation between three-parameter variables. Orthogonalization method was used to obtain an expression for symmetric two-dimensional density satisfying Markov equation and having a linear regression equation. The results of stochastic simulation of water level variations in the drainless Chany Lake, carried out with the use of the proposed model, showed that the use of linear correlation leads to a value of asymmetry of the level distribution that is in a good agreement with sample estimates of these parameter based on observation data.

A New 3D Surface Reconstruction Method

The existing complete orthogonal function system cannot introduce the spectrum analysis into geometry information processing field. This paper proposes complete orthogonal v system reconstruction method based on the 3D data mode, and this method is to map surface data model to parametric triangular domain. The complete orthogonal V system in triangular domain is used to expand the surface data model and the 3D model reconstruction with piecewise polynomial is realized, and it has simple operation and can handle any topology grid. The experimental results show that the error of three experimental models is controlled in 0.25 mm, Compared with Voronoi-based method and Yoshimoto’s method, the accuracy of the proposed method is improved obviously, and the orthogonal v system under reconstruction time compared with Voronoi-based method reconstruction time is shortened by 13%. Therefore, reconstruction precision and speed can meet the requirements of industrial design.

Numerical solutions of hybrid fuzzy differential equations in a Hilbert space

The main goal of this work is to study a numerical method for certain hybrid fuzzy differential equations with an application of a reproducing kernel Hilbert space technique for fuzzy differential equations. Meanwhile, we construct a system of orthogonal functions of the space W22[a,b]⊕W22[a,b] depending on a Gram-Schmidt orthogonalization process to get approximate-analytical solutions of a hybrid fuzzy differential equation. A proof of convergence of this method is also discussed in detail. The exact as well as the approximate solutions are displayed by a series in terms of their α-cut representation form in the Hilbert space W22[a,b]⊕W22[a,b]. To demonstrate behavior, efficiency, and appropriateness of the present technique, two different numerical experiments are solved numerically in this paper.