Best Approximation Of Functions Research Articles

Methods for controlling heavy metals in environmental soils based on artificial neural networks

The problem of heavy metal pollution in soil has become a global environmental problem, and it is very important to predict and manage the heavy metals in the environmental soil in a timely manner. The changes in heavy metal content in soil have characteristics such as nonlinearity and large delay, making it difficult to predict heavy metals in soil using traditional methods. Traditional prediction methods are complex and cumbersome, which can lead to longer treatment time and easy secondary pollution. This article analyzed the Back Propagation neural network (BPNN) in artificial neural networks (ANN) and applied it to the prediction of heavy metals in environmental soils. BPNN has good nonlinear function approximation ability, so it can be well applied to complex problems such as soil heavy metal prediction. The methods of treating soil heavy metals include physical repair method, chemical repair method, microbial repair method, plant repair method, plant microbial combined repair method and so on. The use of BPNN can predict heavy metals in environmental soils through adaptive dynamic learning. However, the training time of the BPNN is relatively long and the convergence speed is relatively slow. Therefore, additional momentum terms were added to adjust the weights and thresholds of the network to improve the BPNN. In the experiment, the prediction performance of the improved BPNN was compared before and after the improvement. This article took 50 monitoring data of heavy metals in the same soil in a certain region in 2021 as sample data and predicted the content of heavy metals in the soil using improved and improved BPNN. Due to time constraints, this article only conducted experimental analysis on heavy metals such as lead and cadmium. In the first experiment, when the soil sample data was 50, the prediction accuracy of the BPNN for cadmium before and after improvement was 75.95% and 89.56%, respectively. In the second experiment, when the soil sample data was 50, the prediction accuracy of the BPNN for cadmium before and after improvement was 77.99% and 89.85%, respectively. The improved BPNN has good prediction accuracy and can effectively predict the status of heavy metals in soil. The analysis in this article can provide scientific basis for the comprehensive prevention and control of heavy metals in regional soil, and also provide reference for the development of pollution-free agriculture and ensuring food safety.

Наилучшее приближение аналитических в единичном круге функций в весовом пространстве Бергмана B_(2,μ)

The paper studies the issues of the best approximation of analytical functions in the Bergman weight space B_(2,μ). In this space, for best approximations of functions analytic in the circle by algebraic complex polynomials we obtain the exact inequalities by means of generalized modules of continuity of higher order derivatives Ω_m (z^r f^((r) ),t),m∈N,r∈Z. For classes of functions analytic in the unit circle defined by the characteristic Ω_m (z^r f^((r) ),t), and the majorant Φ, the exact values of some n-widths are calculated. When proving the main results of this work, we use methods for solving extremal problems in normalized spaces of functions analytic in the circle, N.P. Korneichuk’s method for estimating upper bounds for the best approximations of classes of functions by a subspace of fixed dimension, and a method for estimating from below the n-widths of function classes in various Banach spaces.

OPTIMIZATION METHODS IN CALCULATIONS OF MECHANISMS OF VEHICLES OF TRANSPORT

Purpose. The optimal design of vehicle mechanisms is a detailed review and evaluation of the scientific and technical literature and individual publications on the methodology for optimizing the inherent calculations of these types of mechanical structures. The main purpose of this review material is to consider a number of directions for designing mechanisms of means of transport, especially highlighting the use of optimization methods. Research in the field of designing mechanisms is carried out in two directions: by the synthesis method according to given provisions and by the method of mathematical programming. Methods. Methods for the synthesis of hinged-lever mechanisms can be divided into two groups. In the first one, methods of approximation of functions are used, in the second one, optimization methods. The search for the optimal solution in the second group means the simultaneous search for the correct formulation of the problem, in which the parameters of the mechanisms should be sought under conditions of the best satisfaction of the requirements of the technological process. The problems of optimal design of mechanisms most often continue to be formulated as problems of the best approximation of functions. Optimization synthesis is a relatively new direction in the synthesis of lever mechanisms. Optimization synthesis is carried out on the basis of nonlinear optimization methods. At present, the complexity and complexity of the problems that arise and require solution in the process of designing mechanisms have significantly increased. A certain amount of research has been carried out within the framework of the classical theory of finite and infinitely small plane displacements. In these works, the theory of spatial motion of a general form is developed in detail and applied to the problems of kinematic synthesis. Results. Since the simplex method is quite effective in solving problems of linear programming, it became the founder of a number of methods of sequential linearization or methods of secant planes for solving nonlinear programming. Practical significance. With an extensive list of modern literature and scientific publications on the synthesis of mechanisms and mathematical programming, a complete chronological review of the achievements of the last decade would be quite loud. Therefore, individual trends in the synthesis of mechanisms were only noted, focusing on the desire to obtain a “better” design, taking into account practical requirements. The synthesis of mechanisms is an area in which the application of mathematical programming methods is quite promising due to their free formulation. The constraints and requirements of engineering practical problems can often be taken into account in forms consistent with the data of modern methods. The main purpose of this study is to make widely known the achievements in the two fields with the hope that currently existing optimization tools will be applied to these synthesis problems and that, conversely, the optimization tools will be further improved as a result of research on the design of vehicle mechanisms. This will contribute to a better understanding and accelerated development of both mechanisms calculation methods and optimization methodology.

On the Jackson–Stechkin Theorems for the Best Approximations of Functions in Clifford Algebras

On the Jackson–Stechkin Theorems for the Best Approximations of Functions in Clifford Algebras

The best approximation of closed operators by bounded operators in Hilbert spaces

We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.

Pointwise Estimates for Finding the Error of Best Approximation by Spline, Positive Algebraic Polynomials and Copositive

The first step in this research is to find some of the necessary estimations in approximation by using certain algebraic polynomials, as well as we use certain specific points in approximation. There are many estimations that help to find the best approximation using algebraic polynomials and geometric polynomials. Throughout this research, we deal with some of these estimations to estimate the best approximation error using algebraic polynomials where the basic estimations in approximation are discussed and proven using algebraic polynomials that are discussed and proven using algebraic polynomials that are specified by the following points and if as well as if . For the second step of the work, the estimations in the first step are used to find and estimate the error for the best approximation of the weighted function . This is done through the use of an algebraic polynomial whose degree at most is where the sign of the algebraic polynomial is positive. Further, the error is also found and estimated for the best approximation of the restricted function using the restricted algebraic polynomial , which is copositive with the function in the quasi weighted normed space. In addition, we deal with the created estimations to estimate the error of the best approximation of the function by using pieces of algebraic polynomials that are of the highest degree .These pieces of algebraic polynomials are connected to each other, so they have formed a spline of the highest degree whose knots are considered the contact areas of the algebraic polynomials.

SOME DIRECT AND INVERSE THEOREMS OF APPROXIMATION OF FUNCTIONS IN JACOBI-DUNKL DISCRETE HARMONIC ANALYSIS

In this paper, several direct and inverse theorems in terms of the best approximations of functions and moduli of smoothness are proved concerning the approximation of functions from the space \(\mathbb {L}_{2}^{(\alpha ,\beta )}\) by partial sums of Jacobi-Dunkl series. For this purpose, we use the generalized Jacobi-Dunkl translation operator which was defined by Vinogradov.

Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions

UDC 517.5 In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point in infinity and their Orlicz norms are finite. Special attention is paid to the study of cases when the constants in these theorems are unimprovable.

Path Loss Modeling: A Machine Learning Based Approach Using Support Vector Regression and Radial Basis Function Models

Path loss prediction models are vital for accurate signal propagation in wireless channels. Empirical and deterministic models used in path loss predictions have not produced optimal results. In this paper, we introduced machine learning algorithms to path loss predictions because it offers a flexible network architecture and extensive data can be used. We introduced support vector regression (SVR) and radial basis function (RBF) models to path loss predictions in the investigated environments. The SVR model was able to process several input parameters without introducing complexity to the network architecture. The RBF on its part provides a good function approximation. Hyperparameter tuning of the machine learning models was carried out in order to achieve optimal results. The performances of the SVR and RBF models were compared and result validated using the root-mean squared error (RMSE). The two machine learning algorithms were also compared with the Cost-231, SUI, Egli, Freespace, Cost-231 W-I models. The analytical models overpredicted path loss. Overall, the machine learning models predicted path loss with greater accuracy than the empirical models. The SVR model performed best across all the indices with RMSE values of 1.378 dB, 1.4523 dB, 2.1568 dB in rural, suburban and urban settings respectively and should therefore be adopted for signal propagation in the investigated environments and beyond.

Best approximation of functions by log-polynomials

Lasserre [32] proved that for every compact set K⊂Rn and every even number d there exists a unique homogeneous polynomial g0 of degree d with K⊂G1(g0)={x∈Rn:g0(x)≤1} minimizing |G1(g)| among all such polynomials g fulfilling the condition K⊂G1(g). This result extends the notion of the Löwner ellipsoid, not only from convex bodies to arbitrary compact sets (which was immediate if d=2 by taking convex hulls), but also from ellipsoids to level sets of homogeneous polynomial of an arbitrary even degree.In this paper we extend this result for the class of non-negative log-concave functions in two different ways. One of them is the straightforward extension of the known results, and the other one is a suitable extension with uniqueness of the solution in the corresponding problem and a characterization in terms of some ‘contact points’.

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.

Jackson-type inequalities and widths of functional classes in the Musielak–Orlicz type spaces

In the Musielak–Orlicz-type spaces 𝒮M, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in 𝒮M are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.

Estimates of Functions with Transformed Double Fourier Series

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

On the best $L_2$-approximations of multivariable functions by means of splines

We obtain sharp inequalities of Jackson type for the best approximations of functions in $L_2(\mathbb{R}^m)$ by means of partial sums of wavelet series in case of multidimensional analogues of Shannon-Kotelnikov wavelets.

Some Estimates to Best Approximation of Functions by Fourier-Jacobi Differential Operators

In this paper, an extension of the idea of the best approximation in the Hölder spaces with respect to Fourier-Jacobi operators by moduli of smoothness is studied. A special form of the moduli of smoothness is considered to get a strong convergence. Further, advanced approaches of approximation and some direct and inverse results are proved. Moreover, the Jackson-type estimate of functions in Hölder spaces by Jacobi transformations to algebraic polynomials with generalized de la Vallée Poussin mean are established.

On approximation of functions by algebraic polynomials on the average in real domain with Chebyshev-Hermite weight

Exact inequalities of Jackson's type, connected with the best approximation of functions by algebraic polynomials, have been obtained in the space $L_{2,\rho}(\mathbb{R})$ at the Chebyshev-Hermite weight.

Monotonicity of sequence of the best approximating constants for the function $\sqrt{t}$

It is proved that the sequence of the constants of the best approximation of function $\sqrt{t}$ is monotone decreasing.

Direct and inverse approximation theorems of functions in the Musielak-Orlicz type spaces

In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.

On the constants in the inverse theorems for the first derivative

The known proofs of the inverse theorems of the theory of approximation by trigonometric polynomials and by functions of exponential type are based on the idea of S. N. Bernstein to expand a function in a series containing its functions of best approximation. In this paper, a new method to establish the inverse theorems is introduced. We establish simple identities that immediately imply the inverse theorems mentioned and, moreover, with better constants. This method can be applied to derivatives of arbitrary order (not necessarily an integer one) and (with certain modifications) to estimates of some other functionals in terms of best approximations. In this paper, the case of the first derivative of a function itself and of its trigonometrically conjugate is considered.

EESTIMATES OF BEST APPROXIMATIONS OF FUNCTIONS WITH LOGARITHMIC SMOOTHNESS IN THE LORENTZ SPACE WITH ANISOTROPIC NORM

In this paper, we consider the anisotropic Lorentz space \(L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})\) of periodic functions of \(m\) variables. The Besov space \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) by trigonometric polynomials under different relations between the parameters \(\bar{p}, \bar\theta,\) and \(\tau\).The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function \(f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})\) to belong to the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\) in the case \(1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m},\) in terms of the best approximation and prove its unimprovability on the class \(E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}\) \({n=0,1,\ldots\},}\) where \(E_{n}(f)_{\bar{p},\bar{\theta}}\) is the best approximation of the function \(f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\) by trigonometric polynomials of order \(n\) in each variable \(x_{j},\) \(j=1,\ldots,m,\) and \(\lambda=\{\lambda_{n}\}\) is a sequence of positive numbers \(\lambda_{n}\downarrow0\) as \(n\to+\infty\). In the second section, we establish order-exact estimates for the best approximation of functions from the class \(B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}\) in the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\).