Function Approximation Theory Research Articles

Random vector representation of continuous functions and its applications in quantum mechanics

The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is a random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be obtained. The random vector representation of continuous functions is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomena can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. Then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, and an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.

Orthogonal Polynomials and Operator Convergence in Hilbert Spaces: Norm-Attainability, Uniform Boundedness, and Compactness

This research paper investigates the convergence properties of operators constructed from orthogonal polynomials in the context of Hilbert spaces. The study establishes norm-attainability and explores the uniform boundedness of these operators, extending the analysis to include complex-valued orthogonal polynomials. Additionally, the paper uncovers connections between operator compactness and the convergence behaviors of orthogonal polynomial operators, revealing how sequences of these operators converge weakly to both identity and zero operators. These results advance our understanding of the intricate interplay betweenalgebraic and analytical properties in Hilbert spaces, contributing to fields such as functional analysis and approximation theory. The research sheds new light on the fundamental connections underlying the behavior of operators defined by orthogonal polynomials in diverse Hilbert space settings.

Extended state observer based dynamic iterative learning for trajectory tracking control of a six-degrees-of-freedom manipulator

With the development of industrial automation comes an ever, broadening number of application scenarios for manipulators along with increasing demands for their precise control. However, manipulator trajectory tracking control schemes often exhibit problems such as those related to high levels of coupling, complex calculations, and in various difficulties in application for industrial environments. For the problems of low accuracy in control and poor robustness of multiple-jointed robotic trajectory tracking, iterative learning control (ILC) with model compensation (MC) based on extended state observer (ESO) has been proposed for the trajectory tracking control of six-degrees-of-freedom (six-DOF) manipulators. The scheme has excellent features to overcome uncertainties in repetitive tasks, including unknown bounded perturbations that are external to the model or dynamic perturbations that are internal to the model. The proposed control strategy combines ESO, iterative learning, and MC, for precise control of trajectory tracking. Here, ESO is used to estimate disturbances, iterative learning allows fast and accurate control in repeated tasks, and the model-compensated control algorithm alleviates the necessary for many inverse operations. The convergence of our proposed control scheme is proved through Lyapunov function and time-varying approximation theory. Simulation and experimental results verify the validity of the proposed scheme.

Democracy of quasi-greedy bases in -Banach spaces with applications to the efficiency of the Thresholding Greedy Algorithm in the Hardy spaces

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

Learning Dynamical Systems with Side Information

We present a mathematical and computational framework for learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn, besides trajectory data, and is typically inferred from domain-specific knowledge or basic principles of a scientific discipline. We are interested in explicitly integrating side information into the learning process in order to compensate for scarcity of trajectory observations. We identify six types of side information that arise naturally in many applications and lead to convex constraints in the learning problem. First, we show that when our model for the unknown dynamical system is parameterized as a polynomial, we can impose our side information constraints computationally via semidefinite programming. We then demonstrate the added value of side information for learning the dynamics of basic models in physics and cell biology, as well as for learning and controlling the dynamics of a model in epidemiology. Finally, we study how well polynomial dynamical systems can approximate continuously differentiable ones while satisfying side information (either exactly or approximately). Our overall learning methodology combines ideas from convex optimization, real algebra, dynamical systems, and functional approximation theory, and can potentially lead to new synergies among these areas.

On the Steklov averages in operator cosine function framework

The Steklov averages (Steklov or integral means) are used in approximation theory of functions in different aspects. This article concerns the Steklov averages by using the operator cosine function framework. The operator cosine function offers a counterpart of the translation operator, which forms the basic concept for the modulus of continuity and for some approximation processes as well. We will show that the operator cosine function concept allows to define very general Steklov averages in an abstract Banach space. The approximation properties of these generalized Steklov averages appear to be quite similar to the properties of the Steklov averages in trigonometric approximation.

Finite element/holomorphic operator function method for the transmission eigenvalue problem

The transmission eigenvalue problem arises from the inverse scattering theory for inhomogeneous media. It plays a key role in the unique determination of inhomogeneous media. Furthermore, transmission eigenvalues can be reconstructed from the scattering data and used to estimate the material properties of the unknown object. The problem is posted as a system of two second order partial differential equations and is nonlinear and non-selfadjoint. It is challenging to develop effective numerical methods. In this paper, we formulate the transmission eigenvalue problem as the eigenvalue problem of a holomorphic operator function. The Lagrange finite elements are used for the discretization and the convergence is proved using the abstract approximation theory for holomorphic Fredholm operator functions. The spectral indicator method is employed to compute the eigenvalues. Numerical examples are presented to validate the proposed method.

A method of estimation for the area and direction of pipe crack

Abstract Crack direction will affect the ultrasonic wave reflected from it. Firstly, the correlation between crack direction and the reflected energy of ultrasonic is analyzed. Then, a mathematical model of crack direction and its reflected energy is developed due to the theory and methods of function approximation. Finally, aiming at the problem that the crack area must be known when using the model to estimate the crack direction, By observing the probability distribution of reflected wave energy of cracks with different areas and directions at different measurement distances, Using its law and mathematical model, an algorithm for estimating crack area and crack direction is designed. According to the related experiments, the study confirmed that both the estimation errors of the crack area and direction by the method are 5.25% and 8.21%, respectively.

On the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals

The article is devoted to the study of some data from the theory of functions approximation by trigonometric polynomials with a spectrum from special sets called harmonic intervals. Due to the limited perception range of devices, the perception range of the senses of the person himself, when studying a mathematical model it is often enough to find an approximation of the object so that the error (noise, interference, distortion) is outside the interval of perception. Harmonic intervals model problems of this kind to some extent. In the article the main components of the approximation theory of functions by trigonometric polynomials with a spectrum from harmonic intervals are presented, the theorem on estimating the best approximation of a function by trigonometric polynomials through the best approximations of a function by trigonometric polynomials with a spectrum from harmonic intervals is proved. Theorems on the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals are considered; such a theorem for the Lorentz space is generalized and proved. The article is mainly aimed at scientific researchers dealing with practical applications of the approximation theory of functions by trigonometric polynomials with a spectrum from special sets.

Finite Element Calculation of Photonic Band Structures for Frequency Dependent Materials

Band structure calculation of frequency dependent photonic crystals has important applications. The associated eigenvalue problem is nonlinear and the development of convergent numerical methods is challenging. In this paper, we formulate the band structure problem as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Lagrange finite elements are used to discretize the operators. The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions. Then a spectral indicator method is developed to practically compute the eigenvalues. Numerical examples are presented to validate the theory and show the effectiveness of the proposed method.

Identification of Nonlinear Dynamical System Based on Raised-Cosine Radial Basis Function Neural Networks

In this paper, we present and investigate a new type of radial basis function (RBF) neural networks mechanism using raised-cosine (RC) function to identify nonlinear dynamic system. In this design, the RBF neural networks mechanism utilizes RC function to replace Gaussian function, which is called RCRBF. An N-dimensional RC function has the constant interpolation property, which is benefit for the function approximating errors analysis in the neural networks. Based on multivariable RC function approximation theory, we develop how to select the updated parameters and the distance of adjacent nodes in lattice points. Therefore, the proposed networks can uniformly approximate nonlinear dynamical function. As persistency excitation (PE) plays an important part in neural networks learning system, how does PE condition behave in input sequences is formulated by RC function analysis. The weights updating and errors convergence are concluded by Lyapunov function analysis. To illustrate the effectiveness of the proposed RCRBF method, Van Der Pol and Rossler dynamical system are used as test examples, in comparison with GRBF mechanism. The results show that the proposed method has better accurate identification and approximating effect than that of GRBF mechanism.

Log orthogonal functions: approximation properties and applications

Abstract We present two new classes of orthogonal functions, log orthogonal functions and generalized log orthogonal functions, which are constructed by applying a $\log $ mapping to Laguerre polynomials. We develop basic approximation theory for these new orthogonal functions, and apply them to solve several typical fractional differential equations whose solutions exhibit weak singularities. Our error analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions that have weak singularities at one endpoint and can lead to exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used.

Lightning Solvers for Potential Flows

We present a method for computing potential flows in planar domains. Our approach is based on a new class of techniques, known as “lightning solvers”, which exploit rational function approximation theory in order to achieve excellent convergence rates. The method is particularly suitable for flows in domains with corners where traditional numerical methods fail. We outline the mathematical basis for the method and establish the connection with potential flow theory. In particular, we apply the new solver to a range of classical problems including steady potential flows, vortex dynamics, and free-streamline flows. The solution method is extremely rapid and usually takes just a fraction of a second to converge to a high degree of accuracy. Numerical evaluations of the solutions are performed in a matter of microseconds and can be compressed further with novel algorithms.

On one interpolating rational process of Fejer – Hermite

In this paper, a new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described. In the introduction a brief analysis of the results on the topic of the research is carried out. Herein, the methods of the construction of interpolating processes, in particular, Fejer – Hermite processes, in the polynomial and rational approximation are analysed. A new method to determine the interpolating rational Fejer – Hermite process is proposed. One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions. The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work. Moreover, we present the numerical analysis of the effectiveness of the application of the constructed interpolating Fejer – Hermite process for the approximation of a continuous function with singularities. The choice of parameters, on which the nodes of interpolation depend, is made in several standard ways. The obtained results can be applied to further study the approximating properties of interpolating processes.

Major advancements in kernel function approximation

Kernel based methods have become popular in a wide variety of machine learning tasks. They rely on the computation of kernel functions, which implicitly transform the data in its input space to data in a very high dimensional space. Efficient application of these functions have been subject to study in the last 10 years. The main focus was on improving the scalability of kernel based methods. In this regard, kernel function approximation using explicit feature maps have emerged as a substitute for traditional kernel based methods. Over the years, various advancements from the theoretical perspective have been made to explicit kernel maps, especially to the method of random Fourier features (RFF), which is the main focus of our work. In this work, the major developments in the theory of kernel function approximation are reviewed in a systematic manner and the practical applications are discussed. Furthermore, we identify the shortcomings of the current research, and discuss possible avenues for future work.

On the Banach Spaces Associated with Multi-Layer ReLU Networks: Function Representation, Approximation Theory and Gradient Descent Dynamics

We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this norm, the unit ball in the space for $L$-layer networks has low Rademacher complexity and thus favorable generalization properties. Functions in these spaces can be approximated by multi-layer neural networks with dimension-independent convergence rates. The key to this work is a new way of representing functions in some form of expectations, motivated by multi-layer neural networks. This representation allows us to define a new class of continuous models for machine learning. We show that the gradient flow defined this way is the natural continuous analog of the gradient descent dynamics for the associated multi-layer neural networks. We show that the path-norm increases at most polynomially under this continuous gradient flow dynamics.

A new finite element approach for the Dirichlet eigenvalue problem

We propose a new finite element approach, which is different than the classic Babuška–Osborn theory, to approximate Dirichlet eigenvalues. The problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. The convergence for conforming finite elements is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed to compute the eigenvalues. A numerical example is presented to validate the theory.

Research on a widely applicable function expression of the dry-wet bulb coefficient through deviation control theory

Relying on the experimental data, by using the theory of error control and function approximation to establish new wet and dry bulb coefficientÂ(v, ta). Through searching method and Iterative approximation, it finds an approximation function whose figure is closer to "experimental data" than previous one did. Hence, it increases the measurement accuracy of relative humidity by the wet and dry bulb method under the condition of the relative humidity in the range of 10℃≤t≤70℃, air flow rate 0.1m/s≤v≤4m/s, and relative humidity 30%≤U≤90%.

Approximation of Lyapunov functions from noisy data

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.

An Analytic Expression for the Inverse Involute

This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.