13,965 Results Found

For $j=1,\dots ,r$, let ${\mathcal{P}}_j$ be a system of generalized prime numbers, ${\mathcal{N}}_{{\mathcal{P}}_j}$ the corresponding system of generalized integers, and ${\zeta }_{{\mathcal{P}}_j}\left(s\right)$, $s=\sigma +it$, the Beurling zeta-function. In the paper, we consider simultaneous approximation of a collection...

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand...

Derivative-matching approximations are constructed as power series built from functions. The method assumes knowledge of the special values of the Bell polynomials of the second kind, for which we refer to the literature. The presented ideas may have...

A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel tran...

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters a...

There has been a growing interest in expressivity of deep neural networks. However, most of the existing work about this topic focuses only on the specific activation function such as ReLU or sigmoid. In this paper, we investigate the approximation a...

For $j=1,\dots ,r$, ${\alpha }_j\in (0,1)$ are such that the set $\{\log (m+{\alpha }_j):m\in {\mathbb{N}}_0,j=1,\dots ,r\}$ is linearly independent over $\mathbb{Q}$. In the paper, we consider simultaneous approximation of a tuple of analytic functions by shifts $(\zeta (s+i\tau ,$...

In this article we present the multivariate approximation of time splitting random functions defined on a box or ${\mathbb{R}}^N,N\in \mathbb{N},$ by neural network operators of quasi-interpolation type. We achieve these approximations by obtaining quantitative-type Jack...

Stochastic computing (SC) is a computational paradigm that represents numbers using Bernoulli processes. It is attractive due to its low power, low hardware complexity, and excellent noise tolerance properties. An SC algorithm that approximates arbit...

Approximation of piecewise linear and generalized functions is an important and difficult problem. These functions are widely used in mathematical modeling of various processes and systems, such as: automatic control theory, electrical engineering, r...

In this paper, we prove the theorems on the simultaneous approximation of a pair of analytic functions by discrete shifts $(\zeta (s+ik{h}_1),\zeta (s+ik{h}_2,\alpha ))$, $h_1>0$, $h_2>0$ of the Riemann zeta function $\zeta \left(s\right)$ and Hurwitz zeta function $\mathrm{\&zeta}$...

In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator $M_{q,n}$. These quantum Meyer-König-Zeller (MKZ) fractal functions employ $M_{q,n}f$ as the base function in the iterated function syste...

In this paper, we study the best approximation of a fixed fuzzy-number-valued continuous function to a subset of fuzzy-number-valued continuous functions. We also introduce a method to measure the distance between a fuzzy-number-valued continuous fun...

This article presents a hybrid approach that combines Laguerre orthonormal functions with deep neural networks (DNN) for effective approximation of impulse responses of dynamic systems. Attention is given to key limitations in approximation with Lagu...

Nonlinear functions are widely used as activation functions in artificial neural networks, which have a great impact on the fitting ability of artificial neural networks. Due to the complexity of the activation function, the computation of the activa...

This paper proposes a hybrid architecture for obtaining digital twins of dynamic systems under conditions of parametric uncertainties and unmodeled dynamics through approximation using deep neural networks (DNNs) and orthonormal Kautz functions. In t...

We propose and investigate two new methods to approximate $f\left(\mathbf{A}\right)\mathbf{b}$ for large, sparse, Hermitian matrices $\mathbf{A}$. Computations of this form play an important role in numerous signal processing and machine learning tasks. The main idea behind both methods is t...

This paper proposes an iterative algorithm for the search for common fixed points of two mappings. The properties of approximation and convergence of the method are analyzed in the context of Banach spaces. In particular, this article provides suffic...

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion the...

In this article, pursuit and evasion policies are developed for multi-player pursuit–evasion games, while obstacle avoidance and velocity constraints are considered simultaneously. As minimum and maximum approximation functions are both differe...

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accur...

A spline-based integral approximation is utilized to define a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The real case is considered and the approximations can be im...

In this paper, function approximation is utilized to establish functional series approximations to integrals. The starting point is the definition of a dual Taylor series, which is a natural extension of a Taylor series, and spline based series appro...

The Z-function is the real function given by $Z\left(t\right)=e^{i\theta \left(t\right)}\zeta \left({\displaystyle \frac{1}{2}}+it\right)$, where $\zeta \left(s\right)$ is the Riemann zeta function, and $\theta \left(t\right)$ is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has...

Applying a predefined set of functionals to a function provides a set of numbers which characterize the function. One can then interpret an approximation very generally as a construction of some function form which reproduces these numbers if the sam...

Studying the Bernstein approximations for fuzzy functions is a new attempt. With the help of considering the support functions of fuzzy sets in which the weak* topology for the normed dual space is involved, we are able to approximate continuous fuzz...

We found analytic approximations for the Bessel function of the first kind $J_{\nu }\left(x\right)$, valid for any real value of x and any value of $\nu$ in the interval (−1/2, 3/2). The present approximation is exact for $\nu =-1/2$, $\nu =1/2$, and $\nu =3/2$...

In this paper, a new approximation formula for the gamma function and some of its symmetric inequalities are established. We prove the superiority of our results over Yang and Tian’s approximation formula for the gamma function of order $v^{\&minus}$...

Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier&ndash...

In this paper, we propose an optimized method for nonlinear function approximation based on multiplierless piecewise linear approximation computation (ML-PLAC), which we call OML-PLAC. OML-PLAC finds the minimum number of segments with the predefined...

In this paper, we study the approximation of continuous functions on a subclass of singular space—the subcartesian space. As is well known, the orbit space of the proper action of a Lie group on a smooth manifold is a subcartesian space. We pro...

We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions—enhanced with Padé approximants&m...

Low Density Parity Check (LDPC) codes are currently being deeply analyzed through algorithms that require the capability of addressing their iterative decoding convergence performance. Since it has been observed that the probability distribution func...

This paper presents a novel deep-learning network designed to detect intervals of jump discontinuities in single-variable piecewise smooth functions from their noisy samples. Enhancing the accuracy of jump discontinuity estimations can be used to fin...

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a...

In this paper, two new approximation formulas with monotonic remainders for the gamma function have been presented. Also, we present some numerical comparisons between our new approximation formulas and some known ones, which demonstrate the superior...

Determining the exact number of primes at large magnitudes is computationally intensive, making approximation methods (e.g., the logarithmic integral, prime number theorem, Riemann zeta function, Chebyshev’s estimates, etc.) particularly valuab...

In this paper, we derive an analytical and explicit approximation for the Bessel function $J_2\left(x\right)$ for positive real x with a maximum absolute error of approximately $0.004$, which refines some new published analytic approximation. The absolute errors for...

The discussion revolves around the most recent outcomes in the realm of approximating functional integrals through calculations. Review of works devoted to the application of functional integrals in quantum mechanics and quantum field theory, nuclear...

The Gaussian Q-function has considerable applications in numerous areas of science and engineering. However, the fact that a closed-form expression for this function does not exist encourages finding approximations or bounds of the Q-function. In thi...

Since 1975, it has been known that the Hurwitz zeta-function has a unique property to approximate by its shifts all analytic functions defined in the strip $\mathfrak{D}=\{s=\sigma +it:1/2<\sigma <1\}$. However, such an approximation causes efficiency problems, and apply...

A new simple and accurate expression to approximate the modified Bessel function of the first kind $I_1(x)$ is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA....

Neural networks are powerful universal approximation tools. They have been utilized for functions/data approximation, classification, pattern recognition, as well as their various applications. Uncertain or interval values result from the incompleten...

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper,...

The two-parameter Mittag–Leffler function $E_{\alpha ,\beta }$ is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating thi...

Surface modeling is closely related to interpolation and approximation by using level set methods, radial basis functions methods, and moving least squares methods. Although radial basis functions with global support have a very good approximation ef...

The Gaussian-radial-basis function neural network (GRBFNN) has been a popular choice for interpolation and classification. However, it is computationally intensive when the dimension of the input vector is high. To address this issue, we propose a ne...

Mathematical models and algorithms for maximizing power extraction have become an essential topic in renewable energies in the last years, especially in wind energy conversion systems. This study proposes maximum power point tracking using gain sched...

In this work, we deal with two approximation problems in a finite-dimensional generalized Wendland space of compactly supported radial basis functions. Namely, we present an interpolation method and a smoothing variational method in this space. Next,...

This study proposes a decoupling approach based on first-order exponential function approximation for reliability-based optimization. It establishes an explicit relationship between the failure probability and design parameters within only one implem...