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In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results. [ABSTRACT FROM AUTHOR]

Let ω be a positive, Lebesgue integrable and exponential decay function defined on an infinite interval I and let L1,ω(I) be the space of weighted Lebesgue integrable functions on I. In this paper, we give the relations of the best one-sided approximation and the optimal Hermite–Fejér interpolation by the set of algebraic polynomials of degree not exceeding a given number for the smooth function classes BC2r,+(I), r ∈ ℕ+, in the metric of the space L1,ω(I), and prove that the Hermite–Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal in L1,ω(I). In addition, we show that the approximation error of the optimal Hermite–Fejér interpolation and quadrature errors of the weighted Gaussian quadrature formula are equal, and give the exact constant of the errors. [ABSTRACT FROM AUTHOR]

The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approximation task, the root-polynomial functions of the fourth and fifth order were applied, the main features of which are the ravine character and the presence of one global minimum. As an initial approach to solving the approximation problem, the values of the polynomial coefficients are calculated by solving the interpolation problem. After this, the approximation task is solved iteratively. All necessary polynomial coefficients are calculated multiple times, taking into account the values of the function and its derivative at the reference points. The final values of polynomial coefficients of high-order root-polynomial functions are calculated using the dichotomy method. The article also provides examples of the applying fourth-order and fifth-order root-polynomial functions to approximate sets of numerical data that correspond to the description of ravine functions. The obtained theoretical results are interesting and important for the experts who study the physics of electron beams and design modern industrial electron beam technological equipment.

The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approximation task, the root-polynomial functions of the fourth and fifth order were applied, the main features of which are the ravine character and the presence of one global minimum. As an initial approach to solving the approximation problem, the values of the polynomial coefficients are calculated by solving the interpolation problem. After this, the approximation task is solved iteratively. All necessary polynomial coefficients are calculated multiple times, taking into account the values of the function and its derivative at the reference points. The final values of polynomial coefficients of high-order root-polynomial functions are calculated using the dichotomy method. The article also provides examples of the applying fourth-order and fifth-order root-polynomial functions to approximate sets of numerical data that correspond to the description of ravine functions. The obtained theoretical results are interesting and important for the experts who study the physics of electron beams and design modern industrial electron beam technological equipment. [ABSTRACT FROM AUTHOR]

In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function (λ -greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the λ -greedy algorithm enables us to define target-data-independent interpolation nodes. [ABSTRACT FROM AUTHOR]

Signed distance fields (SDFs) have emerged as an alternative shape representation for real-time collision detection and lighting effects. Computing these for complex models can be expensive, so one popular approach is to prepare an approximation via sampling and interpolation. Then, these may be rendered using sphere marching, which gets close to the surface quickly, but needs several iterations to converge to it. In this paper, we propose an alternative that computes the intersection of a given ray and the surface analytically at a narrow band. This may be combined with other enhancements like having variable error for the approximation depending on the distance to the surface and skipping regions that do not contain the surface to accelerate the outer band ray traversal while reducing the required memory. To achieve smoother representations with minimal computational cost, we propose a method for computing surface intersections and normals from separate interpolants. We evaluate all these to find the optimal combination improving the rendering performance and memory consumption of these SDF approximations. [Display omitted] • Enhancements for rendering SDF piecewise approximations. • Solving narrow band intersections using analytical methods. • Reducing SDF approximations for improving required memory and outer band ray traversal. [ABSTRACT FROM AUTHOR]

In this article, two new techniques of approximation of the reliability of a two-terminal network are developed based on the constructive theory of functions and related methods. Two methods of generating an approximation cubic spline are used: Lagrange-type interpolation procedures and Bernstein approximation operator. A possibility of minimizing the total error of approximation, based on keeping some properties invariant, is described in case of a large class of pairs of dual two-terminal networks. Simulations are included, showing that the error of approximation is negligible in case of some special initial data. [ABSTRACT FROM AUTHOR]

• We construct a new representation of CSRBF, which unites interpolation and approximation. • A further adaptive algorithm to determine interpolation and approximation at each point is presented. • We develop the quasi-interpolation method into a method combining interpolation and approximation. • With our multi-level scheme, we can interpolate all the data points and hold a low shape approximation error. Interpolation and approximation radial basis function (RBF) has been widely used in the implicit surface reconstruction. However, a simple interpolation or approximation method may either have unwanted oscillation around the scattered points, or have low accuracy at scattered points. In this paper, we construct a new representation of RBF, which unites interpolation and approximation. A further adaptive algorithm to determine interpolation and approximation is presented. With the new representation, we can develop quasi-interpolation method from an approximation method into a method combining interpolation and approximation. And we can determine interpolation or approximation at each of scattered points without solving large linear systems. Experiments show that the given method is precision-controlled and fast in implicit surface reconstruction. [ABSTRACT FROM AUTHOR]