Penalized hyperbolic-polynomial splines

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

Download Full-text

Curve and surface construction using variable degree polynomial splines

Computer Aided Geometric Design ◽

10.1016/s0167-8396(00)00010-8 ◽

2000 ◽

Vol 17 (5) ◽

pp. 419-446 ◽

Cited By ~ 54

Author(s):

Paolo Costantini

Keyword(s):

Variable Degree ◽

Polynomial Splines ◽

Degree Polynomial ◽

Surface Construction

Download Full-text

Interpolating Polynomial Splines for Constructing Smooth Curves

Numerical Algorithms with Fortran ◽

10.1007/978-3-642-80043-6_10 ◽

1996 ◽

pp. 251-286

Author(s):

Gisela Engeln-Müllges ◽

Frank Uhlig

Keyword(s):

Polynomial Splines ◽

Smooth Curves

Download Full-text

Non-periodic Polynomial Splines

Spline and Spline Wavelet Methods with Applications to Signal and Image Processing ◽

10.1007/978-3-319-22303-2_4 ◽

2015 ◽

pp. 51-85

Author(s):

Amir Z. Averbuch ◽

Pekka Neittaanmäki ◽

Valery A. Zheludev

Keyword(s):

Polynomial Splines

Download Full-text

A characterization of supersmoothness of multivariate splines

Advances in Computational Mathematics ◽

10.1007/s10444-020-09813-y ◽

2020 ◽

Vol 46 (5) ◽

Author(s):

Michael S. Floater ◽

Kaibo Hu

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Finite Order ◽

Maximal Order ◽

Spline Functions ◽

Multivariate Splines ◽

The Finite Element Method ◽

Polynomial Splines ◽

Element Method

Abstract We consider spline functions over simplicial meshes in $\mathbb {R}^{n}$ ℝ n . We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as supersmoothness, which plays a role in the construction of multivariate splines and in the finite element method. In this paper, we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations.

Download Full-text

Continuous delay estimation with polynomial splines

The Journal of the Acoustical Society of America ◽

10.1121/1.4787607 ◽

2006 ◽

Vol 120 (5) ◽

pp. 3112-3112

Author(s):

Gianmarco Pinton ◽

Gregg Trahey

Keyword(s):

Delay Estimation ◽

Polynomial Splines ◽

Continuous Delay

Download Full-text

Interval estimation of trigonometrical splines

MATEC Web of Conferences ◽

10.1051/matecconf/201929203001 ◽

2019 ◽

Vol 292 ◽

pp. 03001

Author(s):

I.G. Burova ◽

E.G. Ivanova ◽

V.A. Kostin

Keyword(s):

Function Approximation ◽

Order Approximation ◽

Spline Approximation ◽

Approximation Error ◽

Interval Estimation ◽

Polynomial Splines ◽

Third Order ◽

Variation Range ◽

Local Spline ◽

Third Order Approximation

Quite often, it is necessary to quickly determine variation range of the function. If the function values are known at some points, then it is easy to construct the local spline approximation of this function and use the interval analysis rules. As a result, we get the area within which the approximation of this function changes. It is necessary to take into account the approximation error when studying the obtained area of change of function approximation. Thus, we get the range of changing the function with the approximation error. This paper discusses the features of using polynomial and trigonometrical splines of the third order approximation to determine the upper and lower boundaries of the area (domain) in which the values of the approximation are contained. Theorems of approximation by these local trigonometric and polynomial splines are formulated. The values of the constants in the estimates of the errors of approximation by the trigonometrical and polynomial splines are given. It is shown that these constants cannot be reduced. An algorithm for constructing the variation domain of the approximation of the function is described. The results of the numerical experiments are given.

Download Full-text