We constructed linear algorithms of sampling recovery and cubature formulas on Smolyak grids parametrized by $m \in \mathbb{N}$ of periodic $d$-variate functions having Lipschitz-Hölder mixed smoothness $α> 0$ based on B-spline quasi-interpolation, and studied their optimality. We established lower estimates (for $α\le 2$) and upper bounds of the error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit in $d$, $m$ and the number $ν$ of active variables of functions when $d$ and $m$ may be large.
We study approximation of multivariate periodic functions from Besov and Triebel--Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the $L_q$-…
The class $Ts(r,f)$ the trigonometric interpolation splines depending on the parameter vectors, selected convergence factors and interpolation factors is considered. The main properties of simple interpolation trigonometric splines are given, which are also transferred to periodic simple interpolation polynomial splines. These results lead to the possibility of combining the theory of simple polynomial interpolation splines and the basics of the theory of simple trigonometric splines into a single theory - the theory of interpolation splines
Explicit pointwise error bounds for the interpolation of a smooth function by piecewise exponential splines of order four are given. Estimates known for cubic splines are extended to a natural class of piecewise exponential splines which are appearing in the construction of multivariate polysplines. The error estimates are derived in an inductive way using error estimates for the interpolation of a smooth function by exponential splines of order two.
The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on the same grids are trigonometric LS polynomials and LS splines, which allow us to construct the approximation of functions by the least squares method and also depend linearly on the values of the approximate function. Trigonometric splines were considered as splines with polynomial analogues in some cases. The material is illustrated in graphs. The considere…
In this paper we construct Ritz-type projectors with boundary interpolation properties in finite dimensional subspaces of the usual Sobolev space and we provide a priori error estimates for them. The abstract analysis is exemplified by considering spline spaces and we equip the corresponding error estimates with explicit constants. This complements our results recently obtained for explicit spline error estimates based on the classical Ritz projectors in [Numer. Math. 144(4):889--929, 2020].
The construction of smooth spatial paths with Pythagorean-hodograph (PH) quintic spline biarcs is proposed. To facilitate real-time computations of $C^2$ PH quintic splines, an efficient local data stream interpolation algorithm is introduced. Each spline segment interpolates second and first order Hermite data at the initial and final end-point, respectively. In the spline extension of the scheme a $C^2$ smooth connection between successive spline segments is obtained by taking the locally required second-order derivative information from the previous segment. Consequently, the data stream sp…
A class of trigonometric interpolation splines depending on parameter vectors, selected convergence factors and interpolation factors is considered. The concept of crosslink grids and interpolation grids is introduced; these grids can match or y not match. Parseval equality is given and some properties of splines are discussed. The material is illustrated by a numerical example.
We prove quartic convergence of cubic spline interpolation for curves into Riemannian manifolds as the grid size of the interpolation grid tends to zero. In contrast to cubic spline interpolation in Euclidean space, where this result is classical, the interpolation operator is no longer linear. Still, concepts from the linear setting may be generalized to the Riemannian case, where we try to use intrinsic Riemannian formulations and avoid charts as much as possible.
In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D Laplace boundary value problems, using a collocation method within the Isogeometric Analysis paradigm. In such setting the regular part of the integrand can be defined as the product of a tensor product B-spline and a general function. The rules are derived by using first the spline quasi-interpolation approach to approximate such function and then the extens…
One of the possible variants of the classification of trigonometric interpolation splines is considered, depending on the chosen convergence factors, the distribution of signs of the basis functions and the interpolation factors. The concept of crosslinking and interpolation grids is introduced; these grids can either match or not match. The proposed classification is illustrated by an example.
Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. In this paper, we study sufficient conditions to obtain monotone cubic splines based on Hermite cubic interpolators and propose different ways to construct them using non-linea…
Interpolation of classes of differentiated functions given on a finite interval by trigonometric splines using the phantom node method is considered. This method consists in supplementing a given sequence of values of an approximate function with an even number of values of a phantom function, which is constructed in such a way as to eliminate gaps in both the function itself and its derivatives up to and including a certain order; in the General case, these gaps occur with the periodic continuation of the function given at a finite interval. The results of calculations on test examples for tr…
Under consideration methods of constructing trigonometric interpolation splines of two variables on rectangular areas. These methods are easily generalized to the case of trigonometric interpolation splines of several variables on such domains. A numerical example illustrating the main theoretical propositions is considered. The given methods of constructing such splines can be widely used in practice.
In the classical theory of cubic interpolation splines there exists an algorithm which works with only $O\left( n\right)$ arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces their computation to interpolation cubic splines and also performs with $O\left( n\right)$ arithmetic operations. In this paper it is shown that many features of the polynomial cubic spline setting carry over to the larger class of $L$-splines where $L$ is a linear differential operator of order $4$ with constant coefficients. Criteria are given such that the …
