Vldimir Denysiuk
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.
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Generalization of Trigonometric B-splines and Kernels of Interpolating Trigonometric Splines with Riemann Convergence Multipliers
本文推广了在不同拼接与插值网格上提取任意阶Riemann三角B样条及插值样条核的方法，证明了不同网格组合下存在不同的三角B样条，并通过数值算例验证了理论原理及其应用价值。
B-splines and kernels of trigonometric interpolation splines
本文研究任意阶三角Riemann B样条与三角插值样条的Riemann核，证明三角插值样条可表示为三角B样条与相应核的卷积，并通过示例说明其广泛应用价值。
Classes of simple polynomial and simple trigonometric splines given by Fourier series are considered. It is shown that the class of simple trigonometric splines includes the class of simple polynomial splines. For some parameter values, the polynomial splines coincide with the trigonometric ones; this allows to transfer to such trigonometric splines all the results obtained for polynomial splines. Thus, it was possible to combine two powerful theories - the theory of trigonometric Fourier series and the theory of simple polynomial splines. The above material is illustrated by numerous examples…
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
The methods of approximation, regularization and smoothing of trigonometric interpolation splines are considered in the paper. It is shown that trigonometric splines can be considered from two points of view - as a trigonometric Fourier series and as discrete trigonometric Fourier series according to certain systems of functions that are smoothness carriers. It is argued that with approximation and smoothing of trigonometric splines it is expedient to consider as discrete rows, since their differential properties are stored.
Forms of representation of interpolation trigonometric splines
本文研究了三角插值样条的三种表示形式，重点讨论了三角B样条表示，并将多项式系数表示与基本三角样条表示推广至非周期函数情形。
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.
Methods of constructing trigonometric fundamental splines with constant sign and sign-changing convergence factors are given. An example and graphics illustrating the concepts of convergence and interpolation grids are given. Some methods of constructing constant-sign and sign-changing coefficients of convergence of trigonometric splines are considered.
$h$-Trigonometric B-splines
本文引入离散三角B样条，推导递推关系、导数公式和Marsden恒等式，证明经典B样条结果可自然推广至三角及离散三角情形。
The work examines norms in of fundamental trigonometric splines of odd and even degrees, which in some cases coincide with polynomial ones. Fundamental trigonometric splines for the case where the con-vergence factors depend on the parameter are also considered. It is shown that an important characteristic of these splines of odd and even degrees is their complete variation. The semi-norms of trigonometric splines of odd degrees are also studied. The given material is illustrated by graphs; so, in particular, an illustration of the property of the least curvature of polynomial splines is given…
Finite Fourier series. The class of trigonometric splines
本文在离散等距点上引入具有插值与微分性质的有限正交函数系，构建有限傅里叶级数形成三角样条类，包含多项式周期简单样条，并指出其推广方向与后续研究需求。
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…
About one method of construction of interpolation trigonometric splines
本文研究了一种通过构造具有特定递减顺序系数的三角傅里叶级数来构造样条类的方法，该递减顺序决定了级数和的连续导数次数，并通过分组和引入乘数来获得具有插值性质的三角插值样条类。
Some questions of application of trigonometric splines in problems of spectral analysis are considered. The known effects of overlay in the frequency and time domains are discussed; deployment effects in these areas are firstly considered. The relation between discrete Fourier coefficients and Fourier coefficients of trigonometric splines was obtained. The expediency of taking into account the differential properties of the investigated signals that are losing during the primary sampling is shown.
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.
Incomplete (even and odd) trigonometric splines in the problems of constructing approximate solutions of second order linear differential equations
本文研究利用不完全偶与奇三角样条构造二阶线性微分方程第一边值问题近似解的方法，并通过数值算例验证了理论结果。
Generation of the Trigonometric Cubic B-Spline Collocation Solutions for the Kuramoto-Sivashinsky(KS) Equation
本文提出利用三角三次B样条函数进行置留法求解库莫-西瓦斯基方程，通过降阶处理高阶项，结合Crank-Nicolson方法进行时间积分，验证了该方法的有效性。
Trigonometric Splines for Oscillator Simulation
本文研究了使用样条方法模拟振荡器时数值阻尼的影响，提出通过三角样条消除其负面影响，提升自振系统模拟的可靠性。
On the normalization of trigonometric and hyperbolic B-splines
本文研究了三角和双曲B-样条的归一化问题，提出递归算法并展示了其在精确表示圆和近似能力方面的应用。
The method of constructing trigonometric Hermite splines, which interpolate the values of some periodic function and its derivatives in the nodes of a uniform grid, is considered. The proposed method is based on the periodicity properties of trigonometric functions and is reduced to solving only systems of linear algebraic equations of the second order; solutions of these systems can be obtained in advance. When implementing this method, it is necessary to calculate the coefficients of interpolation trigonometric polynomials that interpolate the values of the function itself and the values of …
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…
A Trigonometric Cubic B-spline Finite Element Method for Solving the Nonlinear Coupled Burger Equation
本文提出一种三角余弦三次B样条有限元方法，结合Crank-Nicolson方法求解非线性耦合Burgers方程，并通过变体Thomas算法求解非线性代数方程系统，验证了方法的准确性。
Non-stationary subdivision schemes originated from uniform trigonometric B-spline
提出一种基于均匀三角B样条基函数生成任意m点二进制非平稳细分格式的算法，利用渐近等价理论分析光滑性，该格式能再现圆锥曲线、三角多项式和三角样条。
The application of cubic trigonometric B-spline to the numerical solution of time-fractional telegraph equation
本文提出了一种高效数值技术，利用立方三角B样条解决时间分数电报方程，通过有限差分法和空间导数的立方三角B样条技术，验证了方法的稳定性与准确性。
The Trigonometric Cubic B-spline Algorithm for Burgers' Equation
本文提出基于三角三次B-样条函数的置留法求解Boussinesq方程，通过数值实验验证了该方法在时间离散和空间离散中的有效性。
An algorithm based on DQM with modified trigonometric cubic Bsplines for solving coupled viscous Burgers equations
本文提出一种改进的三角立方B样条差分求积方法，用于求解时间依赖的偏微分方程，特别针对Burgers方程的数值解，通过将初值边界值问题转化为常微分方程初值问题，并采用五阶段四阶强稳定性保存RK方法求解，验证了方法的精度和效率。
An Efficient Computational Technique based on Cubic Trigonometric B-splines for Time Fractional Burgers' Equation
本文提出一种基于三次三角B样条的线性计算技术，用于求解时间分数阶Burgers方程，通过高效线性化技术减少计算成本，结合有限差分法和B样条函数实现稳定性与精度分析，验证了方法的准确性与有效性。
The Numerical Approach to the Fisher's Equation via Trigonometric Cubic B-spline Collocation Method
本文提出一种结合三角三次B样条函数和Crank-Nicolson方法的数值技术，用于求解费舍尔方程，验证了该方法的准确性及有效性。
Numerical Solution of second order hyperbolic telegraph equation via new Cubic Trigonometric B-Splines Approach
本文提出了一种新的方法，利用三次三角B样条格点法求解带有狄利克雷和诺伊曼边界条件的二维超声波方程，验证了该方法在不同参数下的无条件稳定性，并通过测试问题展示了方法的准确性和计算效率。
A Finite Difference Scheme based on Cubic Trigonometric B-splines for Time Fractional Diffusion-wave Equation
本文提出基于三次三角基函数的高效数值方案，用于求解含反应项的时间分数扩散波方程，通过有限差分和三角B样条离散，经稳定性分析与实验验证，证明其精度高、计算成本低，优于现有方法。
A Comperative Numerical Study Based on Cubic Polynomial and Trigonometric B-splines for the Gardner Equation
本文比较了三次多项式与三角B-样条在求解Gardner方程中的性能，研究了孤立波、kink波、波生成及正反向传播的双孤子相互作用，通过离散最大范数和守恒律变化评估数值解的准确性与稳定性。
Numerical Solutions of Reaction-Diffusion Equation Systems with Trigonometric Quintic B-spline Collocation Algorithm
本文提出三角五次B样条置信算法用于求解反应扩散方程组，通过时间离散化和空间离散化方法，验证了该算法在不同问题中的准确性与有效性，适用于高阶微分方程的求解。
Hyperbolic trigonometric functions as approximation kernels and their properties II: Wavelets
本文提出利用双曲三角函数构造的小波进行函数近似，并探讨其在时频分析中的应用及通用性。
Construction of interpolation splines minimizing the semi-norm in the space $K_2(P_m)$
本文基于S.L. Sobolev方法，构造了最小化特定表达式的插值样条，并给出了系数的显式公式，该样条对多项式和三角函数具有精确性。
Generalized T-splines and VMCR T-meshes
本文探讨了T-样条方法向广义B-样条（GB-样条）的扩展，提出了一种VMCR T-网格类，使对应的广义T-样条与多项式T-样条线性无关。
Hyperbolic trigonometric functions as approximation kernels and their properties I: generalised Fourier transforms
该研究提出一类基于双曲三角函数的新型径向基函数，推导其广义傅里叶变换展开式，结合Strang-Fix条件证明拟插值算子的多项式精确性并给出显式系数，通过多维数值算例验证不同构造的效果。
On some transforms of trigonometric series
研究将收敛三角级数变换为等价收敛级数的方法，并给出变换后级数比原级数收敛更快的充分条件。
Univariate spline quasi-interpolants and applications to numerical analysis
本文描述有界区间均匀划分上的新单变量样条拟插值，并将其应用于数值积分、数值微分和零点近似。
Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process. Usually many NURBS pieces are needed to build geometrically continuous CAD models. In this paper, we introduce some splines defined on branched covering of sphere, torus or general domains of $R^2$, which are called branched splines in this paper. A single piece of such splines is enough to build some complex CAD models. Multiresolution analysis on surfaces of hi…
Generalized $C^1$ quadratic B-splines generated by Merrien subdivision algorithm and some applications
本文在Merrien细分算法生成的广义二次样条空间中定义新的全局B样条基，给出细化方程与切角算法，并应用于单调凸插值、控制多边形收敛及精确拟插值算子的构造。
Spline functions have long been used in numerical solution of differential equations. Recently it revives as isogeometric analysis, which offers integration of finite element analysis and NURBS based CAD into a single unified process. Usually many NURBS pieces are needed to build geometrically continuous CAD models. In this paper, we introduce some multiply periodic splines defined on hyperbolic disc. A single piece of such splines is enough to build complex CAD models. Multiresolution analysis on surfaces of high genus built from such splines can be carried out naturally. CAD and FEA are inte…
On the Stability of Some Spline Collocation Implicit Difference Scheme
针对具有多重特征曲线的线性偏微分代数方程组边值问题，通过样条配置方法构造了各变量任意逼近阶的差分格式，并给出了其绝对稳定的充分条件。
We construct a family of monotone and convex $C^1$ integro cubic splines under a strictly convex position of the dataset. Then, we find an optimal spline by considering its approximation properties. Finally, we give some examples to illustrate the convex-preserving properties of these splines.
A note on some bounds between cubic spline interpolants depending on the boundary conditions: Application to a monotonicity property
本文针对审稿人提出的“能否通过设置端点一阶导数值让插值单调数据的三次样条保持单调性”的问题，证明该问题的答案是否定的，补充了三次样条边界条件与单调性关联领域的相关结果。
Normalized B-spline-like representation for low-degree Hermite osculatory interpolation problems
本文研究Hermite密切插值样条，针对具有额外光滑性的最低可能次数样条空间，提出了规范化类B样条表示，并基于开花和控制多项式开发了拟插值算子，数值测试验证了其性能。
Error analysis for local coarsening in univariate spline spaces
本文针对单变量样条空间中移除任意节点导致的误差进行了分析，推导出基于邻近节点和控制点的简单误差公式，并据此提出高效低成本的局部误差指示器与自适应粗化算法，通过数值实验验证了其性能。
Numerical integration rules based on B-spline bases
该研究基于B样条基相关关系，提出作为梯形积分法扰动形式的任意精度阶数新型数值积分公式，证明其逼近阶并推广至多维，通过数值实验验证了理论结果。
Numerical Solutions of the Gardner Equation by Extended Form of the Cubic B-splines
本文利用扩展的三次B-样条置位法求解Gardner方程的初值边界值问题，通过测试问题验证方法的效率和精度，并分析守恒律及方法稳定性。
Complementary results for the spectral analysis of matrices in Galerkin methods with GB-splines
本文研究了基于高光滑性广义B样条的Galerkin方法中矩阵谱分析的新结果，探讨了其核心方法及主要贡献。
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…
On the $L_p$-error of approximation of bivariate functions by harmonic splines
研究调和样条插值作为多项式样条插值的替代方法，并给出自适应逼近的$L_p$误差渐近性。
Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the me…
On Some Generalizations of B-Splines
本文研究了多项式和指数B-样条的推广，提出了一种从整数阶到复数阶的扩展，并构建了基于整数和复数阶多项式和指数B-样条的不可数族自指或分形函数，扩展了已知的分形插值技术以包含这种设置。
Locally refined spline surfaces (LRB) is a representation well suited for scattered data approximation. When a data set has local details in some areas and is largely smooth in other, LR B-splines allow the spatial distribution of degrees of freedom to follow the variations of the data set. An LRB surface approximating a data set is refined in areas where the accuracy does not meet a required tolerance. In this paper we address, in a systematic study, different LRB refinement strategies and polynomial degrees for surface approximation. We study their influence on data volume and accuracy when …
Cubic spline functions revisited
本文为仅用函数值的新三次插值样条 Q-spline 推导出四阶渐近最优误差界，并通过修正 not-a-knot 样条端部条件，使其在小网格尺寸下插值误差接近 Q-spline 且优于原 not-a-knot 样条。
Recent progress on univariate and multivariate polynomial and spline quasi-interpolants
本文综述了单变量与多变量多项式及样条拟插值的最新进展，涵盖其性质、系数泛函类型及应用，强调其无需解线性方程组、均匀有界等优点。
Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness
本文研究混合光滑函数的逼近问题，结合超几何交叉逼近和贪心逼近技术，获得三角系统最佳m项逼近的精确估计，并探讨数值积分与混合光滑函数的近似恢复。
Interpolations: a linear algebra approach
本文提出统一的线性代数框架处理插值问题，通过对偶空间表示插值条件，给出存在唯一性定理及基函数显式公式，并推广至三角插值及异质传感器场景。
This paper reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by (a suitable notion of) continuity in divided differences. As it happens, discrete splines bear connections to a wide array of developments in applied mathematics and statistics, from divided differences and Newton interpolation (dating back to over 300 years ago) to trend filtering (from the last 15 years). We survey these connections, and contribute some new perspect…
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