Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines

This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.     

Another approach to the quintic spline

A set of end conditions for interpolatory quintic spline is derived by use of integration. These end conditions are only in terms of function values at the knots (data), and rise to O(h⁶) spline approximation.     

Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in . Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.

     

Subdivision Schemes of Sets and the Approximation of Set-Valued Functions in the Symmetric Difference Metric

In this work we construct subdivision schemes refining general subsets of ? ⁿ and study their applications to the approximation of set-valued functions. Differently from previous works on set-valued approximation, our methods are developed and analyzed in the metric space of Lebesgue measurable sets endowed with the symmetric difference metric. The construction of the set-valued subdivision schemes is based on a new weighted average of two sets, which is defined for positive weights (corresponding to interpolation) and also when one weight is negative (corresponding to extrapolation). Using the new average with positive weights, we adapt to sets spline subdivision schemes computed by the Lane–Riesenfeld algorithm, which requires only averages of pairs of numbers. The averages of numbers are then replaced by the new averages of pairs of sets. Among other features of the resulting set-valued subdivision schemes, we prove their monotonicity preservation property. Using the new weighted average of sets with both positive and negative weights, we adapt to sets the 4-point interpolatory subdivision scheme. Finally, we discuss the extension of the results obtained in metric spaces of sets, to general metric spaces endowed with an averaging operation satisfying certain properties.     

On spline approximation with a reproducing kernel method

Spline approximation with a reproducing kernel of a semi-Hilbert space is studied. Conditions are formulated that uniquely identify the natural Hilbert space by a reproducing kernel, a trend of the spline, and the approximation domain. The construction of a spline with external drift is proposed. It allows one to approximate functions having areas of large gradients or first-kind discontinuities. The conditional positive definiteness of some known radial basis functions is proved.     

Sobolev estimates for constructive uniform-grid FFT interpolatory approximations of spherical functions

The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications. In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the uniform spherical-polar grid locations \(\left (\frac {j\pi }{N}, \frac {k \pi }{N}\right )\), for j=1,…,N?1, k=0,…,2N?1. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than N. Using the \({\mathcal {O}}(N^{2})\) data, the spherical interpolatory approximation is efficiently constructed by applying the FFT techniques (in both azimuthal and latitudinal variables) with only \({\mathcal {O}}(N^{2} \log N)\) complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework that are well suited for quantification of various computer models. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and strongly-oscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.     

Minimal-norm-derivative spline function in interpolation and approximation

The paper is concerned with applications of quadratic splines with minimal derivative to approximation of functions in approximation and interpolation problems. A smooth spline is constructed on a uniform mesh so as the norm of the spline derivative is minimal; the nodes of the spline and the nodes of interpolations coincide. This approach allows construction of a spline from given values of the function on the mesh without additional assignment of the value of the function derivative at the initial point, because the derivative can be determined from the minimality condition for the norm of the spline derivative in L ₂.     

Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations

In this paper necessary and sufficient optimality conditions for uniform approximation of continuous functions by polynomial splines with fixed knots are derived. The obtained results are generalisations of the existing results obtained for polynomial approximation and polynomial spline approximation. The main result is two-fold. First, the generalisation of the existing results to the case when the degree of the polynomials, which compose polynomial splines, can vary from one subinterval to another. Second, the construction of necessary and sufficient optimality conditions for polynomial spline approximation with fixed values of the splines at one or both borders of the corresponding approximation interval.     

Curvature variation minimizing cubic Hermite interpolants

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain G¹ cubic interpolatory spline.     

A Method Which Generates Splines in H-Locally Convex Spaces and Connections with Vectorial Optimization

In this research paper we present a modality for generating splines in H-locally convex spaces which allows us to solve some problems of best approximation by linear subspaces of spline functions in these spaces. In this way one shows that the elements of best vectorial approximation coincide with the spline functions introduced by us in a previous research work. These splines are also the only elements of best simultaneous approximation by their generated linear subspaces with respect to any family of seminorms which induces the H-locally convex topology and, consequently, they are the only solutions for some frequent strong and vectorial optimization programs. Moreover, as we shall see in the numerical examples, our construction leads to discover orthogonal decompositions for H-locally convex spaces which, in general, are difficult to be identified.     

An O(h6) cubic spline interpolating procedure for harmonic functions

An O(h⁶) method for the interpolation of harmonic functions in rectangular domains is described and analyzed, The method is based on an earlier cubic spline technique [N. Papamichael and J.R. Whiteman, BIT 14 , 452–459 (1974)], and makes use of recent results concerning the a posteriori correction of interpolatory cubic splines.     

RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS

This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1] are chosen as a starting point for characterizations of functions in Besov spaces B _r,r ⁶ (0.1) with 0<σ<∞ and (1+σ)⁻¹相似文献   

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

Restricted nonlinear approximation and singular solutions of boundary integral equations

This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.     

Comparison of Radial Basis Functions

A survey of algorithms for approximation of multivariate functions with radial basis function (RBF) splines is presented. Algorithms of interpolating, smoothing, selecting the smoothing parameter, and regression with splines are described in detail. These algorithms are based on the feature of conditional positive definiteness of the spline radial basis function. Several families of radial basis functions generated by means of conditionally completely monotone functions are considered. Recommendations for the selection of the spline basis and preparation of initial data for approximation with the help of the RBF spline are given.     

Bivariate interpolation based on univariate subdivision schemes

The paper presents a bivariate subdivision scheme interpolating data consisting of univariate functions along equidistant parallel lines by repeated refinements. This method can be applied to the construction of a surface passing through a given set of parametric curves. Following the methodology of polysplines and tension surfaces, we define a local interpolator of four consecutive univariate functions, from which we sample a univariate function at the mid-point. This refinement step is the basis to an extension of the 4-point subdivision scheme to our setting. The bivariate subdivision scheme can be reduced to a countable number of univariate, interpolatory, non-stationary subdivision schemes. Properties of the generated interpolant are derived, such as continuity, smoothness and approximation order.     

Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

变阶唯一可解函数的一致逼近

设F(A,x)是连续依赖于x∈[a,b]和参数A∈P的实值函数,其中P是实n维空间的一个非空子集,并设F(A,x)是Rice意义下的变阶唯一可解函数[1,p.3—6],它关于P具有最大阶n。这类函数最典型的是由Meinardus提出的一种具有局部     

A sinusoidal polynomial spline and its Bezier blended interpolant

Functional polynomials composed of sinusoidal functions are introduced as basis functions to construct an interpolatory spline. An interpolant constructed in this way does not require solving a system of linear equations as many approaches do. However there are vanishing tangent vectors at the interpolating points. By blending with a Bezier curve using the data points as the control points, the blended curve is a proper smooth interpolant. The blending factor has the effect similar to the “tension” control of tension splines. Piecewise interpolants can be constructed in an analogous way as a connection of Bezier curve segments to achieve C¹ continuity at the connecting points. Smooth interpolating surface patches can also be defined by blending sinusoidal polynomial tensor surfaces and Bezier tensor surfaces. The interpolant can very efficiently be evaluated by tabulating the sinusoidal function.