Quadrature Formulas for Refinable Functions and Wavelets II: Error Analysis

This paper is concerned with the construction and the analysis of Gauss quadrature formulas for computing integrals of (smooth) functions against refinable functions and wavelets. The main goal of this paper is to develop rigorous error estimates for these formulas. For the univariate setting, we derive asymptotic error bounds for a huge class of weight functions including spline functions. We also discuss multivariate quadrature rules and present error estimates for specific nonseparable refinable functions, i.e., for some special box splines.     

Approximations by polynomial and trigonometric splines of third order preserving some properties of approximated functions

In this paper with the help of parabolic splines we construct a linear method of approximate recovery of functions by their values on an arbitrary grid. In the method, a spline inherits the properties of monotonicity and convexity from the approximated function, and is sufficiently smooth. In addition, the constructed linear operator as an operator acting from the space of continuous functions to the same space has the norm equal to one. We also obtain similar results for trigonometric splines of third order.     

A topological spline system approach to even degree polynomial splines

The existence of minimum norm properties for even degree polynomial splines, analogous to the. ones known for odd degree splines, is investigated within the framework of the theory of

topological spline systems. It is shown that such properties cannot exist for even degree splines interpolating functions halfway between the partition points. For another class of even

degree spline functions, however, which hterpolate the local integrals of given functions with respect to the partitions, the seeked minimum norm properties can be proved. This is carried out

by first investigating a generalized problem within the theory of spline systems and then deriving corresponding conclusions. As a corollary the existence of spline systems with respect to differential operators of fractional degree is obtained.     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

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.     

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.     

On Spline Approximation for Singular Integral Equations on an Interval

This paper analyses the convergence of spline approximation methods for strongly elliptic singular integral equations on a finite interval. We consider collocation by smooth polynomial splines of odd degree multiplied by a weight function and a Galerkin-Petrov method with spline trial functions of even degree and piecewise constant test functions. We prove the stability of the methods in weighted Sobolev spaces and obtain the optimal orders of convergence in the case of graded meshes.     

Splines and Linear Control Theory

In this work, the relationship between splines and the linear control theory has been analyzed. We show that spline functions can be constructed naturally from the control theory. By establishing a framework based on control theory, we provide a simple and systematic way to construct splines. We have constructed the traditional spline functions including polynomial splines and the classical exponential spline. We have also discovered some new spline functions such as the combination of polynomial, exponential and trigonometric splines. The method proposed in this paper is easy to implement. Some numerical experiments are performed to investigate properties of different spline approximations.     

On interpolatory divergence-free wavelets

We construct interpolating divergence-free multiwavelets based on cubic Hermite splines. We give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields. We also show that the standard interpolatory wavelets, based on the Deslauriers-Dubuc interpolatory scheme or on interpolatory splines, cannot be used to construct compactly supported divergence-free interpolatory wavelets.

     

How to build all Chebyshevian spline spaces good for geometric design?

In the present work we determine all Chebyshevian spline spaces good for geometric design. By Chebyshevian spline space we mean a space of splines with sections in different Extended Chebyshev spaces and with connection matrices at the knots. We say that such a spline space is good for design when it possesses blossoms. To justify the terminology, let us recall that, in this general framework, existence of blossoms (defined on a restricted set of tuples) makes it possible to develop all the classical geometric design algorithms for splines. Furthermore, existence of blossoms is equivalent to existence of a B-spline bases both in the spline space itself and in all other spline spaces derived from it by insertion of knots. We show that Chebyshevian spline spaces good for design can be described by linear piecewise differential operators associated with systems of piecewise weight functions, with respect to which the connection matrices are identity matrices. Many interesting consequences can be drawn from the latter characterisation: as an example, all Chebsyhevian spline spaces good for design can be built by means of integral recurrence relations.     

The evaluation of the conjugate function of a periodic spline on a uniform mesh

We present an O(N log N) algorithm for the simultaneous evaluation (at all N points of a uniform mesh) of a function conjugate to a periodic spline. The algorithm is based on the attenuation factor theory and makes use of the FFT. It can be applied to other function classes with known attenuation factors also. For the approximation by functions conjugate to splines of odd degree we state several optimality criterions and error bounds.     

Multivariate Bernoulli splines and the periodic interpolation problem

Periodic spline interpolation in Euclidian spaceR d is studied using translates of multivariate Bernoulli splines introduced in [25]. The interpolating polynomial spline functions are characterized by a minimal norm property among all interpolants in a Hilbert space of Sobolev type. The results follow from a relation between multivariate Bernoulli splines and the reproducing kernel of this Hilbert space. They apply to scattered data interpolation as well as to interpolation on a uniform grid. For bivariate three-directional Bernoulli splines the approximation order of the interpolants on a refined uniform mesh is computed.     

Small Support Spline Riesz Wavelets in Low Dimensions

In Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006), a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in Han and Shen (J. Fourier Anal. Appl. 11:615–637, 2005). Motivated by these two papers, we develop in this article a general theory and a construction method to derive small support Riesz wavelets in low dimensions from refinable functions. In particular, we obtain small support spline Riesz wavelets from bivariate and trivariate box splines. Small support Riesz wavelets are desirable for developing efficient algorithms in various applications. For example, the short support Riesz wavelets from Han and Shen (SIAM J. Math. Anal. 38:530–556, 2006) were used in a surface fitting algorithm of Johnson et al. (J. Approx. Theory 159:197–223, 2009), and the Riesz wavelet basis from the Loop scheme was used in a very efficient geometric mesh compression algorithm in Khodakovsky et al. (Proceedings of SIGGRAPH, 2000).     

Construction of trivariate compactly supported biorthogonal box spline wavelets

We give a formula for the duals of the masks associated with trivariate box spline functions. We show how to construct trivariate nonseparable compactly supported biorthogonal wavelets associated with box spline functions. The biorthogonal wavelets may have arbitrarily high regularities.     

Generalized Nonparametric Mixed Effects Models

Generalized linear mixed effects models (GLMM) provide useful tools for correlated and/or over-dispersed non-Gaussian data. This article considers generalized nonparametric mixed effects models (GNMM), which relax the rigid linear assumption on the conditional predictor in a GLMM. We use smoothing splines to model fixed effects. The random effects are general and may also contain stochastic processes corresponding to smoothing splines. We show how to construct smoothing spline ANOVA (SS ANOVA) decompositions for the predictor function. Components in a SS ANOVA decomposition have nice interpretations as main effects and interactions. Experimental design considerations help determine which components are fixed or random. We estimate all parameters and spline functions using stochastic approximation with Markov chain Monte Carlo (MCMC). As iteration increases we increase the MCMC sample size and decrease the step-size of the parameter update. This approach guarantees convergence of the estimates to the expected fixed points. We evaluate our methods through a simulation study.     

Approximation by Piecewise Constant Functions on a Square

In this paper we consider several algorithms for approximating functions defined on the unit square I = [0,1]² and ranging in $\mathbb{R}^2 $ . We use functions of zeroth-order Lagrange spline type as the approximation apparatus. They differ from the standard Lagrange splines on the plane by the rule for choosing grid lines according to which the spline is constructed; namely, a set of one-dimensional splines is used instead of a family of parallel lines determining the interpolation nodes.     

Uncertainty constants and quasispline wavelets

In 1996 Chui and Wang proved that the uncertainty constants of scaling and wavelet functions tend to infinity as smoothness of the wavelets grows for a broad class of wavelets such as Daubechies wavelets and spline wavelets. We construct a class of new families of wavelets (quasispline wavelets) whose uncertainty constants tend to those of the Meyer wavelet function used in construction.     

Spline fitting discontinuous functions given just a few Fourier coefficients

This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL ₂ approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.     

Uniform bounds for cardinal Hermite spline operators with double knots

A method is given for computing the uniform norm of the cardinal Hermite spline operator. This is the operator that takes two bounded biinfinite sequences of numbers into the unique bounded spline of degree 2k − 1(k 2) with knots of multiplicity two at the integers and that interpolates the two given sequences for both functional and first derivative values at the integers. The computational schema relies on knowledge of the Bernoulli splines, while the theoretical aspects make use of some properties of zeros of periodic splines.     

Uncertainty products of local periodic wavelets

This paper is on the angle–frequency localization of periodic scaling functions and wavelets. It is shown that the uncertainty products of uniformly local, uniformly regular and uniformly stable scaling functions and wavelets are uniformly bounded from above by a constant. Results for the construction of such scaling functions and wavelets are also obtained. As an illustration, scaling functions and wavelets associated with a family of generalized periodic splines are studied. This family is generated by periodic weighted convolutions, and it includes the well‐known periodic B‐splines and trigonometric B‐splines. This revised version was published online in June 2006 with corrections to the Cover Date.