Minimal splines of Lagrange type

We obtain sufficient conditions for the existence, smoothness, and embedding of spline spaces, construct a biorthogonal system of functionals, find calibration relations, and study the asymptotic behavior of minimal first and second order splines of Lagrange type. Bibliography: 9 titles.     

A general framework of compactly supported splines and wavelets

Let {V_k} be a nested sequence of closed subspaces that constitute a multiresolution analysis of L²( ). We characterize the family Φ = {φ} where each φ generates this multiresolution analysis such that the two-scale relation of φ is governed by a finite sequence. In particular, we identify the ε Φ that has minimum support. We also characterize the collection Ψ of functions η such that each η generates the orthogonal complementary subspaces W_k of V_k, . In particular, the minimally supported ψ ε Ψ is determined. Hence, the “B-spline” and “B-wavelet” pair (, ψ) provides the most economical and computational efficient “spline” representations and “wavelet” decompositions of L² functions from the “spline” spaces V_k and “wavelet” spaces W_k, k . A very general duality principle, which yields the dual bases of both {(·−j):j and {η(·−j):j } for any η ε Ψ by essentially interchanging the pair of two-scale sequences with the pair of decomposition sequences, is also established. For many filtering applications, it is very important to select a multiresolution for which both and ψ have linear phases. Hence, “non-symmetric” and ψ, such as the compactly supported orthogonal ones introduced by Daubechies, are sometimes undesirable for these applications. Conditions on linear-phase φ and ψ are established in this paper. In particular, even-order polynomial B-splines and B-wavelets φ_m and ψ_m have linear phases, but the odd-order B-wavelet only has generalized linear phases.     

A multiresolution analysis for tensor-product splines using weighted spline wavelets

We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing-as a model problem-image compression with black-and-white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.     

Some recurrence formulas for box splines and cone splines

A degree elevation formula for multivariate simplex splines was given by Micchelli [6] and extended to hold for multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box_splines. To this end, we utilize a relation due to Dahmen and Micchelli [4] that connects box splines and cone splines and a degree reduction formula given by Cohen, Lyche, and Riesenfeld in [2].     

SOME RECURRENCE FORMULAS FOR BOX SPLINES AND CONE SPLINES

A degree elevation formula for multivariate simplex splines was given by Micchellis[6] and extended to hold for multivariate Dirichlet splines in [8].We report similar formulae for multivariate cone splines and box splines.To this and ,we utilize a relation due to Dahmen and Micchelli[4] that connects box splines and cone splines and a degree reduction formula given by Cohen,Lyche,and Riesenfeld in [2].     

Some recurrence formulas for box splines and cone splines

A degree elevation formula for multivariate simplex splines was given by Micchelli and extended to hold ]or multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box splsplines andines. To this end, we utilize a relation due to Dahmen and Micchelli that connects box cone splines and a degree reduction formulagiven by Cohen, Lyche, and Riesenfeld in [2].     

Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of order $k\geqslant 2$ spanned by $\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}$ on each subinterval $[x_i,x_{i+1}\rangle\subset [0,1]$ , i?=?0,1, ...l. Most of the paper deals with non-polynomial case m _i ,n _i ?∈?[4,?∞?), and polynomial splines known as VDP–splines are the special case when m _i , n _i are integers. We describe VDP–splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functions p(x), q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator $\frac{d}{dx} p \frac{d}{dx} q \frac{d^{k-2}} {dx^{k-2}}$ . Although the space of splines is not based on an Extended Chebyshev system, we argue that total positivity and variation diminishing still holds. Unlike the abstract results, constructive properties, like Marsden identity, recurrences for quasi-Bernstein polynomials and knot insertion algorithms may be more involved and we prove them only for VDP splines of orders 4 and 5.     

Polynomial splines as examples of Chebyshevian splines

We consider geometrically continuous polynomial splines defined on a given knot-vector by lower triangular connection matrices with positive diagonals. In order to find out which connection matrices make them suitable for design, we regard them as examples of geometrically continuous piecewise Chebyshevian splines. Indeed, in this larger context we recently achieved a simple characterisation of all suitable splines for design. Applying it to our initial polynomial splines will require us to treat polynomial spaces on given closed bounded intervals as instances of Extended Chebyshev spaces, so as to determine all possible systems of generalised derivatives which can be associated with them.     

Histopolating splines

Given an integrable function f, we are concerned with the construction of a spline H_n(f) of degree n with prescribed knots that satisfies the histopolation conditions

for some fixed . Additionally, the resulting spline operator should be local and reproduce all polynomials of degree n. Our approach of generating such a histospline is based on a local spline interpolation operator that is exact for all polynomials of degree n.     

Gibbs-Wilbraham splines

If a function with a jump discontinuity is approximated in the norm ofL ²[–1,1] by a periodic spline of orderk with equidistant knots, a behavior analogous to the Gibbs-Wilbraham phenomenon for Fourier series occurs. A set of cardinal splines which play the role of the sine integral function of the classical phenomenon is introduced. It is then shown that ask becomes large, the phenomenon for splines approaches the classical phenomenon.Communicated by Ronald A. DeVore.     

Inf-convolution splines

Inf-convolution splines have been introduced for the representation of functions presenting singularities (discontinuities, peaks, etc). In these cases, the energy functional to be minimized is the inf-convolution of a semi-Hilbertian function and of the indicator function of a linear subspace containing the singularities. This principle is extended here to interpolating or smoothing splines based on the inf-convolution of a finite number of arbitrary semi-Hilbertian functions. Characterization theorems are given using the notion of semikernel. Several algorithms are proposed.Communicated by Larry L. Schumaker.     

Multivariate splines and polytopes

In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion’s formula for the volume of polytopes. We also prove that the integration of polynomials over polytopes can be dealt with by using the multivariate truncated power. Moreover, we show that the volume of cube slicing can be considered as the maximum value of the box spline. On the basis of this connection, we give a simple proof for Good’s conjecture, which has been settled before by probability methods.     

Band-Limited Wavelets

Using the harmonic method,we get a class of more general band-limited wavelet,which was got by complicate operator interpolation method before.Our result is alittle better than the result by operator interpolation method.The fast band-limitedwavelet transform shall be given in another paper.     

多频率小波

通过方向多分辨分析把由一个函数生成的多频率小波推广到由有限个函数生成的多频率小波，给出由函数φ1,…，φn，…ψn(2^j1 ^j2-1)的平移生成Vj(1)空间的Riesz基的充分必要条件，同时给出该小波的分解式。     

Helix splines as an example of affine Tchebycheffian splines

The present paper summarizes the theory of affine Tchebycheffian splines and presents an interesting affine Tchebycheffian free-form scheme, the “helix scheme”. The curve scheme provides exact representations of straight lines, circles and helix curves in an arc length parameterization. The corresponding tensor product surfaces contain helicoidal surfaces, surfaces of revolution and patches on all types of quadrics. We also show an application to the construction of planarC ² motions interpolating a given set of positions. Because the spline curve segments are calculated using a subdivision algorithm, many algorithms, which are of fundamental importance in the B-spline technique, can be applied to helix splines as well. This paper should demonstrate how to create an affine free-form scheme fitting to certain special applications.     

Mahler's conjecture and wavelets

It is shown that a special case of Mahler's conjecture can be reformulated in terms of the solutions to the scaling equation of wavelet theory.