Monotone and Convex C 1 Hermite Interpolants Generated by a Subdivision Scheme

   Abstract. We propose C ¹ Hermite interpolants generated by the general subdivision scheme introduced by Merrien [17] and satisfying monotonicity or convexity constraints. For arbitrary values and slopes of a given function f at the end-points of a bounded interval, which are compatible with the contraints, the given algorithms construct shape-preserving interpolants. Moreover, these algorithms are quite simple and fast as well as adapted to CAGD. We also give error estimates in the case of interpolation of smooth functions.     

Convergent Vector and Hermite Subdivision Schemes

Hermite subdivision schemes have been studied by Merrien, Dyn, and Levin and they appear to be very different from subdivision schemes analyzed before since the rules depend on the subdivision level. As suggested by Dyn and Levin, it is possible to transform the initial scheme into a uniform stationary vector subdivision scheme which can be handled more easily.With this transformation, the study of convergence of Hermite subdivision schemes is reduced to that of vector stationary subdivision schemes. We propose a first criterion for C⁰-convergence for a large class of vector subdivision schemes. This gives a criterion for C¹-convergence of Hermite subdivision schemes. It can be noticed that these schemes do not have to be interpolatory. We conclude by investigating spectral properties of Hermite schemes and other necessary/sufficient conditions of convergence.     

Hermite Subdivision Schemes and Taylor Polynomials

We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is C ⁰ and ℋ is C ^d . We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.      

C1 monotone cubic Hermite interpolant

Constraining an interpolation to be shape preserving is a well established technique for modelling scientific data. Many techniques express the constraint variables in terms of abstract quantities that are difficult to relate to either physical values or the geometric properties of the interpolant. In this paper, we construct a piecewise monotonic interpolant where the degrees of freedom are expressed in terms of the weights of the rational Bézier cubic interpolant.     

Maximal Monotone Operators, Convex Functions and a Special Family of Enlargements

This work establishes new connections between maximal monotone operators and convex functions. Associated to each maximal monotone operator, there is a family of convex functions, each of which characterizes the operator. The basic tool in our analysis is a family of enlargements, recently introduced by Svaiter. This family of convex functions is in a one-to-one relation with a subfamily of these enlargements. We study the family of convex functions, and determine its extremal elements. An operator closely related to the Legendre–Fenchel conjugacy is introduced and we prove that this family of convex functions is invariant under this operator. The particular case in which the operator is a subdifferential of a convex function is discussed.     

一种细分格式生成极限曲面的法向量的求法

在Levin给出的三角域上生成极限曲面的法向量求法基础上,给了同拟蝴蝶形细分在矩形域上生成极限曲面的情况,并得到了两个自由度,可以对法向量进行优化选取,这对讨论曲面的等距面有广泛的实际意义。     

The Sum of a Maximally Monotone Linear Relation and the Subdifferential of a Proper Lower Semicontinuous Convex Function is Maximally Monotone

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A+?fA+\partial f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying \operatornamedom A?\operatornameint\operatornamedom ?f 1 \varnothing\operatorname{dom} A\cap\operatorname{int}\operatorname{dom} \partial f\neq\varnothing. Moreover, A+?fA+\partial f is of type (FPV). The maximal monotonicity of A+?fA+\partial f when \operatornameint\operatornamedom A?\operatornamedom ?f 1 \varnothing{\operatorname{int}\operatorname{dom}}\, A\cap\operatorname{dom} \partial f\neq\varnothing follows from a result by Verona and Verona, which the present work complements.     

Monotone Operators Representable by l.s.c. Convex Functions

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.     

The Mean Perimeter of Some Random Plane Convex Sets Generated by a Brownian Motion

If C ₁ is the convex hull of the curve of a standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of C ₁ and several rotations of it and compute the mean of the length of their perimeter by elementary calculations. This can be seen geometrically as a study of the exit time by a Brownian motion from certain polytopes having the unit circle as an inscribed one.     

On Bregman-Type Distances for Convex Functions and Maximally Monotone Operators

Given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.     

Locally Convex Spaces of the Sebastian-e-Silva Type Generated by the Operator of Differentiation

It is established in the paper that the spaces of entire functions of exponential type generated by the operator of differentiation defined in the space of summable functions on the axis belong to the well-known Sebastian-e-Silva class of locally convex spaces. Some properties of the spaces relating to the duality and the open map theorem, as well as properties of convolution, are described.     

Rapid Growth Paths in Multivalued Dynamical Systems Generated by Homogeneous Convex Stochastic Operators

The paper examines dynamical systems generated by convex homogeneous multivalued operators in spaces of random vectors. The primary goal is to investigate the growth rates of random trajectories of these dynamical systems. Existence and characterization theorems for rapid trajectories, growing faster in a certain sense than others, are obtained.     

On the Universal C*-Algebra Generated by a Partial Isometry

A universal C*-algebra is constructed which is generated by a partial isometry. Using grading on this algebra we construct an analog of Cuntz algebras which gives a homotopical interpretation of KK-groups. It is proved that this algebra is homotopy equivalent up to stabilization by 2×2 matrices to M ₂(C). Therefore those algebras are KK-isomorphic.     

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

A nonlinear multiresolution scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.     

Convex Potentials and Optimal Shift Generated Oblique Duals in Shift Invariant Spaces

We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in \(L^2(\mathbb {R}^k)\). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.     

Shape-Preserving Widths of Weighted Sobolev-Type Classes of Positive, Monotone, and Convex Functions on a Finite Interval

   Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ ^s ₊ L _q the subset of all functions y∈ L _q such that the s -difference Δ ^s _τ y(t) is nonnegative on I ,

Monotone interpolation of order 3 by C2 cubic splines

We propose a local solution for the problem of interpolatingmonotone data by monotone C² cubic splines with two additionalknots per interval, on an arbitrary partition, and with an approximationof order 3. Our paper extends recent works by Archer & LeGruyer and Pruess.