A new approach to interpolating scaling functions

We present a new method to construct interpolating refinable functions in higher dimensions. The approach is based on the solutions to specific Lagrange interpolation problems by polynomials and applies to a large class of scaling matrices. The resulting scaling functions automatically satisfy certain Strang-Fix conditions. Several examples are discussed.     

Frame scaling function sets and frame wavelet sets in

In this paper, we classify frame wavelet sets and frame scaling function sets in higher dimensions. Firstly, we obtain a necessary condition for a set to be the frame wavelet sets. Then, we present a necessary and sufficient condition for a set to be a frame scaling function set. We give a property of frame scaling function sets, too. Some corresponding examples are given to prove our theory in each section.     

对称尺度函数的参数化设计

In this paper, the factorization for filters with length Km of scaling functions intosimple blocks is considered.     

Critical sets of functions

Summary With the introduction of an alternate definition for critical point, this paper studies critical sets, as defined byW. M. Whyburn, in terms of certain related domains. Critical sets are divided into four classes. Type0 has the limit point property with respect to critical sets which are not type0; type1 and2 critical sets compare, respectively, to classical minimum and maximum points; type3 includes themin-max and flex type. This paper is a result of a study of critical sets made as a dissertation problem under the direction ofW. M. Whyburn, to whom the author is indebted for many helpful suggestions.     

From finite to linear elastic fracture mechanics by scaling

In the setting of finite elasticity we study the asymptotic behaviour of a crack that propagates quasi-statically in a brittle material. With a natural scaling of size and boundary conditions we prove that for large domains the evolution with finite elasticity converges to the evolution with linearized elasticity. In the proof the crucial step is the (locally uniform) convergence of the non-linear to the linear energy release rate, which follows from the combination of several ingredients: the \(\Gamma \) -convergence of re-scaled energies, the strong convergence of minimizers, the Euler–Lagrange equation for non-linear elasticity and the volume integral representation of the energy release.     

From best constants to critical functions

The study of sharp Sobolev inequalities starts with the notion of best constant and leads naturally to the question to know whether or not there exist extremal functions for these inequalities. We restrict ourselves in this paper to the -Sobolev inequality. Then, we extend the notion of best constant to that of critical function, and, with the help of this notion, we answer the question to know whether or not there exist extremal functions for the sharp -Sobolev inequality. Partial answers to the more general question to know whether or not an extremal function always comes with a critical function are also given. Received November 9, 1999; in final form February 21, 2000 / Published online March 12, 2001     

Sub-topical functions and plus-co-radiant sets

In this paper, we study sub-topical functions in the framework of abstract convexity and examine the relevant properties such as support sets, polar sets and sub-differentials for these functions. Plus-radiant and plus-co-radiant sets, and their relations with sub-topical functions are studied. Applying sub-topical functions, we present some separation theorems for both plus-radiant and plus-co-radiant sets.     

On scaling laws for production functions

Summary Solutions of functional equations are used in this paper to develop laws for scaling output under proportional changes in input vectors leading to special classes of production functions, which are of significance for the question of returns to scale.I am grateful to ProfessorShephard with whom I have the great pleasure to work while writing this paper.     

The nonexistence of shearlet scaling functions

Over the past five years, the directional representation system of shearlets has received much attention and has been shown to exhibit many advantageous properties. Over this time period, there have been a number of attempts to associate shearlet systems with a multiresolution analysis (MRA). However, one can argue that, in each of these attempts, the following statement regarding the resulting shearlet MRA notion is inaccurate: “There exist scaling functions satisfying various desirable properties, such as significant amounts of decay or regularity, nonnegativity, or advantageous refinement or representation conditions. Each such scaling function naturally induces an associated shearlet (either traditional or cone-adapted) that satisfies similar desirable properties. Each such scaling function/associated shearlet pair rationally induces a fast decomposition algorithm for discrete data.” In this article, we attempt to provide explanation for this situation by arguing the great difficulty of associating shearlet systems with such an MRA. We do so by considering two very natural and general notions of shearlet MRA—one which leads to traditional shearlets and one which leads to cone-adapted shearlets—each of which seems to be an excellent candidate to satisfy the above quoted statement. For each of these notions, we prove the nonexistence of associated scaling functions satisfying the above mentioned desirable properties.     

Irregular scaling functions with orthogonal translations

For all integers m,k>1 with m≡1modk we construct, among others, a function with dense graph in the set [0,k]×R such that

     

Asymptotic regularity of Daubechies' scaling functions

Let , , be Daubechies' scaling function with symbol , and let , be the corresponding Sobolev exponent. In this paper, we make a sharp estimation of , and we prove that there exists a constant independent of such that

This answers a question of Cohen and Daubeschies ( Rev. Mat. Iberoamericana, 12(1996), 527-591) positively.

     

Meromorphic functions sharing two sets

In the paper we discuss the uniqueness problem for meromorphic functions that share two sets and prove five theorems which improve and supplement some results earlier given by Yi and Yang [13], Lahiri and Banerjee [5].     

Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.