On interpolation families of wavelet sets

The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets.

     

Anisotropic interpolation error estimates via orthogonal expansions

We prove anisotropic interpolation error estimates for quadrilateral and hexahedral elements with all possible shape function spaces, which cover the intermediate families, tensor product families and serendipity families. Moreover, we show that the anisotropic interpolation error estimates hold for derivatives of any order. This goal is accomplished by investigating an interpolation defined via orthogonal expansions.     

Interpolation of several multiparametric approximation spaces

We prove a general interpolation theorem for linear operators acting simultaneously in several approximation spaces which are defined by multiparametric approximation families. As a consequence, we obtain interpolation results for finite families of Besov spaces of various types including those determined by a given set of mixed differences.     

Infinite Dimensional Geometric Properties of Real Interpolation Spaces

Estimates for the moduli of noncompact convexity of l_p-sums and real interpolation spaces for finite families of spaces are given. It is proved that such an interpolation preserves nearly uniform convexity and property (β).     

Extensions of the Minimal Logic and the Interpolation Problem

Under study is the interpolation problem over Johansson’s minimal logic J. We give a detailed exposition of the current state of this difficult problem, establish Craig’s interpolation property for several extensions of J, prove the absence of CIP in some families of extensions of J, and survey the results on interpolation over J. Also, the relationship is discussed between the interpolation properties and the recognizability of logics.     

BCn-symmetric polynomials

We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.     

Real and complex interpolation methods for finite and infinite families of Banach spaces

A comparative study is made of the various interpolation spaces generated with respect to n-tuples or infinite families of compatible Banach spaces by real and complex interpolation methods due to Sparr, Favini-Lions, Coifman-Cwikel-Rochberg-Sagher-Weiss, and Fernandez. Certain inclusions are established between these spaces and examples are given showing that in general they do not coincide. It is also shown that, in contrast to the case of couples of spaces, the spaces generated by the above methods may depend on the structure of the containing space in which the Banach spaces of the n-tuple (nϵ 3) or infinite family are embedded. Finally a construction is given which enables the spaces of Sparr and Favini-Lions, hitherto defined only with respect to n-tuples, to also be defined with respect to infinite families of Banach spaces.     

James constant for interpolation spaces

Estimates for the James constant for various norms in real interpolation spaces for finite families of Banach spaces are given. As a corollary it is shown that if a family contains at least one space which is uniformly nonsquare, then the interpolation space is uniformly nonsquare.     

Multilinear forms and measures of dependence between random variables

Based on an idea of Rosenblatt, the methods of interpolation theory are used to establish moment inequalities and equivalence relations for measures of dependence between two or more families of random variables. A couple of “interpolation” theorems proved here appear to be new.     

A theory of complex interpolation for families of Banach spaces

A detailed development is given of a theory of complex interpolation for families of Banach spaces which extends the well-known theory for pairs of spaces.     

Type,cotype and twisted sums induced by complex interpolation

This paper deals with extensions or twisted sums of Banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension. The results are presented in the context of interpolation of families of Banach spaces, and are applied to the study of submodules of Schatten classes. We also obtain nontrivial extensions of spaces without the CAP which also fail the CAP.     

Families of discrete advection-reaction operators investigated via divided differences

An infinite matrix formulation of the families of discrete advection-reaction operators is given in order to investigate their relevance to interpolation theory. A basic characteristic under study is the connection of each iteration of the operators to a series of interpolation problems for the canonical polynomial base for selected initial conditions. In order to generalize our results, we extend the definition of advection-reaction operators to sequences of polynomials.     

On p-adic families of Hilbert cusp forms of finite slope

Let p be a prime number and F a totally real field. In this article, we obtain a p-adic interpolation of spaces of totally definite quaternionic automorphic forms over F of finite slope, and construct p-adic families of automorphic forms parametrized by affinoid Hecke varieties. Further, as an application to the case where [F:Q] is even, we obtain p-adic analytic families of Hilbert eigenforms having fixed finite slope parametrized by weights. This is an analogue of Coleman's analytic families in [R.F. Coleman, p-Adic Banach spaces and families of modular forms, Invent. Math. 127 (1997) 417-479].     

On the generation of symmetric Lebesgue-like points in the triangle

We compute point sets on the triangle that have low Lebesgue constant, with sixfold symmetries and Gauss–Legendre–Lobatto distribution on the sides, up to interpolation degree 18. Such points have the best Lebesgue constants among the families of symmetric points used so far in the framework of triangular spectral elements.     

Real Interpolation for Families of Banach Spaces and Convexity

We show how the geometrical properties of uniform convexity and uniformly non-?? are inherited by real interpolation spaces for infinite families.     

Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.     

Mapped Finite Elements on Hexahedra. Necessary and Sufficient Conditions for Optimal Interpolation Errors

This paper considers finite elements which are defined on hexahedral cells via a reference transformation which is in general trilinear. For affine reference mappings, the necessary and sufficient condition for an interpolation order O(h ^k+1) in the L ²-norm and O(h ^k ) in the H ¹-norm is that the finite dimensional function space on the reference cell contains all polynomials of degree less than or equal to k. The situation changes in the case of a general trilinear reference transformation. We will show that on general meshes the necessary and sufficient condition for an optimal order for the interpolation error is that the space of polynomials of degree less than or equal to k in each variable separately is contained in the function space on the reference cell. Furthermore, we will show that this condition can be weakened on special families of meshes. These families which are obtained by applying usual refinement techniques can be characterized by the asymptotic behaviour of the semi-norms of the reference mapping.     

Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

We find necessary density conditions for Marcinkiewicz–Zygmund inequalities and interpolation for spaces of spherical harmonics in with respect to the L^p norm. Moreover, we prove that there are no complete interpolation families for p≠2.     

The hypoelliptic Laplacian on the cotangent bundle

In this paper, we construct a new version of Hodge theory, where the corresponding Laplacian acts on the total space of the cotangent bundle. This Laplacian is a hypoelliptic operator, which is in general non-self-adjoint. When properly interpreted, it provides an interpolation between classical Hodge theory and the generator of the geodesic flow. The construction is also done in families in the superconnection formalism of Quillen and extends earlier work by Lott and the author.