Equivalence of a -functional with the approximation behavior of some linear means for abstract Fourier series

Within the setting of abstract Cesàro-bounded Fourier series a -functional is introduced and characterized by the convergence behavior of some linear means. Applications are given within the framework of Jacobi, Laguerre and Hermite expansions. In particular, Ditzian's (1996) equivalence result in the setting of Legendre expansions is covered.

     

Unicity in piecewise polynomial -approximation via an algorithm

Our main result shows that certain generalized convex functions on a real interval possess a unique best approximation from the family of piecewise polynomial functions of fixed degree with varying knots. This result was anticipated by Kioustelidis in [11]; however the proof given there is nonconstructive and uses topological degree as the primary tool, in a fashion similar to the proof the comparable result for the case in [5]. By contrast, the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors. The proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness. This algorithm was suggested by Kioustelidis [10]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the case.

     

Common operator properties of the linear operators

Let and be bounded linear operators defined on Banach spaces, , . When , then the operators and have many basic operator properties in common. This situation is studied in this paper.

     

On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

On linear series on general -gonal projective curves

Let be a general -gonal curve of genus . Here we prove a strong upper bound for the dimension of linear series on , i.e. we prove that .

     

WKB-type approximations for second-order differential equations in -algebras

Rigorous asymptotic approximations of the WKB (or Liouville-Green) type are obtained for a basis of solutions to in the framework of -algebras. Both cases and are included, thus generalizing the classical theory for scalar equations developed by F.W.J. Olver to matrix as well as to infinite-dimensional equations.

     

Compact operators and the geometric structure of -algebras

Given a -algebra and an element , we give necessary and sufficient geometric conditions equivalent to the existence of a representation of so that is a compact or a finite-rank operator. The implications of these criteria on the geometric structure of -algebras are also discussed.

     

A model for invertible composition operators on

A model is obtained for invertible hyperbolic and parabolic composition operators on . This model shows that the adjoints of these composition operators are similar to block Toeplitz matrices constructed with weighted bilateral shifts and rank one operators.

     

Finite and -resolvability

A topological space is -resolvable if has disjoint dense subsets. In this paper, we prove that if is -resolvable for each positive integer , then is -resolvable.

     

Dunford-Pettis composition operators on in several variables

A bounded composition operator on , where is the unit ball in , is Dunford-Pettis if and only if the radial limit function of takes values on the unit sphere only on a set of surface measure zero. A similar theorem holds on bounded strongly pseudoconvex domains with smooth boundary.

     

The length of -pseudodifferential operators

We compute the length of the -algebra generated by the algebra of b-pseudodifferential operators of order on compact manifolds with corners.

     

A critical metric for the

The -norm of the curvature tensor

defines a Riemannian functional on the space of metrics. This work exhibits a metric on which is of Berger type but not of constant ricci curvature, and yet is critical for .

     

On the perturbation theory of -accretive operators in Banach spaces

Let be a real Banach space. Let be -accretive with compact. Let be such that is condensing for some Let and assume that there exists a bounded open set and such that is bounded and

for all Then A basic homotopy result of the degree theory for with condensing and possibly unbounded, is used to improve and/or extend recent results by Hirano and Kalinde.

     

convergence of Picard's successive approximations

A simple, elementary proof of the existence, uniqueness, and
smoothness of solutions to ordinary differential equations is given. In fact, it is shown that for a differential equation of class , the successive approximations of Picard converge in the -sense.

     

A -dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients

A -dimensional version is given of a -dimensional result due to C. Huneke. His result produced a formula relating the length to the difference , where is primary for the maximal ideal of a -dimensional Cohen-Macaulay local ring , is a minimal reduction of , , and is the Hilbert-Samuel polynomial of . We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of . We also include a characterization, in terms of the Hilbert coefficients of , of the condition .

     

The Toda molecule equation and the -algorithm

One of the well-known convergence acceleration methods, the -algorithm is investigated from the viewpoint of the Toda molecule equation. It is shown that the error caused by the algorithm is evaluated by means of solutions for the equation. The acceleration algorithm based on the discrete Toda molecule equation is also presented.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

A group of paths in

We define a group structure on the set of compact ``minimal' paths in . We classify all finitely generated subgroups of this group : they are free products of free abelian groups and surface groups. Moreover, each such group occurs in . The subgroups of isomorphic to surface groups arise from certain topological -forms on the corresponding surfaces. We construct examples of such -forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using we construct a non-polygonal tiling problem in , that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group has applications to combinatorial tiling problems of the type: given a set of tiles and a region , can be tiled by translated copies of tiles in ?

     

Characterizations of contraction -semigroups

A -semigroup is of contractions if for , . Using the Hille-Yosida space, we completely characterize the generators of contraction -semigroups. We also give the Lumer-Phillips theorem for -semigroups in several special cases.