Supporting Functions and The Differentiabilities of the Norms on Banach Spaces

ＳｕｐｐｏｒｔｉｎｇＦｕｎｃｔｉｏｎｓａｎｄＴｈｅＤｉｆｆｅｒｅｎｔｉａｂｉｌｉｔｉｅｓｏｆｔｈｅＮｏｒｍｓｏｎＢａｎａｃｈＳｐａｃｅｓ￥ＷａｎｇＪｉａｎｇｅｎ（王建根）（ＬｕｏｙａｎｇＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓ...     

Wavelets in Banach Spaces

We describe a construction of wavelets (coherent states) in Banach spaces generated by admissible group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example, we consider operator-valued Segal–Bargmann-type spaces and the Weyl functional calculus.     

Norms of Powers in Banach Algebras

By showing the existence of certain functions in A⁺ (the algebraof analytic functions in the unit disc with absolutely convergentTaylor series), we prove that if T is a power bounded operatoron a Banach space X, and x X satisfies Tx x, then      

Flat Chains in Banach Spaces

We extend the notion of a flat chain with coefficients in a normed abelian group from Euclidean space to an arbitrary Banach space and prove a compactness result. We also remove the condition that a flat chain with arbitrary coefficients have finite mass in order for its support to exist. This research was part of the author’s Ph. D. dissertation at Stanford University.     

Constrained Approximation in Banach Spaces

   Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

Banach空间的K—M逼近

设Ｘ是自反、严格凸且具有Ｈ性质的Ｂａｎａｃｈ空间，Ａ是Ｘ的弱序列完备子集。本文证明了Ａ是可逼近紧的Ｃｈｅｂｙｓｈｅｖ集的充分必要条件是Ａ是太阳集；同时，也讨论了Ｏｒｌｉｃｚ序列空间的太阳集。     

Large-Deviation Probabilities in Banach Spaces

Theorems concerning exact asymptotics of large-deviation probabilities of sums of independent random elements in a Banach space are proved. We consider probabilities of the following events: sums of independed elements belong to balls such that their centers deviate from the origin as the sample size increases. The results are a version of Bentkus' and Rachkauskas' theorems proved for the exteriors of balls centered at the origin. Bibliography: 11 titles.     

p-Frames in Separable Banach Spaces

Let X be a separable Banach space with dual X ^*. A countable family of elements {g _i }X ^* is a p-frame (1 p ) if the norm _X is equivalent to the ^p -norm of the sequence {g _i ()}. Without further assumptions, we prove that a p-frame allows every gX ^* to be represented as an unconditionally convergent series g=d _i g _i for coefficients {d _i } ^q , where 1/p+1/q=1. A p-frame {g _i } is not necessarily linear independent, so {g _i } is some kind of overcomplete basis for X ^*. We prove that a q-Riesz basis for X ^* is a p-frame for X and that the associated coefficient functionals {f _i } constitutes a p-Riesz basis allowing us to expand every fX (respectively gX ^*) as f=g _i (f)f _i (respectively g=g(f _i )g _i ). In the general case of a p-frame such expansions are only possible under extra assumptions.     

On Decoupling in Banach Spaces

Journal of Theoretical Probability - We consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional...     

Flat Forms in Banach Spaces

We define a flat partial differential form in a Banach space and show that the space of these forms is isometrically the dual space of the space of flat chains as defined by T. Adams.     

Distance Types in Banach Spaces

We introduce and study the notion of a distance type, on a Banach space, defined by a nested sequence of convex sets. Among other things, we show that there always exist distance types that are not types in the classical sense. Then, we recover the notion of the flat nested sequence of Milman and Milman and show that distance types defined by flat nested sequences coincide with the bidual types of Farmaki. These results are applied to show that a flat nested sequence of convex sets is Wijsman convergent to the intersection of their weak*-closures in bidual space.     

Coefficient Quantization in Banach Spaces

Let (e _i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a ‘finite alphabet’. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c ₀. We also show that, for every ε>0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x _i ) such that the additive group generated by (x _i ) is (3+ε)⁻¹-separated and 1/3-dense in X.      

Constrained Approximation in Banach Spaces

Abstract. We consider the problem of approximating vectors from a complemented subspace Z ₊ of a Banach space X by the projections onto Z ₊ of vectors from a subspace Y ₊ with a norm constraint on their projections onto the complementary subspace. Sufficient conditions are found for the existence of a unique best approximant and a characterization via a critical point equation is provided, thus extending known results on Hilbert spaces. These results are then applied in the case that X is L ^p (T), where T denotes the unit circle, Z ₊ consists of functions supported on a subset of the circle, and Y ₊ is the corresponding Hardy space.     

K—very Smooth Banach Spaces

We introduce the notion of K-very smoothness which is a generalization of very smoothness in Banach spaces. A necessary and sufficient condition for a Banach space to be K-very smooth is obtained. We also consider some relations between K-very smoothness and other geometrical notions.     

Very Convex Banach Spaces

ＶｅｒｙＣｏｎｖｅｘＢａｎａｃｈＳｐａｃｅｓＴｅｇｕｓｉ（特古斯）Ｓｕｙａｌａｔｕ（苏雅拉图）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＩｎｎｅｒＭｏｎｇｏｌｉａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｈｕｈｈｏｔ，０１００２２）ＬｉＹｏｎｇｊｉｎ...     

On asymptotic models in Banach Spaces

A well known application of Ramsey’s Theorem to Banach Space Theory is the notion of a spreading model of a normalized basic sequence (x _i) in a Banach spaceX. We show how to generalize the construction to define a new creature (e _i), which we call an asymptotic model ofX. Every spreading model ofX is an asymptotic model ofX and in most settings, such as ifX is reflexive, every normalized block basis of an asymptotic model is itself an asymptotic model. We also show how to use the Hindman-Milliken Theorem—a strengthened form of Ramsey’s Theorem—to generate asymptotic models with a stronger form of convergence. Research supported by NSF.     

Biholomorphic Functions in Dual Banach Spaces

We present an infinite-dimensional version of Cartan-Carathéodory-Kaup-Wu theorem about the analyticity of the inverse of a given analytic mapping. It is valid for a class of domains in separable Banach dual spaces that includes all bounded convex domains.