Theory Of Function Spaces Research Articles

The Role of Over-Parameterization in Machine Learning – the Good, the Bad, the Ugly

The conventional wisdom of simple models in machine learning misses the bigger picture, especially over-parameterized neural networks (NNs), where the number of parameters are much larger than the number of training data. Our goal is to explore the mystery behind over-parameterized models from a theoretical side. In this talk, I will discuss the role of over-parameterization in neural networks, to theoretically understand why they can perform well. First, I will discuss the role of over-parameterization in neural networks from the perspective of models, to theoretically understand why they can genralize well. Second, the effects of over-parameterization in robustness, privacy are discussed. Third, I will talk about the over-parameterization from kernel methods to neural networks in a function space theory view. Besides, from classical statistical learning to sequential decision making, I will talk about the benefits of over-parameterization on how deep reinforcement learning works well for function approximation. Potential future directions on theory of over-parameterization ML will also be discussed.

On Bergman type projections in bounded strongly pseudoconvex domains

In our note we prove the boundedness of Bergman type projections in two different spaces of analytic functions with mixed norm in general bounded strongly pseudoconvex domains with smooth boundary.The first class of analytic functions was studied previously by many authors,the second function space hovewer is completely new. Our proofs are based on standard known estimates of function space theory in bounded strongly pseudoconvex domains with smooth boundary and on some known estimates of Bergman kernel in such type domains. These estimates are also well- known in the unit disk.This allows us to provide proofs first in one dimensional case which are simpler and then repeating same arguments to show same type results also in more general situation. Note that many results on boundedness of Bergmam type projections are well known and they have also various nice applications in complex function theory in one or several complex variable. Our results may also have various applications in complex function theory in bounded strongly pseudoconvex domains with smooth boundary.

A Pointwise Condition for the Absolute Continuity of a Function of One Variable and Its Applications

An absolutely continuous function in calculus is precisely such a function that, within the framework of Lebesgue integration, can be restored from its derivative, that is, the Newton--Leibniz theorem on the relationship between integration and differentiation is fulfilled for it. An equivalent definition is that the the sum of the moduli of the increments of the function with respect to arbitrary pair-wise disjoint intervals is less than any positive number if the sum of the lengths of the intervals is small enough. Certain sufficient conditions for absolute continuity are known, for example, the Banach--Zaretsky theorem. In this paper we prove a new sufficient condition for the absolute continuity of a function of one variable and give some of its applications to problems in the theory of function spaces. The proved condition makes it possible to significantly simplify the proof of the theorems on the pointwise description of functions of the Sobolev classes defined on Euclidean spaces and Сarnot groups.

Characterizations of predual spaces to a class of Sobolev multiplier type spaces

We characterize preduals and Köthe duals to a class of Sobolev multiplier type spaces. Our results fit in well with the modern theory of function spaces of harmonic analysis and are also applicable to nonlinear partial differential equations. We make use of several tools from nonlinear potential theory, weighted norm inequalities, and the theory of Banach function spaces to obtain our results.

Grand Morrey Type Spaces

The so called grand spaces nowadays are one of the main objects in the theory of function spaces. Grand Lebesgue spaces were introduced by T. Iwaniec and C. Sbordone in the case of sets $\Omega$ with finite measure $|\Omega|<\infty$, and by the authors in the case $|\Omega|=\infty$. The latter is based on introduction of the notion of grandizer. The idea of "grandization" was also applied in the context of Morrey spaces. In this paper we develop the idea of grandization to more general Morrey spaces $L^{p,q,w}(\mathbb{R}^n)$, known as Morrey type spaces. We introduce grand Morrey type spaces, which include mixed and partial grand versions of such spaces. The mixed grand space is defined by the norm $$ \sup_{\varepsilon,\delta} \varphi(\varepsilon,\delta)\sup_{x\in E} \left(\int\limits_{0}^{\infty}{w(r)^{q-\delta}}b(r)^{\frac{\delta}{q}} \left(\,\int\limits_{|x-y|<r}\big|f(y)\big|^{p-\varepsilon} a(y)^{\frac{\varepsilon}{p}}\,dy\right)^{\frac{q-\delta}{p-\varepsilon}} \frac{dr}{r}\right)^{\frac{1}{q-\varepsilon}} $$ with the use of two grandizers $a$ and $b$. In the case of grand spaces, partial with respect to the exponent $q$, we study the boundedness of some integral operators. The class of these operators contains, in particular, multidimensional versions of Hardy type and Hilbert operators.

L0-Convex Compactness and Random Normal Structure in L0 (F, B)

Let (B, ∥ · ∥) be a Banach space, (Ω, F, P) a probability space, and L0 (F, B) the set of equivalence classes of strong random elements (or strongly measurable functions) from (Ω, F, P) to (B, ∥ · ∥). It is well known that L0 (F, B) becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory. Let V be a closed convex subset of B and L0 (F, V) the set of equivalence classes of strong random elements from (Ω, F, P) to V. The central purpose of this article is to prove the following two results: (1) L0 (F, V) is L0-convexly compact if and only if V is weakly compact; (2) L0 (F, V) has random normal structure if V is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules, and Banach space techniques, can be applied in the other related aspects.

Formalization of functional variation in HOL Light

Functional variation is an important fundamental mathematical theory widely used in engineering analysis, especially in dynamics based design. However, its formalization still remains unexplored and this makes the verification of dynamic properties of engineering structures quite cumbersome. The present paper aims at initiating the formalization of this theory. For this purpose, the function-space theory is first formalized and then it is formally proved that this continuous function space is a Banach space. Then, linear functionals based on this function space are formalized. Further, the functional variation in the Fréchet derivative form and its main properties are formalized. Then the necessary condition of functional extrema is formally verified. Lastly, we formally verify the continuous functional and mean value theorem of functional variation for demonstrating the application of the formalized functional variation. The present formalization work can serve as a foundation for the formal verification of dynamic behavior of mechanical systems.

Commutators of Riesz potential in the vanishing generalized weighted Morrey spaces with variable exponent

Let $\Omega \subset \mathbb{R}^n$ be an unbounded open set. We consider the generalized weighted Morrey spaces $\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ and the vanishing generalized weighted Morrey spaces $V\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ with variable exponent $p(x)$ and a general function $\varphi(x,r)$ defining the Morrey-type norm. The main result of this paper are the boundedness of Riesz potential and its commutators on the spaces $\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$ and $V\mathcal{M}^{p(\cdot),\varphi}_{\omega}(\Omega)$. This result generalizes several existing results for Riesz potential and its commutators on Morrey type spaces. Especially, it gives a unified result for generalized Morrey spaces and variable Morrey spaces which currently gained a lot of attentions from researchers in theory of function spaces.

Hardy spaces for ball quasi-Banach function spaces

This article unifies the theory for Hardy spaces built on Banach lattices on R-n satisfying certain weak conditions on indicator functions of balls. The authors introduce a new family of function spaces, named the ball quasi-Banach function spaces, to define Hardy type spaces. The ones in this article extend classical Hardy spaces and include various known function spaces, for example, Hardy-Lorentz spaces, Hardy-Herz spaces, Hardy-Orlicz spaces, Hardy-Morrey spaces, Musielak-Orlicz-Hardy spaces, variable Hardy spaces and variable Hardy-Morrey spaces. Among them, Hardy-Herz spaces are shown to naturally arise in the context of any function spaces above. The example of Hardy-Morrey spaces shows that the absolute continuity of the quasi-norm is not necessary, which is used to guarantee the density of the set of functions having compact supports in Hardy spaces for ball quasi-Banach function spaces, but the decomposition result on these Hardy-type spaces never requires this absolute continuity of the quasi-norm. Moreover, via assuming that the powered Hardy-Littlewood maximal operator satisfies certain Fefferman-Stein vector-valued maximal inequality as well as it is bounded on the associate space, the atomic characterizations of Hardy type spaces are obtained. Although the results are based on the rather abstract theory of function spaces, they improve and extend the results for Orlicz spaces and Musielak-Orlicz spaces. Moreover, local Hardy type spaces and Hardy type spaces associated with operators in this setting are also studied.

Basic Sequences and Unbiased Estimation in Quasi Power Series Distributions

By using results from function space theory we give a characterization of when lacunary quasi power series sequences are basic in $C[0, 1]$. The paper discusses the links with unbiased estimable functions and the subspaces generated by the density of the lacunary quasi power series distributions. The paper also provides the rates of convergence of all the moments of the classic odds ratio estimator. This extends some known results in Bleimann, Butzer and Hahn's approximation operator.

A metrizable X with C p (X) not homeomorphic to C p (X) × C p (X)

We give an example of an infinite metrizable space $X$ such that the space $C_p(X)$, of continuous real-valued function on $X$ endowed with the pointwise topology, is not homeomorphic to its own square $C_p(X)\times C_p(X)$. The space $X$ is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A.V. Arhangel'skii.

SUBNORMALITY OF S2(a, b, c, d) AND ITS BERGER MEASURE

We introduce a 2-variable weighted shift, denoted by <TEX>$S_2$</TEX>(a, b, c, d), which arises naturally from analytic function space theory. We investigate when it is subnormal, and compute the Berger measure of it when it is subnormal. And we apply the results to investigate the relationship among 2-variable subnormal, hyponormal and 2-hyponormal weighted shifts.

Function Spaces, Approximation Theory, and Their Applications

The purpose of this special issue was to present new developments in the theory of function spaces, along with the deep interconnections with approximation theory and the applications in various fields of pure and applied mathematics. The reaction of the mathematical community was very satisfactory. We collected thirty-five submissions, covering a wide range of mathematical topics, ten of which were found to be suitable for publications in this issue. The major part of the accepted papers treats function spaces and their applications. In this respect, in the article by X Yang et al. a new class of function spaces, named “multi-βnormed spaces”, is introduced, in connection with stability properties of certain type of functional equations, while, in the paper by A. A. Bakery, sequential spaces of Orlicz type are studied and connected with the theory of summability. In the review paper by L. Angeloni and G. Vinti, the approximation theory in the space of functions with bounded variation is developed, in view of applications to signal processing. Different notions of variation are considered and several approximation theorems for families of integral or discrete type operators are given. In the more theoretical article by S. Wulede et al., a new class of Banach spaces which generalizes the class of uniformly extremely convex Banach spaces is introduced, and some characterizations of these spaces are given. Another paper by N. Khan treats the convergence of new type of double sequences, here introduced, in n-normed spaces. An interesting abstract approach to the theory of filter convergence is given in the article by A. Boccuto and X. Dimitriou, in which the links with function spaces and approximation theory are also dealt with. Other aspects of the theory of function spaces and their interconnections with calculus of variations, numerical analysis, complex variables, and stochastic processes are discussed, respectively, in the articles by T. Ma and Y. Feng, H. Wang et al., S. Wang and T. Zhan, and finally P. Duan.These four papers point out how certain methods of general approximation theory in function spaces can be employed in order to solve problems coming from a large variety of mathematical fields. We think that these contributions may represent starting points for new researches in the field of function spaces and approximation theory.

Mini-Workshop: Modern Applications of $s$-numbers and Operator Ideals

The main aim of this mini-workshop was to present and discuss some modern applications of the functional-analytic concepts of s -numbers and operator ideals in areas like Numerical Analysis, Theory of Function Spaces, Signal Processing, Approximation Theory, Probability on Banach Spaces, and Statistical Learning Theory.

Recent development of the theory of Slice regular functions

Theory of Slice regular functions is a natural generalization of complex analysis from complex numbers to quaternions. It was introduced initially by Gentili and Struppa in 2006, and has been extensively studied and has found its elegant applications to functional calculus for quaternionic linear operators, operator theory, Schur analysis and quaternionic quantum mechanics. Meanwhile, the notion of Slice regularity was also extended to functions of an octonionic variable and to the setting of Clifford algebras as well as to the setting of alternative real algebras. In this survey, we shall focus mainly on our recent results in the theory of Slice regular functions. Firstly, we establish the Borel-Caratheodory inequality and the sharp growth and distortion theorems for Slice regular extensions of normalized biholomorphic functions on the unit disc in the setting of quaternions. In addition, we study the boundary behavior of Slice regular functions and obtain the Julia lemma, the Julia-Caratheodory theorem as well as the boundary Schwarz lemma. In particular, we find that the boundary Schwarz lemma does not ensure the Slice derivative of a Slice regular self-mappings of the open unit ball B⊂H at its boundary fixed point to be necessarily a real number, in contrast to that in the complex case. Finally, we study theory of function spaces of Slice regular functions and establish the Forelli-Rudin type estimates.

Slice Lebesgue Measure of Quaternions

The slice Lebesgue measure is introduced in this article in the theory of slice regular functions, which replaces the role of Lebesgue measure in the theory of holomorphic functions. It is naturally arisen from the predual of the slice regular Lipschitz space. The slice regular Lipschitz space can be described in terms of the so called σ-distance globally, or alternatively in terms of derivative as a Bloch type space. Its predual turns out to be a new kind of Bergman spaces of slice regular functions in the unit ball of quaternions associated with the slice Lebesgue measure. Associated with the slice Lebesgue measure as well as the related surface measure, the Forelli-Rudin type estimate can be established so that the theory of holomorphic function spaces could be generalized in parallel to the slice regular setting.

相关于微分算子的(Musielak-)Orlicz-Hardy空间的实变理论及其应用

As is well-known, because of their important applications in several branches of mathematics, such as harmonic analysis and partial differential equations, the theory of Calderon-Zygmund singular integral operators and the real-variable theory of various function spaces, which could characterize the boundedness of those operators, turn into one of the main contents of modern harmonic analysis. However, the real-variable theory of classical function spaces has been no longer suitable for characterizing the boundedness of singular integral operators associated with some more general differential operators than Laplace operators. Thus, for different operators, it has become one of the very active research fields of harmonic analysis in recent years to develop the real-variable theory of function spaces, which are suitable to those operators and could characterize the boundedness of singular integral operators associated with those operators. In this article, we study the realvariable theory of (Musielak-)Orlicz-Hardy spaces associated with some differential operators, including secondorder divergence form elliptic operators and Schrodinger operators as special cases, on n -dimensional Euclidean space R n , strongly domains of R n or metric spaces with doubling measure, and its applications to the boundedness of operators.

Recent Developments of Function Spaces With Variable Exponents

In this paper,we give a summary of the boundedness of(local) Hardy-Littlewood maximal operator and(variable) fractional integrals on variable Lebesgue spaces.Then we present recent developments of the theory of function spaces with variable exponents,which include Besov and Triebel-Lizorkin spaces,Morrey type Besov and Triebel-Lizorkin spaces,Herz type Besov and Triebel-Lizorkin spaces,Hardy spaces and Herz type Hardy spaces with variable exponents.Along the way several open problems are also given.

Geometry of Banach Spaces, Operator Theory, and Their Applications

In recent years, with developments of computer, high technique, and life science, more and more requirements were proposed to functional analysis. There were a large number of research articles published in the relative area and many major discoveries have been obtained. At the present time the ideas, terminology, and methods of functional analysis have penetrated deeply, not only into natural sciences, but also into applied disciplines. Many real world problems were analyzed and solved by using the ideas and results of function analysis. To fit the requirements, we edit this special issue. The special issue represented the recent development in geometric structure of Banach spaces, Bergman spaces, and Minkowski spaces, random normed modules on function space, linear and nonlinear operator theory, and the applications of modern analysis in related areas of mathematics, as well as other disciplines, such as economics, finance and risk management, dynamic systems, the natural and life sciences, medicine, physics, and other real world problems. The special issue consisted of twenty research articles contributed by participants of the 4th International Conference on Modern Analysis and Its Applications which was held at the Tianjin University, China, from August 1 to 4, 2013. With over 240 participants fromdifferent countries, the conference proved highly successful in bringing together experts and researchers and provided an opportunity to evaluate and disseminate new ideas and methods in modern analysis. Each paper has been peer-reviewed and the editors thank the referees for their time and efforts to help ensure a high standard for this special issue. We hope that readers interested in function spaces and operator theory will find in this special issue not only new information about development of functional analysis, but also important questions be resolved and directions for further study. At same time, we hope that in the near future, we can see new results to be published based on this special issue.

Function Spaces and Operators with Applications

The theory of function spaces and operators along with their various applications is an active branch of mathematics and there were a large number of papers published in the relative areas and many major discoveries have been obtained. We are happy that this special issue has received the attention of many researchers in these areas. The special issue consisted of twelve research papers covering commutators of multiliner Fourier multiplier operators on weighted Lebesgue spaces, fractional Hardy operators and their commutators on Hardy spaces, Marcinkiewicz integrals in weighted weak Hardy spaces, geometry of normed linear spaces (including Banach spaces and Hilbert spaces), ArzelaAscoli theorem for demilinear mappings in topological linear spaces, Malliavin derivatives in variable exponent spaces, the theory of Young measures in variable exponent Sobolev spaces and applications, the applications of reproducing kernel spaces on binary singular problems and nonlinear fourth-order differential equations with integral boundary conditions, and weighted Lorentz spaces and operators. We hope that readers interested in functional analysis will find useful information in this special issue. At the same time we also hope that in the near future we can see new papers to be published based on this special issue.