Pełczyński-type sets and Pełczyński’s geometrical properties of locally convex spaces

Augmented Lagrangian (AL) methods for solving convex optimization problems with linear constraints are attractive for imaging applications with composite cost functions due to the empirical fast convergence rate under weak conditions. However, for problems such as X-ray computed tomography (CT) image reconstruction, where the inner least-squares problem is challenging and requires iterations, AL methods can be slow. This paper focuses on solving regularized (weighted) least-squares problems using a linearized variant of AL methods that replaces the quadratic AL penalty term in the scaled augmented Lagrangian with its separable quadratic surrogate function, leading to a simpler ordered-subsets (OS) accelerable splitting-based algorithm, OS-LALM. To further accelerate the proposed algorithm, we use a second-order recursive system analysis to design a deterministic downward continuation approach that avoids tedious parameter tuning and provides fast convergence. Experimental results show that the proposed algorithm significantly accelerates the convergence of X-ray CT image reconstruction with negligible overhead and can reduce OS artifacts when using many subsets.

The augmented Lagrangian (AL) optimization method has drawn more attention recently in imaging applications due to its decomposable structure for composite cost functions and empirical fast convergence rate under weak conditions. However, for problems, e.g., X-ray computed tomography (CT) image reconstruction, where the inner least-squares problem is challenging, the AL method can be slow due to its iterative inner updates. In this paper, using a linearized AL framework, we propose an ordered-subsets (OS) accelerable linearized AL method, OS-LALM, for solving penalized weighted least-squares (PWLS) X-ray CT image reconstruction problems. To further accelerate the proposed algorithm, we also propose a deterministic downward continuation approach for fast convergence without additional parameter tuning. Experimental results show that the proposed algo- rithm significantly accelerates the “convergence” of X-ray CT image reconstruction with negligible overhead and exhibits excellent gradient error tolerance when using many subsets for OS acceleration.

We study a visibility relation on the nonempty connected convex subsets of a finite partially ordered set and we investigate the partial orders representable as a visibility relation of such subsets of a weak order. Moreover, we consider restrictions where the subsets of the weak order are total orders or isomorphic total orders.

Weak⁎ derived sets of convex sets in duals of non-reflexive spaces

In recent years, several papers were devoted to the study of (p, q)-summing sequences of operators and their applications in the theory of Banach spaces. In the present paper we initiate a similar study of "integral multiplier functions", thereby introducing these spaces of operator valued multiplier functions as normed spaces, establishing the basic results needed in such a theory and considering applications to the characterization of classical (operator valued) function spaces and spaces of operators on (or into) classical function spaces. We prove some significant "measurability results" for operator valued functions and apply these to establish several "function space versions" of results in the (p, q)-summing sequence setting.

Partial orders and their convex subsets

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.

This chapter is devoted to the classical sequence spaces and function spaces. The sequence spaces are defined explicitly but the function spaces require some measure theoretic machinery to be developed. The approach we take avoids the technicalities of measure theory and instead uses the metric formalism already at hand in order to present the function spaces both rigorously and comprehensibly. Of course, the measure theoretic techniques are of great importance. The aim is to quickly equip the reader with familiarity with these important spaces and pave the future way into measure theoretic analyses.

One of the main goals of functional spaces is to interpret and quantify the smoothness of functions. In this chapter, we discuss the analogs of classical functional spaces with respect to the Gaussian measure. We see that almost all classical spaces with respect to the Lebesgue measure have an analog for the Gaussian measure; nevertheless, we see that in some cases, for instance, Hardy spaces, the analogs to classical spaces are still incomplete and/or imperfect. On the other hand, most of the time, even if the spaces look similar, most of the proofs are different, mainly because the Gaussian measure is not invariant by translation, which implies the need for completely different techniques.

Let > 0. The symbol B 1 denotes the space of all entire functions of exponential type not exceeding that are bounded on the real axis. Various exact descriptions of uniqueness sequences for the Bernstein spaces B 1 are given in terms of and the Poisson and Hilbert transformations. These descriptions lead to completeness criteria for systems of exponentials (up to one or two members) in various classical function spaces on an interval (closed or open) of length d.

In the next two sections, we will consider the classical sequence and function spaces. The main purpose of these sections is to make the necessary definitions and to identify the dual spaces for these classical spaces.

In this thesis we develop a functional analytic framework for shape optimization with elliptic partial differential equation (PDE) constraints in classical function spaces (Holder spaces). This approach is motivated by shape optimization problems, which are subjected to linear elasticity constraints and involve a special class of shape functionals which calculate the failure rate of a mechanically loaded device w.r.t. the component shape. These objectives are ill-defined for H1-solutions of the state equation and the shape derivatives are not defined for H1-material derivatives. Thus, the resulting optimal reliability problems can not be solved by the already existing methods of shape calculus. We develop a general concept on Banach and Hilbert spaces which is based on parameter depending variational equations, classical PDE Solutions, Schauder estimates and compact embeddings and which allows to transfer differentiability in lower Banach space topologies to higher ones. We apply this framework to the linear elasticity equation and its variational formulation, given that the domain is transformed according to the speed method. We prove the existence of material and local shape derivatives in Holder spaces, the existence of shape derivatives and derive adjoint equations. We also give a classification of the L2-shape gradient w.r.t. its regularity and its potential to sustain the domain regularity along a descent flow.

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from L p. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov–Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that L p is the only space where it is possible to change this order.

Marston Morse and William Transue (6, 8) have introduced and studied function spaces, called MT-spaces, for which the elements of the topological dual are of integral type. Their theory does not admit certain classical Banach function spaces including spaces of bounded functions and spaces. The theory of function spaces determined by a length function (λ-spaces) (4, 5), which depends on a fixed measure, admits many of the maximal MT-spaces, the spaces and spaces of locally integrable functions but does not admit certain maximal MT-spaces including the space of complex continuous functions with compact supports.

The criteria for nonsquareness in the classical Orlicz function spaces have been given already. However, because of the complication of Musielak‐Orlicz‐Bochner function spaces, at present the criteria for nonsquareness have not been discussed yet. In the paper, the criteria for nonsquareness of Musielak‐Orlicz‐Bochner function spaces are given. As a corollary, the criteria for nonsquareness of Musielak‐Orlicz function spaces are given.