Constructive Gelfand duality for C*-algebras

We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.Comment: 6page

Korovkin-type Theorems and Approximation by Positive Linear Operators

This survey paper contains a detailed self-contained introduction to Korovkin-type theorems and to some of their applications concerning the approximation of continuous functions as well as of L^p-functions, by means of positive linear operators. The paper also contains several new results and applications. Moreover, the organization of the subject follows a simple and direct approach which quickly leads both to the main results of the theory and to some new ones

Constructive topology of bishop spaces

The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS. Instead of having a common space-structure on a set X and R, where R denotes the set of constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F), where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R. The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions, the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces. The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient, the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact, the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces. The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively the Urysohn extension theorem for Bishop spaces. The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included

A Comprehensive Survey on Functional Approximation

The theory of functional approximation has numerous applications in sciences and industry. This thesis focuses on the possible approaches to approximate a continuous function on a compact subset of R2 using a variety of constructions. The results are presented from the following four general topics: polynomials, Fourier series, wavelets, and neural networks. Approximation with polynomials on subsets of R leads to the discussion of the Stone-Weierstrass theorem. Convergence of Fourier series is characterized on the unit circle. Wavelets are introduced following the Fourier transform, and their construction as well as ability to approximate functions in L2(R) is discussed. At the end, the universal approximation theorem for artificial neural networks is presented, and the function representation and approximation with single- and multilayer neural networks on R2 is constructed

McShane-Whitney extensions in constructive analysis

Within Bishop-style constructive mathematics we study the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of McShane-Whitney theorem, we show that the Lipschitz real-valued functions on a totally bounded space are uniformly dense in the set of uniformly continuous functions. Through the introduced notion of a McShane-Whitney pair we describe the constructive content of the original McShane-Whitney extension and examine how the properties of a Lipschitz function defined on the subspace of the pair extend to its McShane-Whitney extensions on the space of the pair. Similar McShane-Whitney pairs and extensions are established for H\"{o}lder functions and $\nu$-continuous functions, where $\nu$ is a modulus of continuity. A Lipschitz version of a fundamental corollary of the Hahn-Banach theorem, and the approximate McShane-Whitney theorem are shown

The algebraic specification of spatial data types with applications to constructive volume geometry.

Spatial objects are modelled as total functions, mapping a topological space of points to a topological algebra of data attributes. High-level operations on these spatial objects form algebras of spatial objects, which model spatial data types. This thesis presents a comprehensive account of the theory of spatial data types. The motivation behind the general theory is Constructive Volume Geometry (CVG). CVG is an algebraic framework for the specification, representation and manipulation of graphics objects in 3D. By using scalar fields as the basic building blocks, CVG gives an abstract representation of spatial objects, with the goal of unifying the many representations of objects used in 3D computer graphics today. The general theory developed in this thesis unifies discrete and continuous spatial data, and the many examples where such data is used - from computer graphics to hardware design. Such a theory is built from the algebraic and topological properties of spatial data types. We examine algebraic laws, approximation methods, and finiteness and computability for general spatial data types. We show how to apply the general theory to modelling (i) hardware and (ii) CVG. We pose the question "Which spatial objects can be represented in the algebraic framework developed for spatial data types?". To answer such a question, we analyse the expressive power of our algebraic framework. Applying our results to the CVG framework yields a new result: We show any CVG spatial object can be approximated by way of CVG terms, to arbitrary accuracy

Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?

We study the problem of approximating compactly-supported integrable functions while implementing their support set using feedforward neural networks. Our first main result transcribes this "structured" approximation problem into a universality problem. We do this by constructing a refinement of the usual topology on the space $L^1_{\operatorname{loc}}(\mathbb{R}^d,\mathbb{R}^D)$ of locally-integrable functions in which compactly-supported functions can only be approximated in $L^1$-norm by functions with matching discretized support. We establish the universality of ReLU feedforward networks with bilinear pooling layers in this refined topology. Consequentially, we find that ReLU feedforward networks with bilinear pooling can approximate compactly supported functions while implementing their discretized support. We derive a quantitative uniform version of our universal approximation theorem on the dense subclass of compactly-supported Lipschitz functions. This quantitative result expresses the depth, width, and the number of bilinear pooling layers required to construct this ReLU network via the target function's regularity, the metric capacity and diameter of its essential support, and the dimensions of the inputs and output spaces. Conversely, we show that polynomial regressors and analytic feedforward networks are not universal in this space.Comment: 23 Pages: Main Text - 16 pages, Appendix - 7.5 pages, - Bibliography - 5 page