Class Of Real Functions Research Articles

Maximal Families for the Class of Upper and Lower Semi-Quasicontinuous Functions

In this article we investigate the maximal additive (multiplicative) [lattice] families for the class of real functions defined on topological spaces which are upper and lower semi-quasicontinuous at each point.

On Some Properties of the Class of Real Functions with λ and λ′ Graphs

We show that (under CH) there exists a $CIVP$ function with a $\lambda^\prime$ graph. We examine some properties of $\mathcal{M}_\alpha$ and $\mathcal{M}_alpha(\lambda^\prime)$ class of real valued functions

Propriétés oscillatoires des intégrales de [formula omitted]-fonctions

We deal here with a o-minimal class of real functions called x λ -functions. They are finite compositions of subanalytic maps and power maps. These functions have nice properties of non-oscillation: the number of connected components of a 1-parameter family of x λ -functions is uniformly bounded with respect to the parameter if the dependency is analytic (or more generally o-minimal) in the parameter. Our purpose is to study how this kind of property can remain under integration. Some result of this kind have been already proved for the smaller class of subanalytic functions. We show here that for almost all power maps arising in a x λ -function, the integration leads to a non-oscillating function. Nevertheless, this class of functions is not stable under integration.

On Exact Values of n-Widths in a Hilbert Space

The exact values of Kolmogorov n -widths have been calculated for two basic classes of functions. They are, on the one hand, classes of real functions defined by variation diminishing kernels and similar classes of analytic functions, and, on the other hand, classes of functions in a Hilbert space which are elliptical cylinders or generalized octahedra. This second case is surveyed and new results are presented. For n -widths of ellipsoids, elliptic cylinders, and generalized octahedra, upper bounds for the n -widths are based on the Fourier method. The lower bounds are based on the method of “embedded balls” for ellipsoids and the method of averaging for generalized octahedra. General theorems concerning elliptical cylinders and generalized octahedra are proved, various corollaries from these general theorems are considered, and some additional problems (average n -widths, extremal spaces for an ellipsoids and octahedra, etc.) are discussed.

On Sequences of Upper and Lower Semi-Quasicontinuous Functions

In this article we investigate the pointwise, discrete and transfinite convergences in the classes of real functions defined on topological spaces which are upper and lower quasicontinuous at each point.

Mixed Functionals of Convex Bodies

We define a class of real functions on tuples of convex bodies. They are a common generalization of mixed volumes and of certain functionals which have been studied in translative integral geometry. For polytopes, these functionals have various explicit representations in terms of volumes of lower-dimensional faces. For the mentioned functionals from integral geometry, these representations generalize a result of Weil and answer a question posed by Janson.

On the Uniqueness of Solutions of a Nonlinear Elliptic Problem Arising in the Confinement of a Plasma in a Stellarator Device

We study the uniqueness of solutions of a semilinear elliptic problem obtained from an inverse formulation when the nonlinear terms of the equation are prescribed in a general class of real functions. The inverse problem arises in the modeling of the magnetic confinement of a plasma in a Stellarator device. The uniqueness proof relies on an \( L^{\infty} \) -estimate on the solution of an auxiliary nonlocal problem formulated in terms of the relative rearrangement of a datum with respect to the solution.

GLIVENKO-CANTELLI TYPE THEOREMS AND THEIR APPLICATIONS

In this paper, we present a sufficient condition for uniform convergence of means to their expectations over the classes of real functions. Our proof is very simple via connecting this convergence to uniform convergence over the class of indicator functions. Furthermore, we obtain a convergent result concerning a non=VC class with an application to variable transformation for fitting regression model.

Decidability problems for meta-R-functions

In this article we consider classes of functions defined by R-transformers, which are machines that sequentially process real numbers represented in the binary number system. The class of real functions defined by R-transformers includes all continuous and some discontinuous functions. The closure of this class under superposition produces the wider class of real meta-R-functions. Such functions are defined by finite sequences of R-transformers. Here we examine the class of meta-R-meta-R-functions. Such functions are defined by finite sequences of R-transformers. Here we examine the class of meta-R-functions and specifically the class of finite meta-R-functions. The latter are defined by finite sequences of finite R-transformers. Decidability of the equivalent problem in the class of functions defined by finite R-transformers, i.e., the class of finite R-functions, is proved elsewhere. Here we generalize this result to the class of finite meta-R-functions. We investigate not only the equivalence problem, but also the monotonicity and continuity problems. The proof is by reduction to the decidable nonemptiness problem for nondeterministic bounded-mode finite transformers with finite-turnaround counters on labeled trees. We also consider the general properties of the class of meta-R-functions.

Topologizing Different Classes of Real Functions

AbstractThe purpose of this paper is to examine which classesof functions fromcan be topologized in a sense that there exist topologies τ1and τ2onandrespectively, such thatis equal to the class C(τ1, τ2) of all continuous functions. We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functionsfor which the sets {x : f(x) = g(x)} are small in some sense for all f, g ∈f ≠ g. The topologies will be Hausdorff and connected. It will be also shown that in some model of set theory ZFC with GCH these topologies could be completely regular and Baire. One of the corollaries of this theorem is that GCH implies the existence of a connected Hausdorff topology T onsuch that the class L of all linear functions g(x) = ax + b coincides with. This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the classof all polynomials, the classof all analytic functions and the class of all harmonic functions.We will also prove that several other classes of real functions cannot be topologized. This includes the classes of C∞functions, differentiable functions, Darboux functions and derivatives.

Combining Binary Search and Newton′s Method to Compute Real Roots for a Class of Real Functions

We generalize a hybrid algorithm of binary search and Newton′s method to compute real roots for a class of real functions. We show that the algorithm computes a root inside (0, R] with error ϵ in O(log log( R/ϵ)) time, where one function evaluation or one arithmetic operation counts for one unit of time. This work is based on Smale′s criterion for using Newton′s method and Renegar′s result of approximating roots of polynomials.

Dynamic spanning without probabilities

The paper presents a non-probabilistic approach to continuous-time trading where, in analogy to the binomial option-pricing model, terminal payoffs resulting from a given trading strategy are meaningful ‘state-by-state’, i.e., path-by-path. In particular, we obtain results of the form: “If a certain trading strategy is applied and if the realized price trajectory satisfies a certain analytical property, then the terminal payoff is.…” This way, derivation of the Black and Scholes formula and its extension become an exercise in the analysis of a certain class of real functions. While results of the above forms are of great interest if the analytical property in question is believed to be satisfied for almost all realized price trajectories (for example, if the price is believed to follow a certain stochastic process which has this property with probability 1), they are valid regardless of the stochastic process which presumably generates the possible price trajectories or the probability assigned to the set of all paths having this analytical property.

WhenL1of a vector measure is an AL-space

We consider the space of real functions which are integrable with respect to a countably additive vector measure with values in a Banach space. In a previous paper we showed that this space can be any order continuous Banach lattice with weak order unit. We study a priori conditions on the vector measure in order to guarantee that the resulting Lι is order isomorphic to an AL-space. We prove that for separable measures with no atoms there exists a Co-valued measure that generates the same space of integrable functions. Introduction. Given a vector measure v we consider the space of classes of real functions which are integrable with respect to v in the sense of Lewis (L-l), denoted by Lι{y). In (C, Theorem 8) we showed that every order continuous Banach lattice with weak unit can be obtained as Lι(u) for a suitable vector measure v. In particular we have Hubert spaces as L 1 of a vector measure. A natural question arises. Under what conditions on the vector measure, or on the Banach space in which the measure takes its values, is the resulting L1 of the vector measure order isomorphic to an AL-space? Recall that a Banach lattice is an AL-space when the norm is additive for disjoint vectors. An order isomorphism is a linear isomorphism that preserves the lattice operations. So the question can be restated in the following way. When can Lx(u) be equivalently renormed so that endowed with the new norm and the same order is a Banach lattice where the norm is additive for disjoint functions? In § 1 we fix notation and basic definitions. In §2 we show the special role that the space Lι(\u\) plays in the problem we are studying, where \v is the variation of υ . It is shown in Proposition 2 that Lx(v) is an Jϊ?\-space, in the sense of Linden- strauss and Pelczynski (L-P), if and only if it is order isomorphic to Lι(\v\). From this it follows that bounded variation of the measure is a necessary condition. The conditions cannot be placed on the Banach space in which the measure takes its values. This is shown in Example 1, that also shows that neither bounded variation nor domination of the variation by the semivariation are sufficient conditions. In §3 we study measures with values in C(K) spaces. In Theorem 1

The exact Hausdorff dimension for a class of fractal functions

In recent years a number of authors (an incomplete list is [ 1,2, 3,6, 7, 9, 11, 12, 131) have considered a class of real functions whose graphs are, in general, fractal sets in R2. In this paper we give sufficient conditions for the fractal and Hausdorff dimensions to be equal for a certain subclass of fractal functions. The sets we consider are examples of self-affine fractals generated using iterated function systems (i.f.s.). Falconer [S] has shown that for almost all such sets the fractal and Hausdorff dimensions are equal and he gives a formula for the common dimension, due originally to Moran [S]. These results, however, give no information about individual fractal functions, In this paper we extend Moran’s original method and show that if certain conditions on the i.f.s. are satisfied, then the two dimensions are equal. Kono [ 111 and Bedford [ 121 considered special cases of the subclass of fractal functions that we will introduce. Bedford and Urbanski [13] use a nonlinear setting to present conditions for the equality of Hausdorff and fractal dimension. However, their criteria are based on measure-theoretic characterizations and the use of the concept of generalized pressure. Our criterion on the other hand is based on the underlying geometry of the attractor and is easier to verify. We will show this on two specific examples which are more general than the self-ahine functions presented in [13].

ON THE CONVOLUTION INVERSE OF DISCRETE SEQUENCES

The aim of this paper is to prove the existence and unicity of the convolution inverse for a certain class of real functions of the discrete argument. The properties of these inverses, in particular the errors that arise from the truncation of the infinite sequences that represent them, are examined. In the second part of this paper subtitled “Application in solving convolution equations” the possibility of using convolution inverses for determining the solution to the Fredholm equations of the first kind is discussed.

Decompositions into the sum of periodic functions belonging to a given Banach space

Then A, , f /=0 for every i and, as the difference operators commute, we have An1... Aa,f=0. Let ~ be a class of real functions. We say that ~has the decomposition property (d. pr.) if, for every fE~and for every finite system of real numbers al, ..., a,,, the condition A,1...A,,f=O implies that there are functions f~, . . . , f ,E~such that f = f z + . . . + f , and fl is periodic modulo a~ for every i = 1, ..., n. A natural generalization of this notion is the following. Let A be a non-empty set and let Tbe a map of A into itself. A function g: A-~R is said to be T-periodic, if goT=g or, equivalently, if A r g = 0 , where Arg=goT-g . Now let T~ . . . . , T~ be maps of A into itself and let f = f l + .-. + f , , where f~ is Trperiodic for every i=1 . . . . . n, If the maps Ti commute i.e. if T/oTj=TjoTI for every i , j= l , ...,n then the operators At, also commute, and we have

The differentiability of real functions on normed linear space using generalized subgradients

The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.

Algorithmic questions for real functions

Turing machines, in particular, finite Mill automata can be considered as a means for giving corresponding subclasses of the class of real functions. From this point of view, finite Mili automata and generalized sequences of machines were considered in [I]. Here a real function f is defined by a finite automaton A in the following way. To determine the value f(x) of the automaton A one successively transforms the input object, the expansion of the real number x in the binary (decimal) system, into the output object, which is the binary (decimal) expansion of the number y, where y = f(x). Analogously for determining real functions one can use Turing machines. The subclass of the class of real functions arising here is actually a proper subclass of the class of C-computable functions, defined below. The latter is strictly included in the class of R-functions, or real functions, and that, in its own right, is contained in the class of V-functions. Actually, each continuous real functions turns out to be a real function. V-functions are given by indeterminate R-transformers, transforming binary representations of real numbers. R-transformers with finite memory, defining real functions are said to be finite reals. The latter are richer than Mill automata, defining real functions according to [I], in two respects: they can be asynchronous and can define functions which are not strictly real. For example, the function f(x) = 3x is defined by a finitely real, but is not strictly real. For finite reals, on the basis of [2] the solvability of the equivalence problem and of some other algorithmic questions is proved. This generalizes the corresponding results found in [i] for finite Mili automata and generalized succcession machines preserving length. The solvability is also established of the equivalence problem for single-valued indeterminate finite reals with finitely turning accumulators. An algebraic characterization is given of the class of partial C-computable functions. We note that in the papers of Red'ko [3, 4] the problem of describing the algebra of computable (partially computable) functions of a real variable was formulated as open. Bui effected the construction of the algebra of computable operations on the set of rational numbers [5].

Derivatives and Lebesgue Points Via Homeomorphic Changes of Scale

Let $I$ be a closed interval, and suppose $\mathcal {K}$, $\mathcal {H}$, and $\Lambda$ denote, respectively, the class of homeomorphisms of $I$ onto itself, the class of homeomorphisms of the line onto itself, and the class of real functions on $I$ for which each point is a Lebesgue point. Maximoff proved that $\Lambda \circ \mathcal {K}$ is exactly the class of Darboux Baire $1$ functions, where $\Lambda \circ \mathcal {K} = \{ f \circ k:f \in \Lambda ,k \in \mathcal {K}\}$. The present paper is devoted primarily to a study of $\mathcal {H} \circ \Lambda = \{ h \circ f:f \in \Lambda ,h \in \mathcal {H}\}$. The characterizations of this class which are obtained show that a function is a member of $\mathcal {H} \circ \Lambda$ if and only if, in addition to the obvious requirement of approximate continuity, it satisfies certain growth and density-like conditions. In particular, any approximately continuous function with countably many non-Lebesgue points belongs to $\mathcal {H} \circ \Lambda$. It is also established that $\mathcal {H} \circ \Lambda$ is a uniformly closed algebra properly containing the smallest algebra generated from $\Lambda$, and a characterization of the latter algebra is provided.

Sampling theorems in polar coordinates

We investigate the problem of representing an arbitrary class of real functions f(·) in terms of their sampled values along the radius r and at equal angular increments of the azimuthal angle θ. Two different bandwidth constraints on f(r,θ) are considered: Fourier and Hankel. The end result is two theorems which enable images to be reconstructed from their samples. The theorems have potential application in image storage, image encoding, and computer-aided tomography.