Spaces Of Differentiable Functions Research Articles

Application of locally regularized extremal shift to the problem of realization of a prescribed motion

Abstract A controlled system of differential equations under the action of an unknown disturbance is considered. The problem discussed in the paper consists in constructing algorithms for forming a control that provides the realization of a prescribed motion for any admissible disturbance. Namely these algorithms should provide the closeness in the metric of the space of differentiable functions of a phase trajectory of a given controlled system and some etalon trajectory of an analogous system functioning when any outer actions are absent. As the space of admissible disturbances, we take the space of measurable square integrable (with respect to the Euclidean norm) functions. The cases of inaccurate measurements of phase trajectories of both systems at all times and at discrete times are under study. Two computer oriented algorithms for solving the problem are designed. The algorithms are based on the (well-known in the theory of guaranteed control) method of extremal shift. In the process, its local (at each time of control correction) regularization is performed by the method of smoothing functional (the Tikhonov method). In addition, estimates for algorithm’s convergence rate are presented.

Nice operators and nice spaces

In this paper we obtain a necessary and sufficient condition for an operator on a uniform algebra to be nice. We characterize nice operators on an expansive class of Banach spaces. Then as examples, we give a complete description of nice operators on the space of differentiable functions on compact perfect plane sets, Bloch and Zygmund spaces and the space of Lipschitz functions on intervals. We also prove that every surjective nice operators are isometries on some of these Banach spaces.

Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions

In this research work, a new derived optimal quadrature formula is discussed, which includes the sum of the values of the function and its first and second order derivatives at the points located at the same distance on the interval [0,1] in the L2(m)(0,1) space. we first obtain an analytical representation of the error function norm, and a system of equations of the Wiener-Hopf type construct using the method of Lagrange unknown multipliers for finding the conditional extremum of multivariable functions. Optimal coefficients found by solving the system. Using the exact form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula for m=3 and m=4 calculate and the order of approximation was shown to be O(hm). The obtain theoretical conclusions confirmed by numerical experiments.

On the approximation of solutions of one singular differential equation on the axis

In this paper we study the problem of the best approximation by linear methods of solutions to one Triebel-type equation. This problem was solved by using estimates of the linear widths of the unit ball in corresponding spaces of differentiable functions. According to the definition, linear widths give the best estimates for the approximation of compact sets in a given normed space by linear methods which are implemented through finite-dimensional operators. The problem includes answers to the questions about the solvability of the studied equation, the construction of the corresponding weighted space of differentiable functions, the development of a method for estimating linear widths of compact sets in weighted polynomial Sobolev space. In this work, conditions are obtained under which the considered operator has a bounded inverse. The weighted Sobolev space corresponding to the posed problem is determined. Upper estimates are obtained for the counting function for a sequence of linear widths, which correspond to the posed problem. One example is constructed in which two-sided estimates of linear widths are given. The method for solving this problem can be applied to the numerical solution of non-standard ordinary differential equations on an infinite axis.

The Newton product of polynomial projectors. Part 2: approximation properties

We prove that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a regular compact set, on spaces of entire functions of given growth and on spaces of differentiable functions. Efficient explicit new projectors are presented.

Asymptotic linear - nonlinear duality, indeterminism and mathematical intelligence

The formalism of asymptotic duality structure with applications to late time asymptotic states of linear and nonlinear systems is presented. A scale invariant linear differential system is shown to admit higher derivative discontinuous, nonsmooth solutions under the action of asymptotic duality transformations. The relevant asymptotic quantities are self dual and finitely differentiable. The geometrical meaning and applications to higher order nonlinear systems are presented on the space of differentiable functions. A novel duality relationship between linear and nonlinear systems is exposed. A linear system is shown to extend self similarly over nonlinear, classically inaccessible scales thereby extending classical smooth solution space to a nonsmooth one. A nonlinear system, on the other hand, is shown to proliferate into a system of self similar linear equations that could be solved exactly to have a very efficient high precision computation of periodic orbits. An extension of self dual asymptotics to critically self dual case in the space of continuous functions reveals a connection to prime number theorem and the associated correction term respecting the Riemann hypothesis.

Isometries between Spaces of Vector-Valued Differentiable Functions

This article characterizes the isometries between spaces of all differentiable functions from a compact interval of the real line into a strictly convex Banach space.

Matrix Hölder's Inequality and Divergence Formulation of Optimal Transport of Vector Measures

We characterise equality cases in matrix H¨older’s inequality and develop a divergence formulation of optimal transport of vector measures. As an application, we reprove the representation formula for measures in the polar cone to monotone maps. We generalise the last result to a wide class of polar cones, including polar cones to tangent cones to the unit ball in the space of differentiable functions and in the Sobolev spaces.

Continuously Differentiable Functions on Compact Sets

We consider the space C^1(K) of real-valued continuously differentiable functions on a compact set Ksubseteq mathbb {R}^d. We characterize the completeness of this space and prove that the restriction space C^1(mathbb {R}^d|K)={f|_K: fin C^1(mathbb {R}^d)} is always dense in C^1(K). The space C^1(K) is then compared with other spaces of differentiable functions on compact sets.

Quantitative Singularity Theory for Random Polynomials

Abstract Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-$W$ singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset $W$ of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-$W$ singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function $p$ we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center $p$, in which the $W$-singularity structure is constant.

О разрешимости вариационной задачи Дирихле для одного класса вырождающихся эллиптических операторов

В работе исследуется однозначная разрешимость вариационной задачи Дирихле, свя-занной с интегро-дифференциальной полуторалинейной формой????[????, ????] =Σ︁????∈???????????? [????, ????], (*)где???????? [????, ????] =Σ︁|????|=|????|=????∫︁Ω????(????)2???????? ????????????(????)????(????)(????) ????(????)(????)????????,Ω — ограниченная область в евклидовом пространстве ???????? с замкнутой (???? − 1)-мернойграницей ????Ω, ????(????), ???? ∈ Ω, — регуляризованное расстояние от точки ???? ∈ Ω до ????Ω, ???? —мультииндекс, ????(????)(????) — обобщенная производная мультииндекса ???? функции ????(????), ???? ∈ Ω,????????????(????) — ограниченные в Ω комплекснозначные функции, ???? ⊂ {1, 2, . . . , ????} и ???????? , ???? ∈ ????, —вещественные числа. Предполагается, что ???? ∈ ????. Вырождение коэффициентов дифферен-циального оператора, ассоциированного с формой (*), называется согласованным, еслисуществует число ???? такое, что ???????? = ???? + ???? − ???? при всех ???? ∈ ????. В противном случае ононазывается несогласованным.Вариационная задача Дирихле, связанная с формой (*), в случае согласованного вы-рождения коэффициентов хорошо исследована во многих работах, где также предполага-ется, что форма (*) удовлетворяет условию коэрцитивности. Следует отметить, что слу-чай несогласованного вырождения коэффициентов сопряжен с некоторыми техническимисложностями и рассмотрен лишь в некоторых отдельных работах. В этом случае с помо-щью теорем вложения пространств дифференцируемых функций со степенными весамивыделяются старшие формы ???????? [????, ????], ???? ∈ ????2 ⊂ ???? и доказывается, что разрешимость вари-ационной задачи Дирихле в основном зависит от старших форм.В работе рассматривается случай несогласованного вырождения коэффициентов ис-следуемого оператора и, в отличие от ранее опубликованных работ по этому направлению,допускается случай, когда основная форма (*) может не удовлетворять условию коэрци-тивности.

Topological properties of operations on spaces of differentiable functions

‎‎In this paper‎, ‎we consider different notions of openness for the scalar multiplication on sequence spaces and spaces of continuous functions‎. ‎We apply existing techniques to derive weak openness of multiplication on spaces of differentiable functions‎, ‎endowed with a large collection of quasi-algebra norms‎.

The approximation property for weighted spaces of differentiable functions

We study spaces $\mathcal{CV}^{k}(\Omega,E)$ of $k$-times continuously partially differentiable functions on an open set $\Omega\subset\mathbb{R}^{d}$ with values in a locally convex Hausdorff space $E$. The space $\mathcal{CV}^{k}(\Omega,E)$ is given a weighted topology generated by a family of weights $\mathcal{V}^{k}$. For the space $\mathcal{CV}^{k}(\Omega,E)$ and its subspace $\mathcal{CV}^{k}_{0}(\Omega,E)$ of functions that vanish at infinity in the weighted topology we try to answer the question whether their elements can be approximated by functions with values in a finite dimensional subspace. We derive sufficient conditions for an affirmative answer to this question using the theory of tensor products.

Дифференциальные свойства обобщённых потенциаловтипа Бесселя и типа Рисса

In this paper we study differential properties of convolutions of functions with kernels thatgeneralize the classical Bessel-Macdonald kernels ... The theory ofclassical Bessel potentials is an important section of the general theory of spaces of differentiablefunctions of fractional smoothness and its applications in the theory of partial differentialequations. The properties of the classical Bessel-Macdonald kernels are studied in detail in thebooks of Bennett and Sharpley, S. M. Nikolskii, I. M. Stein, V. G. Mazya. The local behavior ofthe Bessel-Macdonald kernels in the neighborhood of the origin is characterized by the presenceof a power-type singularity ||-. At infinity, they tend to zero at an exponential rate. Therecent work of M. L. Goldman, A. V. Malysheva, and D. Haroske was devoted to the investigationof the differential properties of generalized Bessel-Riesz potentials.In this paper we study the differential properties of potentials that generalize the classicalBessel-Riesz potentials. Potential kernels can have nonpower singularities in the neighborhoodof the origin. Their behavior at infinity is related only to the integrability condition, so thatkernels with a compact support are included. In this connection, the spaces of generalized Besselpotentials generated by them belong to the so-called spaces of generalized smoothness. The casewith the satisfied criterion for embedding potentials in the space of continuous bounded functionsis considered. In this case, the differential properties of the potentials are expressed in termsof the behavior of their module of continuity in the uniform metric. Criteria for embedding ofpotentials in Calderon spaces are established and explicit descriptions of the module of continuityof potentials and optimal spaces for such embeddings are obtained in the case when the basespace for potentials is the Lorentz weight space. These results specify the general constructionsestablished in previous works.

G1-smooth splines on quad meshes with 4-split macro-patch elements

We analyze the space of differentiable functions on a quad-mesh M, which are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges, that satisfy conditions which ensure that the space of differentiable functions is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of G1 spline functions of bi-degree ⩽(k,k) on each quadrangular face of M. We determine the dimension of this space of G1 spline functions for k big enough and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. This construction requires the analysis of the module of syzygies of univariate b-spline functions with b-spline function coefficients. New results on their generators and dimensions are provided. Examples of bases of G1 splines of small degree for simple topological surfaces are detailed and illustrated by parametric surface constructions.

Local approximations of differentiable functions

Spaces of differentiable functions constructed from the spaces Lp, 0 < p < ∞, and the interdependence of the differential properties of functions with their local approximations are considered.

Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology

We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G1 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework does not require manifold constructions and is general enough to allow non-orientable surfaces. We describe compatibility conditions on the transition maps so that the space of differentiable functions is ample and show that these conditions are necessary and sufficient to construct ample spline spaces. We determine the dimension of the space of G1 spline functions which are of degree ⩽k on triangular pieces and of bi-degree ⩽(k,k) on rectangular pieces, for k big enough. A separability property on the edges is involved to obtain the dimension formula. An explicit construction of basis functions attached respectively to vertices, edges and faces is proposed; examples of bases of G1 splines of small degree for topological surfaces with boundary and without boundary are detailed.

Differential Calculus for Differential Equations, Special Functions, Laplace Transform

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Nonlinear order isomorphisms on function spaces

Let X be a topological space. A subset of C(X), the space of continuous real-valued functions on X, is a partially ordered set in the pointwise order. Suppose that X and Y are topological spaces, and A(X) and A(Y ) are subsets of C(X) and C(Y ) respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between A(X) and A(Y ). Under some general assumptions on A(X) and A(Y ), and when X and Y are compact Hausdorff, it is shown that existence of an order isomorphism between A(X) and A(Y ) gives rise to an associated homeomorphism between X and Y . This generalizes a classical result of Kaplansky concerning linear order isomorphisms between C(X) and C(Y ) for compact Hausdorff X and Y . The class of near vector lattices is introduced in order to extend the result further to noncompact spaces X and Y . The main applications lie in the case when X and Y are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, “local” and “bounded local” versions of these spaces, characterizations of when spaces of one type can be order isomorphic to spaces of another type are obtained.

Optimal average approximations for functions mapping in quasi-Banach spaces

In 1994, M.M. Popov [6] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the ℓp-spaces for 0<p<1, and the question arose whether such a significant result might hold in some other non-Banach spaces. In this article we completely settle the problem by proving that the fundamental theorem of calculus breaks down in the context of any non-locally convex quasi-Banach space. Our approach introduces the tool of Riemann-integral averages of continuous functions, and uses it to bring out to light the differences in behavior of their approximates in the lack of local convexity. As a by-product of our work we solve a problem raised in [1] on the different types of spaces of differentiable functions with values on a quasi-Banach space.