Continuous Functionsf Research Articles

Paley – Wiener type theorem for functions with values in Banach spaces

UDC 517.5 Let ( 𝕏 , ‖ . ‖ 𝕏 ) denote a complex Banach space and L ( 𝕏 ) = B C ( ℝ → 𝕏 ) be the set of all 𝕏 -valued bounded continuous functions f : ℝ → 𝕏 . For f ∈ L ( 𝕏 ) we define ‖ f ‖ L ( 𝕏 ) = sup { ‖ f ( x ) ‖ 𝕏 : x ∈ ℝ } . Then ( L ( 𝕏 ) , ‖ . ‖ L ( 𝕏 ) ) itself is a Banach space. The Beurling spectrum S p e c ( f ) of a function f ∈ L ( 𝕏 ) is defined by S p e c ( f ) = { ζ ∈ ℝ : ∀ ϵ > 0 ∃ φ ∈ 𝒮 ( ℝ ) : supp φ ^ ⊂ ( ζ - ϵ , ζ + ϵ ) , φ * f ≢ 0 } . } . We obtain the following Paley-Wiener type theorem for functions with values in Banach spaces: Let f ∈ L ( 𝕏 ) and K be an arbitrary compact set in ℝ . Then Spec ( f ) ⊂ K if and only if for any τ > 0 there exists a constant C τ < ∞ such that < b r > ‖ P ( D ) f ‖ L ( 𝕏 ) ≤ C τ < b r > ‖ f ‖ L ( 𝕏 ) sup x ∈ K ( τ ) | P ( x ) | < b r > for all polynomials with complex coefficients P ( x ) , where the differential operator P ( D ) is obtained from P ( x ) by substituting x → - i d ⅆ x , d ⅆ x is the usual derivative in L ( 𝕏 ) and K ( τ ) is the τ -neighborhood in ℂ of K . Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts K are also given.

Externally definable quotients and NIP expansions of the real ordered additive group

Let R \mathscr {R} be an N I P \mathrm {NIP} expansion of ( R , > , + ) (\mathbb {R},>,+) by closed subsets of R n \mathbb {R}^n and continuous functions f : R m → R n f : \mathbb {R}^m \to \mathbb {R}^n . Then R \mathscr {R} is generically locally o-minimal. This follows from a more general theorem on N I P \mathrm {NIP} expansions of locally compact groups, which itself follows from a result on quotients of definable sets in ℵ 1 \aleph _1 -saturated N I P \mathrm {NIP} structures by equivalence relations which are both externally definable and ⋀ \bigwedge -definable. We also show that R \mathscr {R} is strongly dependent if and only if R \mathscr {R} is either o-minimal or ( R , > , + , α Z ) (\mathbb {R},>,+,\alpha \mathbb {Z}) -minimal for some α > 0 \alpha > 0 .

Porosity of 𝓢-approximately continuous functions

AbstractConsidering the natural topology or 𝓢-density topology on the domain and on the range we obtain different families of continuous functionsf: ℝ → ℝ. In this paper we compare these families in porosity terms. In particular, we obtain strengthening of some recent results by J. Hejduk, A. Loranty, R. Wiertelak.

An algebra of absolutely continuous functions and its multipliers

The aim of this paper is to study the algebraACp of absolutely continuous functionsf on [0,1] satisfying f(0) = 0,f ’ ∈ Lp[0, 1] and the multipliers ofACp.

On the predictability of discrete dynamical systems II

Given a metrizable compact topological n n -manifold X X with boundary and a metric d d compatible with the topology of X X , we prove that “most” continuous functions f : X → X f : X \to X are non-sensitive at “most” points of X X but are sensitive at every point of an infinite set which is dense in the set of all periodic points of f f . We also establish some results concerning sets of periodic points and non-wandering points.

A number theoretic estimate of Davenport from a functional analytic viewpoint

We consider the problem of the identification of continuous functionsf∶[0, 1]→R, by means of the sums $$\sum\limits_{a = 1}^n {f\left( {\frac{a}{n}} \right)} $$ . This is not possible, in general, but we prove that it may be the case under auxiliary conditions. We also study the behaviour of a well known exceptional function.

On Some Degenerate Differential Operators on Weighted Function Spaces

We deal with the degenerate differential operatorAu(x)≔α(x)u″(x) (x≥0) defined for everyu∈W20∩C2(]0,+∞[) satisfying limx→0+α(x)u″(x)=limx→+∞(α(x)/(1+x2))u″(x)=0. HereW20denotes the Banach space of all continuous functionsf:[0,+∞[→R such thatf(x)/(1+x2) vanishes at infinity, endowed with the weighted norm ‖f‖2≔supx≥0(|f(x)|/(1+x2)) (f∈W20). Moreover, we assume that the function α is continuous and positive on [0,+∞[, it is differentiable at 0, and satisfies the inequalities 0<α0≤α(x)/x≤α1(x>0) for suitable constants α0and α1. We show that the operatorAgenerates aC0-semigroup (T(t))t≥0of positive operators onW20. Moreover, we prove that everyT(t) can be represented as a limit of powers of suitable discrete-type positive linear operators that are constructed by means of the coefficient α.

Summation rational operators of the Jackson type

Summation rational positive operatorsD4n−n(x; f) of the Jackson type are constructed on the real axis. The corresponding approximations of continuous functionsf onℝwith coinciding finite limits limx→−∞f(x) and limx→+∞f(x) are estimated.

Some functional inequalities and their Baire category properties

In the paper we consider, from a topological point of view, the set of all continuous functionsf:I → I for which the unique continuous solutionψ:I − [0, ∞) ofψ(f(x)) ≤ β(x, ψ(x)) andα(x, ψ(x)) ≤ ψ(f(x)) ≤ β(x, ψ(x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equationϕ(f(x)) = g(x, ϕ(x)).

On the theorems of Y. Mibu and G. Debs on separate continuity

Using a game‐theoretic characterization of Baire spaces, conditions upon the domain and the range are given to ensure a fat set C(f) of points of continuity in the sets of type X × {y}, y ∈ Y for certain almost separately continuous functions f : X × Y → Z. These results (especially Theorem B) generalize Mibu′s. First Theorem, previous theorems of the author, answers one of his problems as well as they are closely related to some other results of Debs [1] and Mibu [2].

Uniformly maldistributed sequences in a strict sense

It is shown that the following three limits $$\begin{gathered} \mathop {\lim }\limits_{n \to \infty } \frac{1}{{N^2 }}\sum\limits_{m,n = 1}^N {|x_m - x_n | = 0,} \hfill \\ \mathop {\lim \inf }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {x_n = 0,\mathop {\lim \sup }\limits_{N \to \infty } } \frac{1}{N}\sum\limits_{n = 1}^N {x_n = 1} \hfill \\ \end{gathered} $$ are a necessary and sufficient condition for the given sequence ω=(xn) n =1/∞ ⊄[0, 1] to have its only distribution functions be all one-jump functions. As an application, such sequences can also be used in deriving estimates of maxf for continuous functionsf defined in [0, 1].

Theory of optical identification of submicron particles in high atmosphere

Interaction between planetary atmospheres and small bodies is connected with radiation effects. Submicron particles in the Earth's upper atmosphere strongly influence the scattering of the shortwave solar radiation. Based on the mutual connection between the environmental and radiation field structures it is possible to determine the physical characteristics of the particle set in this environment. Generaly, the diffused radiation field in the real atmosphere is given by a sum of elementary and multiple scattering components. Solving the inverse problems always leads to complicated integral equations. A major part of the diffused radiation field in the upper atmosphere is due to the first order scattering. The paper presents a new method for determination of the effective complex refractive index and size distribution of the particles based on the radiance data. The solution of integral equations is to be found in the space of quadratically integrable and continuous functionsf ∈L2.

On the Durrmeyer-type modification of some discrete approximation operators

In [10], for continuous functionsf from the domain of certain discrete operatorsLn the inequalities are proved concerning the modulus of continuity ofLnf. Here we present analogues of the results obtained for the Durrmeyer-type modification\(\tilde L_n \) ofLn. Moreover, we give the estimates of the rate of convergence of\(\tilde L_n f\) in Holder-type norms

On some local topological semigroups

We determine all continuous functionsf, defined on a real intervalI with 0∈ I, taking values in ℝ and such that the operationAf:I × I → ℝ given by $$A_f (x,y) = xf(y) + yf(x)$$ is locally associative, i.e., for allx, y, z ∈ I, ifAf(x, y) ∈ I andAf(y, z) ∈ I, thenAf(Af(x, y), z) = Af(x, Af(y, z)). The problem leads to the following functional equation $$f(A_f (x,y)) = f(x)f(y) + cxy$$ wherec is a real constant andx, y ∈ I are such thatAf(x, y) ∈ I. We solve this equation generalizing thus some earlier results obtained by N. Brillouet and J. Dhombres [3] who solved it in the caseI = ℝ andc = 0, as well as those obtained by P. Volkmann and H. Weigel [7] who were dealing with an equivalent form of this equation in the caseI = ℝ andc >; 0. Some partial results concerning the problem can be found in our paper [6].

Effective Łojasiewicz inequalities in semialgebraic geometry

The main result of this paper can be stated as follows: letV ⊂ ℝn be a compact semialgebraic set given by a boolean combination of inequalities involving only polynomials whose number and degrees are bounded by someD > 1. LetF, G∈∝[X1,⋯, Xn] be polynomials with degF, degG ≦ D inducing onV continuous semialgebraic functionsf, g:V→R. Assume that the zeros off are contained in the zeros ofg. Then the following effective Łojasiewicz inequality is true: there exists an universal constantc1∈ℕ and a positive constantc2∈∝ (depending onV, f,g) such that\(|g(x)|^{D^{c_1 .n} } \leqq c_2 \cdot |f(x)|\) for allx∈V. This result is generalized to arbitrary given compact semialgebraic setsV and arbitrary continuous functionsf,g:V → ∝. An effective global Łojasiewicz inequality on the minimal distance of solutions of polynomial inequalities systems and an effective Finiteness Theorem (with admissible complexity bounds) for open and closed semialgebraic sets are derived.

On a generalized asymptotic mean value property

In this paper we study the spaces of continuous functionsf on ℝ2 satisfying $$\int_0^{2\pi } {f(z + \xi e^{r\theta } )} d\mu (\theta ) \to f(z)$$ (4) uniformly on compact sets as ξ → ∞, where μ is a finite complex Borel measure supported on the unit circle {z ∈ ℂ: ¦z¦=1} with ∫2π0 dμ(θ)=1. Our main result is the following:

The continuous solution of a functional equation of Abel

The functional equationϕ(x) + ϕ(y) = ψ(xf(y) + yf(x)) (1) for the unknown functionsf, ϕ andψ mapping reals into reals appears in the title of N. H. Abel's paper [1] from 1827 and its differentiable solutions are given there. In 1900 D. Hilbert pointed to (1), and to other functional equations considered by Abel, in the second part of his fifth problem. He asked if these equations could be solved without, for instance, assumption of differentiability of given and unknown functions. Hilbert's question was recalled by J. Aczel in 1987, during the 25th International Symposium on Functional Equations in Hamburg-Rissen. In particular Aczel asked for all continuous solutions of (1). An answer to his question is contained in our paper. We determine all continuous functionsf: I → ℝ,ψ: Af(I × I) → ℝ andϕ: I → ℝ that satisfy (1). HereI denotes a real interval containing 0 andAf(x,y) := xf(y) + yf(x), x, y ∈ I. The list contains not only the differentiable solutions, implicitly described by Abel, but also some nondifferentiable ones.

Direct and inverse theorems for Bernstein polynomials in the space of riemann integrable functions

Equivalence theorems concerning the convergence of the Bernstein polynomialsB n f are well known for continuous functionsf in the sup-norm. The purpose of this paper is to extend these results for functionsf, Riemann integrable on [0, 1], We have therefore to consider the seminorm $$\left\| f \right\|_\delta : = \int_0^1 {\mathop {\sup }\limits_{y \in U_\delta (x)} |f(y)|dx,U_\delta (x): = \{ y \in [01]:|x - y \leqslant \delta \} ,}$$ depending also on the increment δ>0. In terms of correspondingτ-moduli direct and inverse theorems are established, e.g., $$\left\| f \right\|_\delta : = \int_0^1 {\mathop {\sup }\limits_{y \in U_\delta (x)} |f(y)|dx,U_\delta (x): = \{ y \in [01]:|x - y \leqslant \delta \} ,}$$ Moreover, a similar result is obtained for the error functional appearing in the recently introduced definition of Riemann convergence.

On the martingale problem for Banach space valued stochastic differential equations

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet≥0 and the continuous functionsf:[0,t] →E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ∫ 0 . Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ∫ 0 . Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY =∫ 0 . Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.

Some dual statements concerning Wiener measure and Baire category

This paper deals with the duality between Wiener measure and Baire category on C 0 1 C_0^1 , the set of continuous functions f : [ 0 , 1 ] → [ 0 , 1 ] f:[0,1] \to [0,1] endowed with the supremum norm. We prove that some properties shared by a residual set of continuous functions, e.g. the typical level set structure, and the existence of periodic points of order 3 hold a.e. with respect to the Wiener measure.