Continuous Functionsf Research Articles

Approximation by Nörlund means of double fourier series to continuous functions in two variables

We study the rate of uniform approximation by Norlund means of the rectangular partial sums of double Fourier series of continuous functionsf(x, y), 2π-periodic in each variable. The results are given in terms of the modulus of symmetric smoothness defined by $$\begin{gathered} \omega _2 \left( {f,\delta _1 ,\delta _2 } \right) = \mathop {\sup }\limits_{x,y} \mathop {\sup }\limits_{\left| u \right| \leqslant \delta _1 ,\left| v \right| \leqslant \delta _2 } \left| {f\left( {x + u,y + v} \right)} \right. + f\left( {x + u,y - v} \right) + f\left( {x - u,y + v} \right) \hfill \\ + \left. {f\left( {x - u,y - v} \right) + 4f\left( {x,y} \right)} \right| for \delta _1 ,\delta _2 \geqslant 0. \hfill \\ \end{gathered} $$ As a special case we obtain the rate of uniform approximation to functionsf(x,y) in Lip({α, β}), the Lipschitz class, and inZ({α, β}), the Zygmund class of ordersα andβ, 0<α,β ≤ l, as well as the rate of uniform approximation to the conjugate functions\(\tilde f^{(1,0)} (x,y), \tilde f^{(0,1)} (x,y)\) and\(\tilde f^{(1,1)} (x,y)\).

Fractal functions and interpolation

Let a data set {(x i,y i) ∈I×R;i=0,1,⋯,N} be given, whereI=[x 0,x N]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x i)=y i fori e {0,1,⋯,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.

Dual spaces of C∞ (X)

ForX a locally compact Stonian Space, letC ∞ (X) denote the universally complete Riesz space of all extended-real-valued continuous functionsf onX for which {x∈X| |f (x)|=∞} is nowhere dense. In this paper the dual spaces ofC ∞ (X) (i.e. the spaces of order bounded; of σ-order continuous; of order continuous linear forms onC ∞ (X), and the extended order dual ofC ∞ (X) denote here byC ∞ (X)ρ (introduced by W.A.J. Luxemburg and J.J. Masterson)) are characterized. It is shown thatC ∞ (X)ρ can be identified in a canonical way with the inductive limitM q (X) of the Riesz spaces of all normal Radon measures defined on the dense open subsets ofX. More generally, ifY is a locally compact space thenM q (Y) is the extended order dual of the inductive limit of the Riesz spaces of all real-valued continuous functions defined on the dense open subsets ofY. IfX is locally compact and hyperstonian, then it is proved thatC ∞ (X) andC ∞ (X)ρ are isomorphic, and a criterion forC ∞ (X)ρ to be the universal completion of the space of order continuous linear forms onC ∞ (X) is given.

The typical structure of the sets {𝑥:𝑓(𝑥)=ℎ(𝑥)} for 𝑓 continuous and ℎ Lipschitz

Let R R be the space of real numbers and C C the space of continuous functions f : [ 0 , 1 ] → R f:[0,1] \to R with the uniform norm. Bruckner and Garg prove that there exists a residual set B B in C C such that for every function f ∈ B f \in B there exists a countable dense set Λ f {\Lambda _f} in R R such that: for λ ∉ Λ f \lambda \notin {\Lambda _f} the top and bottom levels in the direction λ \lambda of f f are singletons, in between these levels there are countably many levels in the direction λ \lambda of f f that consist of a nonempty perfect set together with a single isolated point, and the remaining levels in the direction λ \lambda of f f are all perfect; for λ ∈ Λ f \lambda \in {\Lambda _f} the level structure in the direction λ \lambda of f f is the same except that one (and only one) of the levels has two isolated points instead of one. In this paper we show that the analogue of the above theorem holds: if we replace the family of straight lines { λ x + c } \{ \lambda x + c\} by a 2 2 -parameter family H H that is almost uniformly Lipschitz; and if we replace { λ x + c } \{ \lambda x + c\} by a homeomorphical image of a certain 2 2 -parameter family H H that is almost uniformly Lipschitz.

Connected projections of blocking sets of 𝐼ⁿ×𝑀

Sufficient conditions are given for each minimal blocking set K of I n × M {I^n} \times M to have the closure of its projection, p ( K ) p(K) , into I n {I^n} connected. The construction of some examples of almost continuous functions f : I n → M f:{I^n} \to M in the literature depends on knowing each p ( K ) ¯ \overline {p(K)} is connected or perfect.

Product-integration with the Clenshaw-Curtis and related points

This paper is concerned with the theoretical properties of a productintegration method for the integral $$\int\limits_{ - 1}^1 {k(x)f(x)dx}$$ , wherek is absolutely integrable andf is continuous. The integral is approximated by $$\sum\limits_{i = 0}^n {w_{ni} f(x_{ni} )}$$ , where the points are given byx ni =cos(i?/n, 0?i?n, and where the weightsw ni are chosen to make the rule exact iff is any polynomial of degree ?n. The principal result is that ifk?L p [?1, 1] for somep>1, then the rule converges to the exact result asn?? for all continuous (or indeed R-integrable) functionsf, and moreover that the sum of the absolute values of the weights converges to the least possible value, namely $$\int\limits_{ - 1}^1 {|k(x)|dx}$$ . A limiting expression for the individual weights is also obtained, under certain assumptions. The results are exteded to other point sets of a similar kind, including the classical Chebyshev points.

On the numerical evaluation of singular integrals

Product-integration rules of the form ∫ −1 1 k(x)f(x)dx≅Σ =1 w ni f(x ni ) are studied, with the points {w ni } chosen to be the zeros of certain orthogonal polynomials, and the weights {w ni } chosen to make the rule exact iff is any polynomial of degree less thann. If, in particular, the points are the Chebyshev points, and ifk eL p [−1, 1] for somep>1, then it is shown that the rule converges to the exact result for all continuous functionsf. With this choice of points, the practical application of the rule is shown to be straightforward in many cases, and to yield satisfactory rates of convergence. The casek(x)=|λ−x|α, α>−1, is studied in detail. Results of a similar, but weaker, kind are also obtained for other choices of the points {x ni }.

Optimization of lipschitz continuous functions

This paper contains basic results that are useful for building algorithms for the optimization of Lipschitz continuous functionsf on compact subsets of En. In this settingf is differentiable a.e. The theory involves a set-valued mappingxźźźf(x) whose range is the convex hull of existing values of źf and limits of źf on a closedź-ball,B(x, ź). As an application, simple descent algorithms are formulated that generate sequence {xk} whose distance from some stationary set (see Section 2) is 0, and where {f(xk)} decreases monotonously. This is done with the aid of anyone of the following three hypotheses: Forź arbitrarily small, a point is available that in arbitrarily close to:(1)the minimizer off onB(x, ź),(2)the closest point inźźf(x) to the origin,(3)ź(h) ź źźf(x), where [ź(h), h] = max {[ź, h]: ź ź źźf(x)}. Observe that these three problems are simplified iff has a tractable local approximation. The minimax problem is taken as an example, and algorithms for it are sketched. For this example, all three hypotheses may be satisfied. A class of functions called uniformly-locally-convex is introduced that is also tractable.

The completeness of systems of functions of the Mittag-Leffler type for weighted uniform approximation in a complex domain

For a givenρ(1/2 <ρ < + ∞) let us set L ρ = {z: |arg z| = π/(2ρ)} and assume that a real valued measurable function ϕ(t) such that ϕ(t) ≥ 1(t ∈ L ρ ) and $$\mathop {\lim }\limits_{|t| \to + \infty } \varphi (t) = + \infty (t \in L_\rho )$$ is defined on L ρ . Let C ϕ (L ρ ) denote the space of continuous functionsf(t) on L ρ such that $$\lim \tfrac{{f(t)}}{{\varphi (t)}} = 0$$ , where the norm of an elementf is defined as: $$\parallel f\parallel = \mathop {\sup }\limits_{t \in L_\rho } \tfrac{{|f(t)|}}{{\varphi (t)}}$$ . In this note we pose the question about the completeness of the system of functions of the Mittag-Leffler type {Eρ(ut; μ)} (μ ≥ 1, 0 ≤ u ≤a) or, what is the same thing, of the system of functions $$p(t) = \int_0^a {E_\rho (ut;\mu )d\sigma (u)} $$ in C ϕ (L ρ ). The following theorem is proved: The system of functions of the Mittag-Leffler type is complete in C ϕ (L ρ ) if and only if sup |p(z)| ≡ +∞, z ∈ L ρ , where the supremum is taken over the set of functions p(t) such that ∥p(t) (t + 1)−1 ∥ ≤ 1.

On best approximation in classes of periodic functions defined by integrals of a linear combination of absolutely monotonic kernels

In the metrics C and L we solve the problem of best approximation by trigonometric polynomials in classes of continuous periodic functionsf(x) of the form $$f(x) = \frac{1}{\pi }\int_0^{2\pi } {K(t)} \varphi (x - t)dt,$$

Weighted approximations by means of entire functions of exponential type

We study the approximation of continuous real functionsf(x)=0((1+|x|)σ) (|x|→∞,σ≥0) by means of entire functions of exponential type in some metric of Hausdorff type. We generalize a theorem due to N. I. Akhiezer concerning uniform weighted approximations.

Γ-compact maps on an interval and fixed points

We characterize the Γ \Gamma -compact continuous functions f : X → X f:X \to X where X X is a possibly-noncompact interval. The map f f is called Γ \Gamma -compact if the closed topological semigroup Γ ( f ) \Gamma (f) generated by f f is compact, or equivalently, if every sequence of iterates of f f under functional composition ( ∗ ) (\ast ) has a subsequence which converges uniformly on compact subsets of X X . For compact X X the characterization is that the set of fixed points of f ∗ f f\ast f is connected. If X X is noncompact an additional technical condition is necessary. We also characterize those maps f f for which iterates of distinct orders agree ( Γ ( f ) \Gamma (f) finite) and state a result on common fixed points of commuting functions when one of the functions is Γ \Gamma -compact.

The topologies of separate continuity. I

A functionf( , ) of two variables is said to beseparately continuousif for each fixedyand each fixedx, the functionsf(,y) andf(x,) are continuous functions of one variable. Separately continuous functions seem to be playing a larger rôle in analysis than formerly. In (1) there is an account of the theory of semigroups in which the multiplication is separately continuous, and in (9) various analytic theorems are obtained by the study of separately continuous functionsf(s,t), wheretranges over a setT, andsranges over a set of functions onT.

Linear difference operators on periodic functions

Let p > 0 and B the Banach space of continuous functionsf: R1--RI of period p, with Ifll =max{ If(x)I ; 0 0 for all x, and let t be a real number. Define the bounded linear operator L: B ->B by Lf (x) =f (x+t) -a(x)f(x). We shall obtain results concerning the solutions in B of the equation Lf =g. We say that L is regular if it is one-to-one and onto B, and that it is singular otherwise. The limit limEO(1/n) ZkoIf(x+kt), where fEB, will be denoted by otf(x). If t/p is irrational, the existence of the limit follows from the fact that for every x the sequence xk x + kt (mod p) is uniformly distributed in [0, p], so that otf(x) (1/p)fPf(x)dx. If t/p is rational, say qt = rp, with q and r integers and q>O, then otf(x) -(1/q) El-'f(x+kt). Note that otf f(x) =o-tf(x) =otf(x+t). The case a(x) = 1, i.e., the equation