Cutting sets of continuous functions on the unit interval

For a function $f:[0,1]\to \mathbb{R}$, we consider the set $E\left(f\right)$ of points at which $f$ cuts the real axis. Given $f:[0,1]\to \mathbb{R}$ and a Cantor set $D?[0,1]$ with $\{0,1\}?D$, we obtain conditions equivalent to the conjunction $f\in C[0,1]$ (or $f\in C^{\infty }[0,1]$) and $D?E\left(f\right)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E\left(f\right)$ is a closed nowhere dense subset of $f^{?1}\left[\left\{0\right\}\right]$. Additionally, if $Int{f}^{?1}\left[\left\{0\right\}\right]=0?$, each $x\in \{0,1\}\cap E\left(f\right)$ is an accumulation point of $E\left(f\right)$. Our main result states that, for a closed nowhere dense set $F?[0,1]$ with each $x\in \{0,1\}\cap F$ being an accumulation point of $F$, there exists $f\in C^{\infty }[0,1]$ such that $F=E\left(f\right)=f^{?1}\left[\left\{0\right\}\right]$.     

Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval

The matrix-valued Weyl-Titchmarsh functions M(λ) of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(λ)) and the residues of M(λ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N×N Weyl-Titchmarsh functions) corresponding to N×N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.     

Dimension prints for continuous functions on the unit square

Acta Mathematica Hungarica - Recently, we have proved that the rectangular pointwise Lipschitz regularity of a continuous function on the unit square is directly related with the local suprema of...     

On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

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This paper was written during the author's scholarship at the State University of Odessa in the USSR.     

Convergence theorems for continuous functions on an arbitrary interval

In this paper, we introduce a new iterative scheme for finding a fixed points of continuous functions on an arbitrary interval. The convergence theorems are also established. Further, the numerical examples comparing with Mann, Ishikawa and Noor iterations are demonstrated. Main results generalize and unify the corresponding ones announced in the literature.     

A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval

A method for construction of CF approximants in some cases of rational approximation of a rational function f on the unit disk and on the unit interval is presented. The inverted square root of the greatest positive eigenvalue and a corresponding eigenvector of an eigenvalue problem defined by the coefficients of f gives the solution.     

Estimates for the Lebesgue functions and the Nevai formula for the sinc-approximations of continuous functions on an interval

We obtain some upper and lower estimates for the sequences of the Lebesgue functions and constants of the Whittaker operators

$L_n (f,x) = \sum\limits_{k = 0}^n {\frac{{\sin (nx - k\pi )}}{{nx - k\pi }}} f\left( {\frac{{k\pi }}{n}} \right)$

for continuous functions. We give an analog of Nevai’s formula for the Lagrange-Chebyshev and Lagrange-Laguerre interpolation polynomials for the operators under consideration. Its “local” version is established.

     

There is no continuous projection from the measurable functions on the square onto the measurable functions on the interval

It is proved that the space of functions constant on vertical lines is not complemented in the space of all measurable functions on the unit square (with the topology of convergence in measure). The analogous results is proved for the space of all measurable functions on the product of two probability spaces, one of which is atomless. This research was partially supported by National Science Foundation grant N.S.F. GP 28577.     

Step functions from the half-open unit interval into a topological space

The set of continuous-from-the-right step functions from the half-open unit interval[0, 1[into a topological space X is denoted by X¹. Elsewhere a topology has been defined which makes X¹ a contractible, locally contractible space with the subspace of constant functions being homeomorphic to X. When X has a bounded metric ?, the topology of X¹ may be described by the metric $\text{d>(f,g)=}\text{∫}\text{0}\text{1}\text{ρ(f(t),g(t))dt}$.It is shown here that if X is separable, then X¹ is separable and if X satisfies the first (or second) axiom of countability, then X¹ satisfies it too. In contrast, it is shown that properties such as normality do not extend from X to X¹. This follows from the main result: X¹ is homeomorphic to its square, and thus contains a copy of X×X (which is closed when X is Hausdorff). The final theorem states that if X has at least two points then X¹ is not complete metrizable.     

Localization principle for expansions of generalized functions with respect to the eigenfunctions of the Sturm-Liouville operator on a finite interval

For linear methods of summation of the expansions of generalized functions into a series with respect to the eigenfunctions of a Sturm-Liouville operator one establishes conditions under which the Riemann localization principle holds.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 43, No. 5, pp. 703–706, May, 1991.     

Lototsky-Schabl operators on the unit interval

In questo lavoro si studiano versioni uni-dimensionali degli operatori di Lototsky-Schnabl associati a proiezioni di Altomare. Si discutono la conservazione della monotonia, la convessità, le classi di Lipschitz ed il primo modulo di continuità. Si dimostra pure che, in condizioni molto generali, questi operatori convergono, in un senso opportuno, all'operatore classico di Szász, e si danno allo stesso tempo stime dell'ordine di approssimazione. Alcune delle proprietà si dimostrano utilizzando rappresentazioni probabilistiche degli operatori in termini di opportuni processi stocastici.     

Absolutely continuous Hardy–Sobolev functions in the unit disc

Let f(z) be an analytic function defined in the unit disc whose fractional derivative of order belongs to H^p, 0<p1. We show that as a consequence of a monotonicity condition on the decay of the Taylor coefficients, it is possible to improve the usual radial boundary growth estimate for H^p functions by a logarithmic factor. As a consequence we show that under certain regularity conditions imposed on the decay and oscillations of the absolute values of the function's Taylor coefficients, it is possible to estimate the function's modulus of continuity and modulus of absolute continuity and that a consequence of this is that as p→0, these functions will be generally smoother. Examples are also given of Hardy–Sobolev functions having modulus of absolute continuity different than modulus of continuity.     

Mixing examples in the class of piecewise monotone and continuous maps of the unit interval

A process (T, P) is said to have the “ ” property if there is a uniform, positive lowerbound δ on the separation between theT-P names of (almost) every pair of pointsx≠y. A finite group rotation with partition into distinct points provides a trivial example. Given any process having the property we show that there exists a Bernoulli shiftB so thatT×B is measurably isomorphic to the natural extension of a piecewise monotone, continuous, and expanding map of the unit interval. This construction is applied to produce interval maps which are ergodic but not weak-mixing, weak-mixing but not mixing, and mixing but not exact with respect to their unique absolutely continuous invariant measures, in contrast with the results known for piecewiseC ^1+∈ expansive interval maps. In obtaining these examples we identify a number of nontrivial classes of automorphismsT which admit processes having the property. Supported by NSERC grant OGP0046586 90.     

Estimates for the equiconvergence rate of spectral expansions on an interval

We study the convergence rate of biorthogonal expansions of functions in series in systems of root functions of a broad class of second-order ordinary differential operators on a finite interval. The above-mentioned expansions are compared with the expansions of the same functions in trigonometric Fourier series in an integral or uniform metric on any interior compact set of the basic interval and on the entire interval. We prove the dependence of the equiconvergence rate of the expansions in question on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.     

Asymptotic expansions of logarithmic-exponential functions

The aim of this paper is to study the asymptotic expansion of real functions which are finite compositions of globally subanalytic maps with the exponential function and the logarithmic function. This is done thanks to a preparation theorem in the spirit of those that exist for analytic functions (Weierstrass) or subanalytic functions (Parusinśki). The main consequence is that logarithmic-exponential functions admit convergent asymptotic expansion in the scale of real power functions. We also deduce a partial answer to a conjecture of van den Dries and Miller. Received: 19 March 2002     

Quadratic expansions of spectral functions

A function, F, on the space of n×n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F(A)=F(UAU^T) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on : those that are invariant under arbitrary swapping of their arguments. In this paper we show that a spectral function has a quadratic expansion around a point A if and only if its corresponding symmetric function has quadratic expansion around λ(A) (the vector of eigenvalues). We also give a concise and easy to use formula for the ‘Hessian' of the spectral function. In the case of convex functions we show that a positive definite ‘Hessian' of f implies positive definiteness of the ‘Hessian' of F.     

Asymptotic expansions of generalized functions

The definition is formulated of the asymptotic expansion of a generalized function depending on a parameter. A number of theorems are proved about the properties of asymptotic expansions and operations on them, in particular, theorems on differentiation and integration. For generalized functions of the formf (x)e^ixt,f (x) S', t ± the relation is investigated between the singularity carrierf and the carrier of coefficient functionals.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 131–138, August, 1972.