For more than a century, the constructive description of functional classes in terms of the possible rate of approximation of its functions by means of functions chosen from a certain set remains among the most important problems of approximation theory. It turns out that the nonuniformity of the approximation rate due between the points of the domain of the approximated function is substantial. For instance, it was only in the mid-1950s that it was possible to constructively describe Holder classes on the segment [–1; 1] in terms of the approximation by algebraic polynomials. For that particular case, the constructive description requires the approximation at neighborhoods of the segment endpoints to be essentially better than the one in a neighborhood of its midpoint. A possible approximation quality test is to find out whether the approximation rate provides a possibility to reconstruct the smoothness of the approximated function. Earlier, we investigated the approximation of classes of smooth functions on a countable union of segments on the real axis. In the present paper, we prove that the rate of the approximation by the entire exponential-type functions provides the possibility to reconstruct the smoothness of the approximated function, i.e., a constructive description of classes of smooth functions is possible in terms of the specified approximation method. In an earlier paper, that result is announced for Holder classes, but the construction of a certain function needed for the proof is omitted. In the present paper, we use another proof; it does not apply the specified function.
In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.
The problem to approximate functions continuous on subsets of the real line by entire functions has a long history that started from the Jackson–Bernstein theorem on the approximation of 2- periodic functions by trigonometric polynomials naturally treated as exponential-type entire functions. In this paper, we deal with the problem referring to the concept of this theorem describing classes of functional spaces via the rate of their possible approximation by entire functions. A key example is the Bernstein theorem describing the class of bounded functions from Holder classes over the whole axis by exponential-type entire functions. The key point is that the approximation rate at a neighborhood of the segment edge exceeds the one that originally appeared in the theory of approximation functions from Holder classes on segments (this allows us to coordinate the direct and inverse theorems for that case, i.e., to recover the holder smoothness from the approximation rate in the said scale). In the present paper, we present a direct theorem on the possibility of a prescribed-rate approximation of functions from Holder classes on countable unions of segments by entire functions. Earlier, such approximations were not considered. Also, we provide general definitions and important lemmas used for further constructing approximating functions. In the second part of the work, we provide a proof of the direct theorem. In our further papers, to obtain a constructive description of the smoothness class by means of the approximation rate, we will prove the corresponding inverse theorem. Usually, to deduce such assertions, one needs a fact similar to the Bernstein theorem on the estimate of the norm of an entire function via the norm of the function itself. In our case, we need an assertion similar to the Akhiezer–Levin theorem estimating an entire function on the axis via its values on a subset of the axis.
In this article a problem of sorting cars in a freight train is considered and its mathematical formulation is given. The sorting is executed on a so called sorting yard hump. The yard hump consists of a small hump with a tree of railroads starting on the top of the hump and branching out to k rail ends with controls that allow to direct a car moving from the top to a given end. On each run, the train is distributed to the end paths and after that collected back from fragments on the paths. The maximum number of runs of the whole train to sort it and overall complexity of the algorithm are found. Some features of this method are similar to the main idea of the well know procedure of sorting perfocards in the classical Hollerith set of hardware. The initial data is a permutation of the set 1: n, where each car is associated with its position in the desired train sequence. In solution we use a partition of the permutation of 1: n to monotone segments. For example, the permutation 3, 8, 2, 6, 1, 4, 5, 7 with n = 8 is divided to segments (3, 2, 1), (4), (6, 5), (8, 7). These segments are easier defined in terms of the permutation inverse to the given one. It is treated as a string consisting of numbers, and each part of the partition corresponds to a maximal monotonically decreasing set of numbers in the list of elements. In our example the inverse permutation 5, 3, 1; 6; 7, 4; 8, 2 is divided to 4 substrings by separators “;”. Let p be the number of parts in this partition, and k is the number of roads in the hump. Then the number of runs of the train through the hump to get n, n–1, …, 2, 1 does not exceed ┌logkp┐ ≤ ┌logkn┐, and this estimation is reached by the proposed algorithm.
Generalization of one of the classical Rцssler systems are considered. It is shown that, to estimate the dimensions of the attractors of these systems, Lyapunov functions can be effectively used. By using these functions, estimates of the Lyapunov dimensions of the attractors of generalized Rцssler systems are obtained. For the local Lyapunov dimensions of the attractors of these systems, exact expressions are given. In the limit case, the coincidence of the topological, Hausdorff, fractal, and Lyapunov dimensions of attractors is proved. It is shown that, for standard values of Rцssler parameters, the values given by expressions for local Lyapunov dimensions at zero coincide with those obtained in numerical experiments.