Unbounded functions with almost periodic differences

The structure of continuous real-valued functions F(t) on the real line, such that for any fixed y the difference F(t+y)–F(t) is an almost periodic Bohr function, is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1409–1413, October, 1991.     

On mean periodic functions

Summary. Mean periodic functions play an important role in the theory of functional equations. Based on fundamental results on spectral synthesis it is possible to introduce a Fourier type transformation for mean periodic functions on the real line. This transformation can be used to solve convolution-type functional equations. It turns out that this transformation has a close relation to some kind of conditional expectation.     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

Approximation of almost periodic functions by periodic ones

It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on =(+).     

Modified Taylor approximation of functions with periodic behaviour

We approximate a function with periodic behaviour by means of a small modification of its Taylor polynomial. This modification is based on the work of Scheifele and will simplify the construction of special numerical methods for differential equations with near periodic solutions.     

Quadrature formulae with free nodes for periodic functions

Summary. The problem of existence and uniqueness of a quadrature formula with maximal trignonometric degree of precision for 2-periodic functions with fixed number of free nodes of fixed different multiplicities at each node is considered. Our approach is based on some properties of the topological degree of a mapping with respect to an open bounded set and a given point. The explicit expression for the quadrature formulae with maximal trignometric degree of precision in the 2-periodic case of multiplicities is obtained. An error analysis for the quadrature with maximal trigonometric degree of precision is given. Received April 16, 1992/Revised version received June 21, 1993     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that f is nondecreasing on [y _i ,y _i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \parallel f - P_n \parallel \leqslant c(s) \omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω₂(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Coconvex approximation of periodic functions

The Jackson inequality relates the value of the best uniform approximation E _n (f) of a continuous 2π-periodic function f: ℝ → ℝ by trigonometric polynomials of degree ≤ n − 1 to its third modulus of continuity ω ₃(f, t). In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity on [−π, π) only at every point of a fixed finite set consisting of an even number of points are approximated by polynomials coconvex to them. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 29–43, January, 2007.     

On periodic matrix-valued Weyl-Titchmarsh functions

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl-Titchmarsh matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension which generate a periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.     

Copositive approximation of periodic functions

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y _i ∈ Y:= {y _i } _i∈ℤ such that for x ∈ [y _i , y _i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P _n of order ≦ n, changing its sign at the same points y _i ∈ Y as f, and