Continuous 2π-periodic Functions Research Articles

Comonotone approximation of periodic functions

Let C˜ be the space of continuous 2π-periodic functions f, endowed with the uniform norm ‖f‖≔maxx∈R|f(x)|, and denote by ωm(f,t), the mth modulus of smoothness of f. Denote by C˜r, the subspace of r times continuously differentiable functions f∈C˜, and let Tn, be the set of trigonometric polynomials Tn of degree <n. If f∈C˜r, has 2s, s≥1, extremal points in (−π,π], denote by En(1)(f)≔infTn∈Tn:f′Tn′≥0‖f−Tn‖, the error of its best comonotone approximation. We prove, that if f∈C˜r, then for either m=1, or m=2 and r=2s, or m∈N and r>2s, En(1)(f)≤c(m,r,s)nrωm(f(r),1/n),n≥1,where the constant c(m,r,s) depends only on m, r and s.

Comonotone approximation by hybrid polynomials

Let f(x):=g(x)+ax, where g∈C˜, the space of continuous 2π-periodic functions and a∈R. Denote ‖g‖:=maxx∈R⁡|g(x)|, and let ω(f,t), denote the modulus of continuity of f. Let Tn be the set of trigonometric polynomials Tn of degree <n. We call Qn(x):=Tn(x)+ax, a∈R, a hybrid polynomial.If f is monotone, then g was called, by Salem and Zygmund, of monotone type.If f has an even number of extremal points in (−π,π], then we estimateinf⁡{‖f−Qn‖:Qns.t.f′(x)Qn′(x)≥0, a.e. in R}, the error of its best comonotone approximation, in the uniform norm, by hybrid polynomials.We obtain Jackson-type estimates for the approximation of f by hybrid polynomials for a wide class of functions f. We also show cases where such estimates are invalid.

An operator version of the Korovkin theorem for continuous periodic functions

The following operator version of the Korovkin theorem for continuous 2π-periodic functions is proved: Let Ln:C2π(R)→C2π(R) be a sequence of linear positive operators and U:C2π(R)→C2π(R) a linear positive operator such that [U(sin⁡)]2+[U(cos⁡)]2=[U(1)]2. If limn→∞⁡Ln(1)=U(1), limn→∞⁡Ln(sin⁡)=U(sin⁡), limn→∞⁡Ln(cos⁡)=U(cos⁡) all uniformly on R, then for every φ∈C2π(R) we have limn→∞⁡[Ln(φ)U(1)−Ln(1)U(φ)]=0 uniformly on R. If in addition, U(1)(x)>0 for every x∈R then for every φ∈C2π(R), limn→∞⁡Ln(φ)=U(φ) uniformly on R. As application we give various concrete examples.

Fejér Sums and Fourier Coefficients of Periodic Measures

The Fejer sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejer sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous 2π-periodic function is uniquely determined by its sequence of Fejer sums at any two points whose difference is incommensurable with π.

Deviations of Fejer Sums and Rates of Convergence in the von Neumann Ergodic Theorem

It turns out that the deviations of the Fejer sums for continuous 2π-periodic functions and the rates of convergence in the von Neumann ergodic theorem can both be calculated using, in fact, the same formulas (by integrating the Fejer kernels). As a result, for many dynamical systems popular in applications, the rates of convergence in the von Neumann ergodic theorem can be estimated with a sharp leading coefficient of the asymptotic by applying S.N. Bernstein’s more than hundred-year old results in harmonic analysis.

Two-Sided Estimates for Some Functionals in Terms of the Best Approximations

Let C be the space of continuous 2π-periodic functions. For some integrals of the form $$ \underset{0}{\overset{\pi }{\int }}{\omega}_r\left(f,t\right)\Phi (t) dt, $$ where ω r (f, t) is the modulus of continuity of order r of a function f in C, two-sided bounds in terms of the best approximations by trigonometric polynomials are established.

Modified Steklov Functions

We obtain estimates for nonnegative semiadditive functionals on the space of continuous 2π-periodic functions defined in terms of Steklov functions.

Summability of Fourier series for almost-periodic functions on locally compact Abelian groups

Some results concerning the summability of Fourier series of continuous 2π-periodic functions are generalized for the case of almost-periodic functions defined on locally compact Abelian groups.

A new approach to the existence of quasiperiodic solutions for a class of semilinear Duffing-type equations with time-periodic parameters

Let \(q(t)\) be a continuous 2π-periodic function with \(\frac{1}{2\pi}\int_{0}^{2\pi}q(t)\,dt>0\). We propose a new approach to establish the existence of Aubry-Mather sets and quasi-periodic solutions for the following time-periodic parameters semilinear Duffing-type equation: $$x''+q(t) x+f(t,x)=0, $$ where \(f(t,x)\) is a continuous function, 2π-periodic in the first argument and continuously differentiable in the second one. Under some assumptions on the functions q and f, we prove that there are infinitely many generalized quasi-periodic solutions via a version of the Aubry-Mather theorem given by Pei. Especially, an advantage of our approach is that it does not require any high smoothness assumptions on the functions \(q(t)\) and \(f(t,x)\).

Approximation by Fourier means and generalized moduli of smoothness

The quality of approximation by Fourier means generated by an arbitrary generator with compact support in the spaces L p , 1 ≤ p ≤ +∞, of 2π-periodic pth integrable functions and in the space C of continuous 2π-periodic functions in terms of the generalized modulus of smoothness constructed froma 2π-periodic generator is studied. Natural sufficient conditions on the generator of the approximation method and values of smoothness ensuring the equivalence of the corresponding approximation error and modulus are obtained. As applications, Fourier means generated by classical kernels as well as the classical moduli of smoothness are considered.

Quasi-periodic solutions for second order differential equation with superlinear asymmetric nonlinearities and nonlinear damping term

In this paper, the existence of Aubry-Mather sets and quasi-periodic solutions of the oscillator $x''+\alpha{x^{+}}^{3}-\beta{x^{-}}^{3}+f(x)q(x')+\psi(x)=p(t) $ are established, where f, q, and ψ belong to the class $C^{1}(R) $ , p is a continuous 2π-periodic function. Under some assumptions on the parities of f, ψ, and p, we prove that there are infinitely many generalized quasi-periodic solutions by a result of Shuinee Chow and Mingliang Pei from the Aubry-Mather theorem of reversible mappings.

Constants in Jackson Type Inequalities for the Best Approximation of Periodic Functions Such that Some of Their Fourier Coefficients Vanish

We consider the class of continuous 2π-periodic functions such that some of their Fourier coefficients vanish. For such functions we study constants in a generalized Jackson theorem providing an estimate for the best approximation by trigonometric polynomials with the help of the moduli of continuity of an arbitrary order.

A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean

The main object of this paper is to prove a Korovkin type theorem for the test functions 1, cosx,sinx in the space C2π(R) of all continuous 2π-periodic functions on the real line R. Our analysis is based upon the statistical summability involving the idea of the generalized de la Vallée Poussin mean. We also investigate the rate of the de la Vallée Poussin statistical summability of positive linear operators in the space C2π(R). Finally, we provide an interesting illustrative example in support of our result.

Korovkin second theorem via statistical summability ( C , 1 )

Korovkin type approximation theorems are useful tools to check whether a given sequence ( L n ) n ≥ 1 of positive linear operators on C[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions, which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x and x 2 in the space C[0,1] as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of statistical summability (C,1) to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.MSC:41A10, 41A25, 41A36, 40A30, 40G15.

Approximation for periodic functions via weighted statistical convergence

Korovkin type approximation theorems are useful tools to check whether a given sequence (Ln)n⩾1 of positive linear operators on C[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1,x and x2 in the space C[0,1] as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of weighted statistical convergence to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.

On approximation of periodic functions by Fourier sums

Let L p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm $$ {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} $$ , and let C = L ∞ be the space of continuous 2π-periodic functions with the norm $$ {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| $$ . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that $$ P(f) \leqslant M\left\| f \right\| $$ for every f ∈ C. By $$ \sum\limits_{k = 0}^\infty {{A_k}} (f) $$ denote the Fourier series of the function f, and let $$ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty $$ be a sequence of real numbers for which $$ \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) $$ is the Fourier series of a certain function f λ ∈ L p . The paper considers questions related to approximating the function f λ by its Fourier sums S n (f λ) on a point set and in the spaces L p and CP. Estimates for $$ {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} $$ and P(f λ − S n (f λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions f and f λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f. Bibliography: 11 titles.

Approximation of the classes C β ψ H ω by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.

On the best approximation by trigonometric polynomials on convolution classes of analytic periodic functions

For a continuous 2π-periodic real-valued function K(t), whose amplitudes decrease as a geometric progression with a denominator q ∈ (0, 1) starting from a given number n ∈ ℕ, we find sharp upper bounds for q ensuring that K(t) satisfies the Nagy condition N*n.

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 $$ \left\| {f - P_n } \right\| \leqq c(s)\omega _3 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω 3(f, t) is the third modulus of smoothness of f and ∥ · ∥ is the max-norm.

The univalent polynomial of Suffridge as a summability kernel

A positive summability trigonometric kernel is generated through a sequence of univalent polynomials constructed by Suffridge. We prove that the convolution {K n *f } approximates every continuous 2π-periodic function f with the rate ω( f,1 / n), where ω( f,δ) denotes the modulus of continuity, and this provides a new proof of the classical Jackson's theorem. Despite that it turns out that Kn (θ) coincide with positive cosine polynomials generated by Fejér, our proof differs from others known in the literature.