Approximation characteristics of the isotropic Nikol'skii-Besov functional classes

In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,\theta}(\mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,\theta}B(\mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,\theta}(\mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(\mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 \leqslant p \leqslant q \leqslant \infty$, $(p,q)\neq \{(1,1), (\infty, \infty)\}$, $d\geq 1$. In the case $2<p=q<\infty$, $d=1$, the established estimate is also new for the classes $S^{r}_{p,\theta}B(\mathbb{R})$.

The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functions

In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.

Fractal functions of exponential type that is generated by the $\mathbf{Q_2^*}$-representation of argument

We consider function $f$ which is depended on the parameters $0<a\in R$, $q_{0n}\in (0;1)$, $n\in N$ and convergent positive series $v_1+v_2+...+v_n+...$, defined by equality $f(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=a^{\varphi(x)}$, where $\alpha_n\in \{0,1\}$, $\varphi(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=\alpha_1v_1+...+\alpha_nv_n+...$, $q_{1n}=1-q_{0n}$, $\Delta^{Q_2^*}_{\alpha_1...\alpha_n...}=\alpha_1q_{1-\alpha_1,1}+\sum\limits_{n=2}^{\infty}\big(\alpha_nq_{1-\alpha_n,n}\prod\limits_{i=1}^{n-1}q_{\alpha_i,i}\big)$.In the paper we study structural, variational, integral, differential and fractal properties of the function $f$.

A variation of functional equations related to sine type functions

In this paper, we consider a class of functional equation Q(λ)Y (λ) −P(λ)Z(λ) = η related to sine type functions, where the known P,Q are appropriate entire functions of exponential type. We are concerned with the existence and uniqueness of the solution (Y,Z) under certain circumstances. Furthermore, we modify the Lagrange interpolation to deal with the situation of the interpolation nodes being counted by multiplicities, which is significant to solve the above functional equation.

Sharp estimates of approximation of classes of differentiable functions by entire functions

In the paper, we find the sharp estimate of the best approximation, by entire functions of exponential type not greater than $\sigma$, for functions $f(x)$ from the class $W^r H^{\omega}$ such that $\lim\limits_{x \rightarrow -\infty} f(x) = \lim\limits_{x \rightarrow \infty} f(x) = 0$,$$A_{\sigma}(W^r H^{\omega}_0)_C = \frac{1}{\sigma^{r+1}} \int\limits_0^{\pi} \Phi_{\pi, r}(t)\omega'(t/\sigma)dt$$for $\sigma > 0$, $r = 1, 2, 3, \ldots$ and concave modulus of continuity.Also, we calculate the supremum$$\sup\limits_{\substack{f\in L^{(r)}\\f \ne const}} \frac{\sigma^r A_{\sigma}(f)_L}{\omega (f^{(r)}, \pi/\sigma)_L} = \frac{K_L}{2}$$

The Density of Polynomials in a Special Space of Entire Functions of Exponential Type

The traditional approach to solving a specific problem of spectral synthesis in a complex domain involves reducing it to the problem of local description of closed submodules in a certain space of entire functions. The last problem is split into checking the stability and saturation of the submodule under study. This approach turned out to be very effective, for example, in the study of submodules of local rank 1 and in the study of submodules in topological modules associated with unbounded convex domains. Recent studies on spectral synthesis in the complex domain are based on a different scheme. This scheme involves reducing the problem of local description to checking the density of polynomials in a special module of entire functions of exponential type. Moreover, the space under study is a separable locally convex space of type (LN)*. Polynomial approximation in such a space is understood by us as sequential approximation, that is, we are talking about the approximation of space elements by ordinary (not generalized) sequences of polynomials. In this article, we study a special locally convex module of entire vector functions over the ring of polynomials in the degree of the independent variable. The theorem proved in the article can serve as a source of new results on spectral synthesis in the complex domain.

Asymptotics of Polynomial Interpolation and the Bernstein Constants

It is well known that the interpolation error for $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in $$L_{\infty }\left[ -1,1\right] $$ L ∞ - 1 , 1 by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when $$\alpha $$ α tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in the $$L_{\infty }$$ L ∞ norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.