Spaces of functions of positive smoothness on irregular domains

The paper is devoted to constructing and studying spaces of functions of positive smoothness on irregular domains of the n-dimensional Euclidean space. We prove embedding theorems that connect the spaces introduced with the Sobolev and Lebesgue spaces. The formulations of the theorems depend on geometric parameters of the domain of definition of functions.     

Spaces of functions of mixed smoothness and approximation from hyperbolic crosses

We compare function spaces of dominating mixed smoothness and spaces of best approximation with respect to hyperbolic crosses.     

Function spaces of Lizorkin-Triebel type on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce function spaces of fractional smoothness s > 0 that are similar to the Lizorkin-Triebel spaces. We prove embedding theorems that show how these spaces are related to the Sobolev and Lebesgue spaces W _p ^m (G) and L _p (G). Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 32–43.     

Kolmogorov widths of Sobolev classes on an irregular domain

We establish asymptotic estimates for the Kolmogorov widths of Sobolev classes W _p ^s (K) in the metric of L _q (K) for a power-law peak K ? ? ^d . These estimates are sharp in the order and coincide with order estimates for the unit cube.     

Approximation of differentiable functions by functions of large smoothness

The order of the quantity \(\delta (L) = \mathop {\sup }\limits_{\chi _{_{_{_1 } } } } \mathop {\inf }\limits_{\chi _2 } \parallel \chi _1 - \chi _2 \parallel L_S [0,2 = ]\) as L → ∞ is studied for the classes of periodic functionsx ₁εW _p ⁿ (I), andx ₂ εW _q ^m (L). Necessary and sufficient conditions under which the inequality $$\parallel x^{(n)} \parallel _{L_p } \leqslant C\parallel x\parallel _{L_q }^x \parallel x^{(m)} \parallel _{L_s }^\beta $$ with the constant independent of x holds for all periodic functions x(t) with \(\int_0^{2\pi } \chi (l)dl = 0\) andx ^(m) (t εL _s [0, 2π] are found.     

An unstructured mesh finite element method for solving the multi-term time fractional and Riesz space distributed-order wave equation on an irregular convex domain

In this paper, the numerical analysis for a multi-term time fracstional and Riesz space distributed-order wave equation is discussed on an irregular convex domain. Firstly, the equation is transformed into a multi-term time-space fractional wave equation using the mid-point quadrature rule to approximate the distributed-order Riesz space derivative. Next, the equation is solved by discretising in time using a Crank–Nicolson scheme and in space using the finite element method (FEM) with an unstructured mesh, respectively. Furthermore, stability and convergence are investigated by introducing some important lemmas on irregular convex domain. Finally, some examples are provided to show the effectiveness and correctness of the proposed numerical method.     

Minimum fractional dominating functions and maximum fractional packing functions

The fractional analogues of domination and 2-packing in a graph form an interesting pair of dual linear programmes in that the feasible solutions for both are functions from the vertices of the graph to the unit interval; efficient (fractional) domination is accomplished when one function simultaneously solves both LPs. We investigate some structural properties of the functions thus defined and classify some families of graphs according to how and whether the sets of functions intersect, developing tools that have proven useful in approaching problems in domination theory.     

Estimates of maximal functions measuring local smoothness

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I ⁿ ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t ^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions ${\mathcal{N}}_\eta f$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ . Furthermore, we obtain estimates of ${\mathcal{N}}_\eta f$ in terms of theL ^p -modulus of continuity off. We find sharp conditions for ${\mathcal{N}}_\eta f$ to belong toL ^p (I ⁿ ) and the Orlicz class?(L), too.     

Differential and smoothness properties of continuous functions

Necessary and sufficient conditions on functions and are found in order for the classes of functions andH ^+k to coincide (k and r are natural numbers).These results are generalizations of [1–4].Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 785–794, December, 1977.The authors thank V. V. Salaev for his unflagging interest in their work.     

Constructive properties of functions with variable smoothness

Analogs of theorems of Jensen and Weierstrass are proven for functions of variable smoothness, defined on the real line.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 443–449, October, 1970.In conclusion I wish to express my gratitude to S. M. Nikol'skii and S. B. Stechkin for their valuable advice concerning this work.