Weak Type Inequalities for Moduli of Smoothness: The Case of Limit Value Parameters

In this paper we obtain Ul’yanov type inequalities for fractional moduli of smoothness/K-functionals for the limit value parameters: p=1 or q=∞. Needed versions of Nikol’skii type inequalities for trigonometric polynomials are given. We show that these estimates are sharp. Corresponding embedding theorems for the Lipschitz spaces are investigated.     

Embedding Nikol’skiĭ classes into Lorentz spaces

We consider the embeddings of the Nikol’skii classes of Lorentz spaces into Lorentz spaces. We obtain necessary and sufficient conditions for these embeddings under some restrictions on the fundamental functions of a Lorentz space.     

Relations between mixed moduli of smoothness,and embedding theorems for Nikol’skii classes

We prove (L _p , L _q ) inequalities for mixed moduli of smoothness of positive orders. As corollaries, we obtain embedding theorems for the Nikol’skii classes.     

Trigonometric and orthoprojection widths of classes of periodic functions of many variables

We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes B ^r _p,θ and Nikol’skii classes Hr p of periodic functions of many variables in the space L _q for certain relations between the parameters p and q.     

On some inequalities for polynomials

We prove the equivalence between analogs of the Paley and Nikol’skii inequalities for any orthonormal system of functions and for almost periodic polynomials with arbitrary spectrum. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1289–1292, September, 1998.     

Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B _p,θ ^r ) and Nukol’skii (H _p ^r ) classes of periodic functions of many variables in the metric of L _q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L ₁ and L _∞ by trigonometric polynomials with the corresponding spectrum.     

Polyharmonic dirichlet problems

A certain Dirichlet problem for the inhomogeneous polyharmonic equation is explicitly solved in the unit disc of the complex plane. The solution is obtained by modifying the related Cauchy-Pompeiu representation with the help of a polyharmonic Green function. Dedicated to Prof. S.M. Nikol’skii on the occasion of his 100th birthday and to the memory of P.G.L. Dirichlet on the occasion of his 200th birthday     

Weighted extrapolation in Iwaniec—Sbordone spaces. Applications to integral operators and approximation theory

We prove extrapolation theorems in weighted Iwaniec–Sbordone spaces and apply them to one-weight inequalities for several integral operators of harmonic analysis. In addition, in weighted grand Lebesgue spaces, we establish Bernstein and Nikol’skii type inequalities and prove direct and inverse theorems on the approximation of functions.     

Variable 2-Microlocal Besov–Triebel–Lizorkin-Type Spaces

This article is devoted to the study of variable 2-microlocal Besov-type and Triebel–Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of φ-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some related Sobolev-type embeddings and trace theorems of these spaces are also established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel–Lizorkin spaces.     

Frequency methods in the theory of bounded solutions of nonlinear <Emphasis Type="Italic">n</Emphasis>th-order differential equations (existence,almost periodicity,and stability)

We obtain new conditions for the existence of bounded solutions of higher-order nonlinear differential equations. In addition to the classical contraction mapping principle, A.N. Tikhonov’s fixed-point principle is used in the proof of existence theorems. Assertions dealing with the stability of a bounded solution are derived directly from the corresponding results obtained by M.A. Krasnosel’skii and A.V. Pokrovskii.     

Direct and inverse theorems on approximation of functions defined on a sphere in the space <Emphasis Type="Italic">S</Emphasis><Superscript>(<Emphasis Type="Italic">p,q</Emphasis>)</Superscript>(σ<Superscript>m</Superscript>)

We prove direct and inverse theorems on the approximation of functions defined on a sphere in the space S ^(p,q)(σ^m), m ≥ 3, in terms of the best approximations and moduli of continuity. We consider constructive characteristics of functional classes defined by majorants of the moduli of continuity of their elements. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 901–911, July, 2007.     

Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer’s theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.     

Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros

It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis, are proved. Similar results are established for polynomials with zeros in the closed unit disk. The classical theorems by Turan and Ankeny-Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of a polynomial is proved. Bibliography: 16 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 101–112.     

Approximation in weighted Orlicz spaces

We prove some direct and converse theorems of trigonometric approximation in weighted Orlicz spaces with weights satisfying so called Muckenhoupt’s A _p condition.     

Approximation of functions by certain nonlinear integral operators

We study approximation properties of certain nonlinear integral operators L _n * obtained by a modification of given operators L _n . The operators L _n;r and L _n;r * of r-times differentiable functions are also studied. We give theorems on approximation orders of functions by these operators in polynomial weight spaces.     

On elliptic equations and systems with critical growth in dimension two

We consider nonlinear elliptic equations of the form −Δu = g(u) in Ω, u = 0 on ∂Ω, and Hamiltonian-type systems of the form −Δu = g(v) in Ω, −Δv = f(u) in Ω, u = 0 and v = 0 on ∂Ω, where Ω is a bounded domain in ℝ² and f, g ∈ C(ℝ) are superlinear nonlinearities. In two dimensions the maximal growth (= critical growth) of f and g (such that the problem can be treated variationally) is of exponential type, given by Pohozaev-Trudinger-type inequalities. We discuss existence and nonexistence results related to the critical growth for the equation and the system. A natural framework for such equations and systems is given by Sobolev spaces, which provide in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results. Dedicated to Professor S. Nikol’skii on the occasion of his 100th birthday     

Discontinuous Galerkin methods for the linear Schrödinger equation in non-cylindrical domains

The convergence of a discontinuous Galerkin method for the linear Schrödinger equation in non-cylindrical domains of ${\mathbb{R}^m}The convergence of a discontinuous Galerkin method for the linear Schr?dinger equation in non-cylindrical domains of \mathbbR^m{\mathbb{R}^m}, m ≥ 1, is analyzed in this paper. We show the existence of the resulting approximations and prove stability and error estimates in finite element spaces of general type. When m = 1 the resulting problem is related to the standard narrow angle ‘parabolic’ approximation of the Helmholtz equation, as it appears in underwater acoustics. In this case we investigate theoretically and numerically the order of convergence using finite element spaces of piecewise polynomial functions.     

Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class W _loc ^m,p (ℝ ⁿ ) or W ^m,p (Q) for some cube Q ⊂ ℝ ⁿ if and only if the formal (m − 1)-Taylor remainder R ^m−1 f(x,y) of f satisfies the pointwise inequality |R ^m−1 f(x,y)| ≤ |x − y| ^m [a(x) + a(y)] for some a ε L ^p (Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝ ⁿ . Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005.     

k-systems, k-networks and k-covers

The concepts of k-systems, k-networks and k-covers were defined by A. Arhangel’skii in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among k-systems, k-networks and k-covers are further discussed and are established by mk-systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of mk-systems.     

Equazioni quasi ellittiche e spaziL p, θ (ω, δ) (I)

Summary Regularity theorems inL ^{2, θ} (ω, δ) spaces are proved for weak solutions of quasielliptic differential equations. In particular, regularization results are obtained in the class of holder continuous functions (with respect to a suitable metric related to the operator). As a consequence, we obtain results and estimates in L^p andL ^{p, θ} spaces for the solution of the Dirichlet problem.

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Lavoro eseguito nell’ambito del Gruppo di Ricerca n^o 46 del Comitato per la Matematica del C N.R.