9 citations · 47 across the 25 of their papers we have counts for
33 papers
A unified approach to inequalities for K-functionals and moduli of smoothness
Amiran Gogatishvili, Bohumir Opic, Sergey Tikhonov +1
The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified…
Truncated smooth function spaces
Oscar Domínguez, Sergey Tikhonov
We introduce truncated Besov and Triebel--Lizorkin function spaces and investigate their main properties: embeddings, interpolation, duality, lifting, traces. These new scales allo…
On the cardinality of lower sets and universal discretization
F. Dai, A. Prymak, A. Shadrin +2
A set in is a lower set if implies whenever for all . We derive new and refine known results…
Hardy-Littlewood-type theorems for Fourier transforms in
Mikhail Dyachenko, Erlan Nursultanov, Sergey Tikhonov +1
We obtain Fourier inequalities in the weighted spaces for any involving the Hardy-Cesàro and Hardy-Bellman operators. We extend these results to product Hardy sp…
Sparse grid approximation in weighted Wiener spaces
Yurii Kolomoitsev, Tetiana Lomako, Segey Tikhonov
We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grids methods constructed with the help of quasi-interpolation operators.…
New estimates for the maximal functions and applications
Oscar Domínguez, Sergey Tikhonov
In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore's inequality for the moduli of smoothness and a logarithmic variant o…
Sharp approximation theorems and Fourier inequalities in the Dunkl setting
D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov
In this paper we study direct and inverse approximation inequalities in , , with the Dunkl weight. We obtain these estimates in their sharp form…
Embeddings and characterizations of Lipschitz spaces
Oscar Domínguez, Dorothee D. Haroske, Sergey Tikhonov
In this paper we give a thorough study of Lipschitz spaces. We obtain the following new results: (1) Sharp Jawerth-Franke-type embeddings between the Besov and Lipschitz spaces ext…
Sobolev embeddings, extrapolations, and related inequalities
Oscar Domínguez, Sergey Tikhonov
In this paper we propose a unified approach, based on limiting interpolation, to investigate the embeddings for the Sobolev space $(\dot{W}^k_p(\mathcal{X}))_0, \, \mathcal{X} \in…
Estimates of the asymptotic Nikolskii constants for spherical polynomials
Feng Dai, Dmitry Gorbachev, Sergey Tikhonov
Let denote the space of spherical polynomials of degree at most on the unit sphere that is equipped with the surface Lebesgue mea…
Properties of moduli of smoothness in
Yurii Kolomoitsev, Sergey Tikhonov
In this paper, we discuss various basic properties of moduli of smoothness of functions from , . In particular, complete versions of Jackson-, Mar…
Nikolskii inequality for lacunary spherical polynomials
Feng Dai, Dmitry Gorbachev, Sergey Tikhonov
We prove that for , the asymptotic order of the usual Nikolskii inequality on (also known as the reverse Hölder's inequality) can be significantly improved i…
Uncertainty principles for eventually constant sign bandlimited functions
D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov
We study the uncertainty principles related to the generalized Logan problem in . Our main result provides the complete solution of the following problem: for a fix…
Function spaces of logarithmic smoothness: embeddings and characterizations
Oscar Domínguez, Sergey Tikhonov
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: Sharp embe…
Weighted gradient inequalities and unique continuation problems
laura De Carli, Dmitry Gorbachev, Sergey Tikhonov
We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality $$ \|e^{-τ\ell(\cdot)} u^{\frac 1q} f\|_q\leq c_τ\| e^{-τ\ell(\cdot)} v^{\fra…
Hardy-Littlewood and Ulyanov inequalities
Yurii Kolomoitsev, Sergey Tikhonov
We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness and for . A similar proble…
Nikolskii constants for polynomials on the unit sphere
Feng Dai, Dmitry Gorbachev, Sergey Tikhonov
This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space of spherical polynomials of degree at most on the unit sph…
Positive -bounded Dunkl-type generalized translation operator and its applications
D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov
We prove that the spherical mean value of the Dunkl-type generalized translation operator is a positive -bounded generalized translation operator . As application,…
Doubling condition at the origin for non-negative positive definite functions
Dmitry Gorbachev, Sergey Tikhonov
We study upper and lower estimates as well as the asymptotic behavior of the sharp constant in the doubling-type condition at the origin \[ \frac{1}{|V|}\int_{V}f(x)\,…
Remez-type inequalities for the hyperbolic cross polynomials
V. Temlyakov, S. Tikhonov
In this paper we study the Remez-type inequalities for trigonometric polynomials with harmonics from hyperbolic crosses. The interrelation between the Remez and Nikolskii inequalit…
Jacobi weights, fractional integration, and sharp Ulyanov inequalities
Polina Glazyrina, Sergey Tikhonov
We consider functions L_p-integrable with Jacobi weights on [-1,1] and prove Hardy--Littlewood type inequalities for fractional integrals. As applications, we obtain the sharp (L_p…
Pitt inequalities and restriction theorems for the Fourier transform
Laura De Carli, Dmitriy Gorbachev, Sergey Tikhonov
We prove new Pitt inequalities for the Fourier transforms with radial and non-radial weights using weighted restriction inequalities for the Fourier transform on the sphere. We als…
Nikolskii inequality and functional classes on compact Lie groups
E. D. Nursultanov, M. V. Ruzhansky, S. Yu. Tikhonov
In this note we study Besov, Triebel-Lizorkin, Wiener, and Beurling function spaces on compact Lie groups. A major role in the analysis is played by the Nikolskii inequality.
Pitt's inequalities and uncertainty principle for generalized Fourier transform
Dmitry Gorbachev, Valery Ivanov, Sergey Tikhonov
We study the two-parameter family of unitary operators \[ \mathcal{F}_{k,a}=\exp\Bigl(\frac{iπ}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{iπ}{2a}\,Δ_{k,a}\Bigr), \] w…