arXiv subjects
Publications and source records attributed to Sergey Tikhonov.
We answer Totik's question on weighted Bernstein's inequalities showing that $$ \|T_n'\|_{L_p(ω)} \le C(p,ω)\, {n}\,\|T_n\|_{L_p(ω)},\qquad 0<p\le \infty, $$ holds for all trigonometric polynomials $T_n$ and certain nondoubling weights $ω$. Moreover, we find necessary conditions on $ω$ for Bernstein's inequality to hold. We also prove weighted Bernstein-Markov, Remez, and Nikolskii inequalities for trigonometric and algebraic polynomials.
In this paper, we prove analogues of O'Neil's inequalities for the convolution in the weighted Lebesgue spaces. We also establish the weighted two-sided norm inequalities for the potential operator.
This paper studies the following weighted, fractional Bernstein inequality for spherical polynomials on $\sph$: \begin{equation}\label{4-1-TD-ab} \|(-Δ_0)^{r/2} f\|_{p,w}\leq C_w n^{r} \|f\|_{p,w}, \ \ \forall f\in Π_n^d, \end{equation} where $Π_n^d$ denotes the space of all spherical polynomials of degree at most $n$ on $\sph$, and $(-Δ_0)^{r/2}$ is the fractional Laplacian-Beltrami operator on $\sph$. A new class of doubling weights with conditions weaker than the $A_p$ is introduced, and used to fully characterize those doubling weights $w$ on $\sph$ for which the weighted Bernstein inequality \eqref{4-1-TD-ab} holds for some $1\leq p\leq \infty$ and all $r>τ$. In the unweighted case, it is shown that if $0 0$ is not an even integer, then \eqref{4-1-TD-ab} with $w\equiv 1$ holds if and only if $r>(d-1)(\f 1p-1)$. As applications, we show that any function $f\in L_p(\sph)$ with $0<p<1$ can be approximated by the de la Vallée Poussin means of a Fourier-Laplace series, and establish a sharp Sobolev type Embedding theorem for the weighted Besov spaces with respect to general doubling weights.