We prove the exact asymptotics of the multidimensional normalized $L^1$ Nikolskii constant \[ \mathcal L^*(d)=\Bigl(\frac{\pi}{2}+o(1)\Bigr)2^{-d}, \quad d\to\infty. \] In addition, for each fixed dimension we obtain asymptotics for the positive zeros of the extremal function $\varphi_d$, and determine their limiting distribution as $d\to\infty$. The lower bound follows from the construction of an admissible function and its asymptotic analysis. For the upper bound, we represent $x^{d+1}\varphi_d(x)$ as the product of two solutions of the equation $u''+V_d(x)u=0$, compare this equation with a Bessel model, and analyze relative canonical products.
We study the radial extremal function $\varphi(|{\,\cdot\,}|)$ arising in the problem of finding the sharp Nikolskii constant $\mathcal C_d$ in $\mathit{PW}_1^1(\mathbb R^d)$ for arbitrary dimension $d\ge1$. We prove a factorization $\varphi=\Phi_1\Phi_2$, where $\Phi_1$ and $\Phi_2$ are entire functions of exponential...
Let $\varphi_{d}(|{\,\cdot\,}|)$ be the radial extremal function in the problem for the sharp Nikolskii constant $\mathcal C_d^{-1}=\inf \|f\|_{1}$ over functions $f\in\mathit{PW}\,_{1}^{1}(\mathbb R^{d})$, $f(0)=1$. For every odd dimension, we construct an entire function $\Phi$ of exponential type $1/2$ such that $\v...
We investigate the asymptotic behavior of Jacobi-Pi\~neiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x) = x^{\alpha_j}(1-x)^{\beta}$, $j=1,2$ where $\alpha_1,\alpha_2, \beta>-1$, and $\alpha_1-\alpha_2 \notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert probl...
We prove a three-term asymptotic formula for the eigenvalues of the fractional Laplacian on the bounded interval $(-1,1)$. This improves the eigenvalue asymptotics of Kulczycki--Kwa\'snicki--Ma{\l}ecki--St\'os and Kwa\'snicki, and confirms the conjectural $O_\alpha(n^{-2})$ remainder suggested by the numerical simulati...
We consider an elliptic differential operator $A_\varepsilon=- \frac{d}{dx} g(\frac x\varepsilon) \frac{d}{dx}$, $\varepsilon > 0$, with a periodic coefficient that acts in $L_2(\mathbb{R})$. We study the behaviour of solutions of the Cauchy problem for the hyperbolic equation $partial_\tau^2 w_\varepsilon (x,\tau)=- (...
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of t...
Long Li, Wei Wang, Shi-Wen Zhang· 0 citations
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