arXiv subjects

Jiaxin Geng

Publications and source records attributed to Jiaxin Geng.

5 recordsLinked to original sources

arXiv2025

Average Nikolskii factors for random trigonometric polynomials

For $1\le p,q\le \infty$, the Nikolskii factor for a trigonometric polynomial $T_{\bf a}$ is defined by $$\mathcal N_{p,q}(T_{\bf a})=\frac{\|T_{\bf a}\|_{q}}{\|T_{\bf a}\|_{p}},\ \ T_{\bf a}(x)=a_{1}+\sum\limits^{n}_{k=1}(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx).$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,σ^{2})$ coefficients and obtain that the exact order. For $1\leq p<q<\infty$, the average Nikolskii factor is order degree to the 0, as compared to the degree $1/p-1/q$ worst case bound. We also give the generalization to random multivariate trigonometric polynomials.

math.CA↗

arXiv2024

Exact $L_2$ Bernstein-Markov inequalities for generalized weights

In this paper, we obtain some exact $L_2$ Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem $$M_n^2(L_2(W_λ),{\rm D}):=\sup_{0\neq p\in\mathcal{P}_n}\frac{\int_I\left|{\rm D} p(x)\right|^2W_λ(x){\rm d}x}{\int_I| p(x)|^2W_λ(x){\rm d}x},\ λ>0,$$ where $\mathcal{P}_n$ denotes the set of all algebraic polynomials of degree at most $n$, ${\rm D}$ is the differential operator given by $${\rm D}=\Bigg\{\begin{aligned}&\frac {\rm d}{{\rm d}x}\ {\rm or}\ \mathcal{D}_λ, &&{\rm if}\ W_λ(x)=|x|^{2λ}e^{-x^2}\ {\rm and}\ I=\mathbb R, \\&(1-x^2)^{\frac12}\,\frac {\rm d}{{\rm d}x}\ {\rm or}\ (1-x^2)^{\frac12}\,\mathcal{D}_λ, &&{\rm if}\ W_λ(x):=|x|^{2λ}(1-x^2)^{μ-\frac 12},μ>-\frac12\ {\rm and}\ I=[-1,1],\end{aligned} $$ and $\mathcal{D}_λ$ is the univariate Dunkl operator, i.e., $\mathcal{D}_λf(x)=f'(x)+λ{(f(x)-f(-x))}/{x}$. Furthermore, the corresponding extremal polynomials are also obtained.

math.CA↗

arXiv2024

Weighted least $\ell_p$ approximation on compact Riemannian manifolds

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gröchenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we develop weighted least $\ell_p$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian manifold $\Bbb M$ with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in $L_q,\,1\le q\le\infty$, and least quadrature errors for both Sobolev spaces $H_p^r(\Bbb M), \, r>d/p$ generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces $B_{p,τ}^r(\Bbb M),\, 0<τ\le \infty, r>d/p $ defined by best polynomial approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces.

math.NA↗

arXiv2024

Optimal quadrature errors and sampling numbers for Sobolev spaces with logarithmic perturbation on spheres

In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{α,β}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb R^{d+1}$. First we obtain strong equivalences of the approximation numbers for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>0$, which gives a clue to Open problem 3 as posed by Krieg and Vybíral in \cite{KV}. Second, for the optimal quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$, we use the "fooling" function technique to get lower bounds in the case $α>d/2$, and apply Hilbert space structure and Vybíral's theorem about Schur product theory to obtain lower bounds in the case $α=d/2,\,β>1/2$ of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin in \cite{GS} and solves Open problem 2 in \cite{KV}. Finally, we employ the weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for ${\rm H}^{α,β}(\Bbb S^d)$ with $α>d/2$ or $α=d/2,\,β>1/2$, which are order optimal.

math.NA↗

arXiv2023

On the power of standard information for tractability for $L_\infty$ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic function in the worst case setting with the error measured in the $L_\infty$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{\rm all}$ under the absolute or normalized error criterion, and show that the power of $Λ^{\rm std}$ is the same as the one of $Λ^{\rm all}$ for some notions of algebraic and exponential tractability. Our result can be applied to weighted Korobov spaces and Korobov spaces with exponential weight. This gives a special solution to Open problem 145 as posed by Novak and Woźniakowski (2012).

math.NA↗