2024/11/25 by Jiansong Li, Jiaxin Geng, Li, Jiansong +5
Mathematics · #33C45 #41A17 #41A44 #42C05 #Advanced Optimization Algorithms Research #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Inequalities and Applications
paper · pdf · doi:10.48550/arxiv.2411.16359
openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we obtain some exact L2 Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem Mn2(L2(Wλ),\rm D):=sup0≠ p\inPn\frac∫I|\rm D p(x)|2Wλ(x)\rm dx∫I| p(x)|2Wλ(x)\rm dx, λgt;0, where Pn denotes the set of all algebraic polynomials of degree at most n, \rm D is the differential operator given by \rm D=\\beginalignedamp;\frac \rm d\rm dx \rm or Dλ, amp;amp;\rm if Wλ(x)=|x|2λe-x2 \rm and I=\mathbb R,
amp;(1-x2)\frac12 \frac \rm d\rm dx \rm or (1-x2)\frac12 Dλ, amp;amp;\rm if Wλ(x):=|x|2λ(1-x2)μ-\frac 12,μgt;-\frac12 \rm and I=[-1,1],\endaligned and Dλ is the univariate Dunkl operator, i.e., Dλf(x)=f'(x)+λ(f(x)-f(-x))/x. Furthermore, the corresponding extremal polynomials are also obtained.