Best weighted approximation of some kernels on the real axis

TL;DR

We address the problem of finding the exact best weighted polynomial approximation in the mean-square metric on $\mathbb{R}$ for kernels $\mathcal{K}_{\lambda,s}(t) = \frac{A+Bt}{(t^2+\lambda^2)^{s+1}}$. Our approach extends the Dzhrbashyan–Takenaka–Malmquist framework by using an orthonormal system $\{\Phi_j\}$ and Blaschke products to derive a deviation representation and a closed-form error expression in terms of $G_k(\lambda,\mathbf a)$. The results identify the extremal polynomial $p_{n-1}^{(\lambda,s)}$ that attains the infimum $\mathcal{E}_n^2({\mathcal{K}}_{\lambda,s})_{2,\rho_n}$ and provide exact formulas for the best weighted mean-square error for arbitrary $s\in\mathbb{N}$, generalizing prior $s=0$ and $s=1$ cases. The findings connect to Poisson and biharmonic Poisson integrals and offer precise solutions to a broad class of kernel-approximation problems with potential applications in harmonic analysis.

Abstract

We calculate the exact value and find the polynomial of the best weighted polynomial approximation of kernels of the form $\frac {A+Bt}{(t^2+λ^2)^{s+1}}$, where $A$ and $B$ are fixed complex numbers, $λ>0$, $s\in {\mathbb N}$, in the mean square metric.

Theorems & Definitions (11)

• Proposition 2.1: Dzhrbashyan_1974
• Proposition 2.2: Dzhrbashyan_1974
• Theorem 3.1
• proof : Proof of Theorem \ref{['Theo:Main_Results1']}
• Remark 3.1
• Corollary 3.1
• Theorem 3.2
• Remark 3.2
• Remark 3.3
• Remark 3.4