Nikola Naidenov | Searcharxiv

arXiv2022

Estimates for the largest critical value of $T_n^{(k)}$

Here we study the quantity $$ τ_{n,k}:=\frac{|T_n^{(k)}(ω_{n,k})|}{T_n^{(k)}(1)}\,, $$ where $T_n$ is the $n$-th Chebyshev polynomial of the first kind and $ω_{n,k}$ is the largest zero of $T_n^{(k+1)}$. Since the absolute values of the local extrema of $T_n^{(k)}$ increase monotonically towards the end-points of $[-1,1]$, the value $τ_{n,k}$ shows how small is the largest critical value of $\,T_n^{(k)}\,$ relative to its global maximum $\,T_n^{(k)}(1)$. This is a continuation of the recent paper \cite{NNS2018}, where upper bounds and asymptotic formuae for $τ_{n,k}$ have been obtained on the basis of Alexei Shadrin's explicit form of the Schaeffer--Duffin pointwise majorant for polynomials with absolute value not exceeding $1$ in $[-1,1]$. We exploit a result of Knut Petras \cite{KP1996} about the weights of the Gaussian quadrature formulae associated with the ultraspherical weight function $w_λ(x)=(1-x^2)^{λ-1/2}$ to find an explicit (modulo $ω_{n,k}$) formula for $τ_{n,k}^2$. This enables us to prove a lower bound and to refine the upper bounds for $τ_{n,k}$ obtained in \cite{NNS2018}. The explicit formula admits also a new derivation of the assymptotic formula in \cite{NNS2018} approximating $τ_{n,k}$ for $n\to\infty$. The new approach is simpler, without using deep results about the ordinates of the Bessel function, and allows to better analyze the sharpness of the estimates.

math.CA↗

arXiv2017

On the largest critical value of $T_n^{(k)}$

We study the quantity $$ τ_{n,k}:=\frac{|T_n^{(k)}(ω_{n,k})|}{T_n^{(k)}(1)}\,, $$ where $T_n$ is the Chebyshev polynomial of degree $n$, and $ω_{n,k}$ is the rightmost zero of $T_n^{(k+1)}$. Since the absolute values of the local maxima of $T_n^{(k)}$ increase monotonically towards the end-points of $[-1,1]$, the value $τ_{n,k}$ shows how small is the largest critical value of $\,T_n^{(k)}\,$ relative to its global maximum $\,T_n^{(k)}(1)$. In this paper, we improve and extend earlier estimates by Erdős--Szegő, Eriksson and Nikolov in several directions. Firstly, we show that the sequence $\,\{τ_{n,k}\}_{n=k+2}^{\infty}$ is monotonically decreasing in $n$, hence derive several sharp estimates, in particular $$ τ_{n,k} \le \begin{cases} τ_{k+4,k} = \frac{1}{2k+1}\,\frac{3}{k+3}\,, & n \ge k+4\, τ_{k+6,k} = \frac{1}{2k+1}\, (\frac{5}{k+5})^2 β_k\,, & n \ge k+6\,, \end{cases} $$ where $β_k < \frac{2+\sqrt{10}}{5} \approx 1.032$. We also obtain an upper bound which is uniform in $n$ and $k$, and that implies in particular $$ τ_{n,k} \approx \big(\frac{2}{e}\big)^k, \quad n \ge k^{3/2}; \qquad τ_{n,n-m} \approx \big(\frac{em}{2}\big)^{m/2} n^{-m/2}; \qquad τ_{n,n/2} \approx \big(\frac{4}{\sqrt{27}}\big)^{n/2}. $$ Finally, we derive the exact asymptotic formulae for the quantities $$ τ_k^{*} := \lim_{n\to\infty}τ_{n,k} \quad \mbox{ and }\quad τ_m^{**} := \lim_{n\to\infty} n^{m/2} τ_{n,n-m}\,, $$ which show that our upper bounds for $τ_{n,k}$ and $τ_{n,n-m}$ are asymptotically correct with respect to the exponential terms given above.

math.CA↗