Estimate of the Lebesgue function of Fourier sums in terms of modified Meixner polynomials (Q2291211)

Consider the grid \(\Omega_\delta:=\{0,\delta,2\delta,\ldots\}\) with \(\delta=N^{-1}\) for a positive integer \(N\). The collection of modified Meixner polynomials is defined as \(M_{n,N} ^\alpha=M_n ^\alpha(Nx,e^{-\delta})\), where \(M_n ^\alpha\) are the classical Meixner polynomials \[ M_n ^\alpha(x,q)=\binom{n+\alpha}{n}\sum_{k=0} ^n\frac{n^{[k]x^{[k]}}}{(\alpha+1)_k k!}\left(1-\frac1q\right)^k,\] with \(x^{[k]}:=x(x-1)\cdots(x-k+1)\) and \((a)_k:=\alpha(\alpha+1)\cdots(\alpha+k-1)\) for \(\alpha,q\in\mathbb R\) and \(q\neq 0\). The relevance of these is that they satisfy the weighted orthogonality relation \[ \sum_{x\in\Omega_\delta} M_{n,N} ^\alpha(x)M_{k,N} ^\alpha(x)\rho_N(x)=h_{n,N} ^\alpha \delta_{nk},\qquad \alpha>-1, \] where \(h_{n,N} ^\alpha\) is a constant, \(\delta_{nk}\) is the Kronecker \(\delta\) and \(\rho_N\) is the weight function \[ \rho_N(x):=e^{-x}\frac{\Gamma(Nx+\alpha+1)}{\Gamma(Nx+1)}(1-e^{-\delta})^{(\alpha+1)}. \] Denoting by \(m_{n,N}(x):=(h_{n,N} ^\alpha)^{-\frac12}\) the normalized polynomials, the author studies the problem of approximation of continuous functions \(f\) on \([0,\infty)\) such that \[ \|f\|_{C_0}:=\sup_{x\geq 0} e^{-x/2}|f(x)|<\infty \] by the partial sums of their Fourier series with respect to \(\{m_{n,N}\}\): \[ S_{n,N} ^\alpha f (x):=\sum_{k=0} ^n f_{k} ^\alpha m_{k,N} ^\alpha (x) \] with \(f_k ^\alpha\) being the corresponding Fourier coefficients on \(\Omega_\delta\) with respect to the weight \(\rho_N\). The estimate \[ e^{-x/2}|S_{n,N} ^\alpha f(x) -f(x)|\leq (1+\lambda_{n,N} ^\alpha)E_n(f) \] is well known, with \(E_n(f)\) denoting the distance in \(C_0\) of \(f\) from the space of algebraic polynomials of degree \(n\), and \(\lambda_{n,N} ^\alpha\) denoting the Lebesgue function. The Lebesgue function has an explicit (and complicated expression) and is the central object of this paper. The main theorem of the paper gives explicit upper bounds for the Lebesgue function, at least away from \(0\), thus quantifying the estimate above.