Bernstein-Nikolskii-Stechkin inequality and Jackson's theorem for the index Whittaker transform (Q6545024)

In this paper, the author considers the index Whittaker transform given by \N\[\NF_W(f)(\lambda)=\int_0^{+\infty}f(x)K_\alpha(\lambda,x)d\mu_\alpha(x)\,,\quad\lambda\geq0,\quad f\in\mathrm{L}^1(\mu_\alpha),\N\]\Nwhere \(\mu_\alpha\) is a weight Lebesgue measure and \(K_\alpha(\lambda,x)\) is the so-called index Whittaker kernel. The author defines the index Whittaker translation operator \(\tau_x\) as follows \N\[\N\tau_xf(y)=\int_0^{+\infty}f(z)d\gamma_{x,y}(z)\,,\quad x,y>0,\N\]\Nwhere \(d\gamma_{x,y}(z)=q(x,y,z)d\mu_\alpha(z)\)\ ,\ \(q(x,y,z)\) is a suitable kernel and \(f\) is a bounded continuous function on \(\mathbb{R}_+\). According to the operator \(\tau_x\)\,, the author defines, for any function \(f\in\mathrm{L}^2(\mu_\alpha)\), the \(m\)th-order finite differences operator with step \(h\) by \N\[\N\Delta_h^mf(x)=\sum_{l=0}^m(-1)^l\left(\begin{array}{cc}m\\\Nl\end{array}\right)\tau_{lh}f(x).\N\]\NConsequently, the author defines the modulus of smoothness of \(m\)th-order as follows \N\[\N\omega_{2,m}^\alpha(f,\delta)=\sup_{0<h\leq\delta}\|\Delta_h^mf\|_{\alpha,2},\N\]\NThe author proves a Bernstein-Nikolskii-Stechkin inequality for the index Whittaker transform which is \N\[\N\|L^m(P_\beta(f))\|_{\alpha,2}\leq C\beta^m\|\Delta_{1/\beta}^mf\|_{\alpha,2}\,,\qquad \beta>0,\ \ m\in\mathbb{N},\N\]\Nwhere \(f\in\mathrm{L}^2(\mu_\alpha)\), and \(L\) is the index Whittaker operator given by \N\[\NL=-\frac{1}{4}\left[x^2\frac{d^2}{dx^2}+\frac{[x^2\mu_\alpha(x)]^\prime}{\mu_\alpha(x)}\,\frac{d}{dx} \right],\N\]\N\(\mu_\alpha(x)\) being the density of the Lebsgue measure \(\mu_\alpha\) and \[P_\beta(f)(x)=F_W^{-1}(F_W(f)(x)1\hskip-1mm\mathrm{I}_{[0,\beta]}(x))\,.\] As an application, by introducing \(K\)-functionals, the author establishes an equivalence theorem between \(K\)-functionals and a modulus of smoothness in \(\mathrm{L}^2(\mu_\alpha)\). At the end of the paper, the author studies Jackson's theorem for the index Whittaker transform in \(\mathrm{L}^2(\mu_\alpha)\).