Frequency-dependent interpolation rules using first derivatives for oscillatory functions

We construct frequency-dependent rules to interpolate oscillatory functions y(x)$y(x)$ with frequency ω$\omega$ of the form,

y(x)=_f1(x)cos(ωx)+_f2(x)sin(ωx),$y(x)=f_1(x)\cos (\omega x)+f_2(x)\sin (\omega x)\text{,}$

at equidistant nodes on the interval of interest where the functions _f1$f_1$ and _f2$f_2$ are smooth. Error analysis of the rules is investigated and numerical results are discussed. We provide numerical illustrations to compare the accuracy of classical Hermite polynomials and newly constructed frequency-dependent rules.