Filon quadrature from nodal interpolation and phase moments

Input $a<b$, $\omega \in \mathbb{R}\setminus \{0\}$, continuous $f:[a,b]\to \mathbb{C}$, continuous real $g:[a,b]\to \mathbb{R}$, an integer $n\ge 1$, distinct nodes $a=c_0<\cdots <c_n=b$, and for each Lagrange basis polynomial ${\mathrm{\ell }}_j$ a complex moment estimate ${\widehat{\mu }}_j\in \mathbb{C}$ with real error bound ${ϵ}_j\ge 0$ satisfying $\mathrm{\mid }{\widehat{\mu }}_j-{\int }_a^b{\mathrm{\ell }}_j(x)e^{i\omega g(x)}dx\mathrm{\mid }\le {ϵ}_j$. Put $f_j=f(c_j)$. Form $p_n={\sum }_{j=0}^n{f}_j{\mathrm{\ell }}_j$ using Lagrange interpolation at distinct real nodes, then return $Q_n={\sum }_{j=0}^n{f}_j{\widehat{\mu }}_j,{\eta }_n={\sum }_{j=0}^n\mathrm{\mid }{f}_j\mathrm{\mid }{ϵ}_j\mathrm{.}$ For $g(x)=x$, expand each real-node Lagrange polynomial as ${\mathrm{\ell }}_j(x)={\sum }_{k=0}^n{a}_{jk}{x}^k$, run Exact recurrence for polynomial Fourier moments through $k=n$, and set ${\widehat{\mu }}_j={\sum }_{k=0}^n{a}_{jk}{\mu }_k$ and ${ϵ}_j=0$. For general nonstationary $g$, a user may obtain bounded moments by applying Bounded integration-by-parts expansion for an oscillatory moment to each ${\mathrm{\ell }}_j$ with a certified derivative bound. The exact-moment branch for a general phase is conditional on supplied exact moments; no general exact moment solver for arbitrary nonlinear $g$ is asserted here.