On the error term of the Filon quadrature formulae
The author considers the following quadrature formulas \[ (1)\quad \int^{x_{k+1}}_{x_{k-1}}f(x)\cos qx dx=h[\alpha (\theta)(h_{k+1}-h_{k-1})+\beta (\theta)(g_{k-1}+g_{k+1})+2\gamma (\theta)g_ k]+R^ c_ k(f;h), \] \[ \int^{x_{k+1}}_{x_{k- 1}}f(x)\sin qx dx=h[\alpha (\theta)(g_{k-1}-g_{k+1})+\beta (\theta)(h_{k-1}+h_{k+1})+2\gamma (\theta)h_ k]+R^ s_ k(f;h), \] where \(R^ c_ k(f;h)\), \(R^ s_ k(f;h)\) are the remainder terms, and \(x_ k=a+kh,b-a=2nh\), \(\theta =qh\), \(h_ k=f(x_ k)\sin qx_ k\), \(g_ k=f(x_ k)\cos qx_ k\), \(\alpha (\theta)=1/\theta +\cos \theta \sin \theta /\theta^ 2-2 \sin^ 2 \theta /\theta^ 3,\quad \beta (\theta)=(1+\cos^ 2 \theta)/\theta^ 2-\sin 2\theta /\theta^ 3,\quad \gamma (\theta)=2(\sin \theta /\theta^ 3-\cos \theta /\theta^ 2).\) If \(f\in C^{(4)}[a,b]\), then it is proved that the remainders have the following representation: \[ (2)\quad R^ c_ k(f;h)=h.\cos qx_ k\int^{1}_{-1}f^{(4)}(x_ k+th)K_ c(th)dt- \] \[ -h.\sin qx_ k\int^{1}_{-1}f^{(3)}(x_ k+th)K_ s(th)dt, \] \[ R^ 2_ k(f;h)=h.\sin qx_ k\int^{1}_{-1}f^{(4)}(x_ k+th)K_ c(th)dt+ \] \[ +h.\cos qx\int^{1}_{-1}f^{(3)}(x_ k+th)K_ s(th)dt, \] where \(K_ s\), \(K_ c\) are certain kernels. The paper includes a deep analysis of definiteness of these kernels. The above equalities (1)-(2) are to be regarded as fundamental corrections of the numerical integration methods given by \textit{T. Håvie} [``Remarks on an expansion for integrals of rapidly oscillating functions'', BIT 13, 16-29 (1973; Zbl 0273.65019)].
scientific article; zbMATH DE number 4030166
scientific article