Error analysis for frequency-dependent interpolation formulas using first derivatives

We call attention to the entire errors which result when frequency-dependent interpolation formulas are utilized to approximate oscillatory functions f(x) at some x on the domain of interest with a frequency ω of the form,f(x)=f1(x)cos(ωx)+f2(x)sin(ωx),where the functions f1 and f2 are smooth enough to be approximated by polynomials. The interpolation formulas to be considered utilize not only the pointwise values of the function f but also of its derivative f′ at two or three nodes on a closed and bounded interval. In particular, investigations about the interpolation formulas I (or I∼) using three equally spaced nodes (or three unequally spaced nodes) enable us to construct I (or I∼)-related composite formulas which are obtained from applying the formulas I (or I∼) onto subintervals where the union of all the subintervals is the domain of interest. Numerical results show that newly constructed composite formulas are superior in their accuracy to other approximations to interpolate the oscillatory functions. Finally, the entire errors with respect to the interpolation formulas using the derivative information at two (or three) nodes are obtained.