Optimal Quadrature Formulas for the Sobolev Space $H^1$
Provides theoretical guidance for choosing quadrature nodes in numerical integration of oscillatory functions, particularly for Fourier coefficients.
The paper derives optimal quadrature formulas for weighted integrals in the Sobolev space H^1, showing that equidistant nodes are optimal for Fourier coefficient computation when n ≥ 2.7|k|+1, but suboptimal for small n, with nodes at j/|k| being worst possible.
We study optimal quadrature formulas for arbitrary weighted integrals and integrands from the Sobolev space $H^1([0,1])$. We obtain general formulas for the worst case error depending on the nodes $x_j$. A particular case is the computation of Fourier coefficients, where the oscillatory weight is given by $ρ_k(x) = \exp(- 2 πi k x)$. Here we study the question whether equidistant nodes are optimal or not. We prove that this depends on $n$ and $k$: equidistant nodes are optimal if $n \ge 2.7 |k| +1 $ but might be suboptimal for small $n$. In particular, the equidistant nodes $x_j = j/ |k|$ for $j=0, 1, \dots , |k| = n+1$ are the worst possible nodes and do not give any useful information. To characterize the worst case function we use certain results from the theory of weak solutions of boundary value problems and related quadratic extremal problems.