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Optimal Quadrature Formulas for the Sobolev Space $H^1$ | | Erich Novak
; Shun Zhang
; | | Date: |
5 Sep 2016 | | Abstract: | 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
$
ho_k(x) = exp(- 2 pi i 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. | | Source: | arXiv, 1609.1146 | | Services: | Forum | Review | PDF | Favorites | |
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