Bramble-Hilbert lemma  From Encyclopedia of Mathematics
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search  Hilbert–Bramble lemma  An abstract theoretical tool for
studying the approximation error of functions in Sobolev spaces
(cf. also Approximation of functions; Sobolev space) by
algebraic polynomials. The general formulation of the lemma was
given and proven first by J.H. Bramble and S.R. Hilbert [a1] in
terms of a class of linear functionals on Sobolev spaces that
annihilate the set of polynomials (see the definition below), an
intermediate to (polynomials of degree ) and , i.e. . This
formulation was motivated by the work of G. Birkhoff, M. Schultz
and R.S. Varga [a3] and applied to estimate the error of the
Hermite interpolation formula [a1], spline interpolation and
Fourier transformation (cf. Fourier transform) [a2].  The
Bramble–Hilbert lemma has a wide range of applications. It is
used in the analysis of projection operators in the function
spaces and for designing optimal domain d