Minimum Norm Quadrature in the Sobolev Spaces W ^m_q

Abstract

We consider the Sobolev space W ^m_q of real functions f defined on the interval [-1,+1] which admit a Taylor’s representation

$$ f\left( x \right) = \sum\limits_{j = 0}^{m - 1} {{f^{\left( j \right)}}\left( { - 1} \right)\frac{{{{\left( {x - 1} \right)}^j}}}{{j!}}} + \mathop \smallint \limits_{ - 1}^{ + 1} \frac{{\left( {x - t} \right)_ + ^{m - 1}}}{{\left( {m - 1} \right)!}}{f^{\left( m \right)}}\left( t \right)dt,\;{f^{\left( m \right)}} \in {L_q}\left[ { - 1, + 1} \right]; $$

((1.1))

; here, m ≥ 2 is a fixed natural number and q ∈ [1,∞] . W ^m_q is endowed with the usual semi-norm

$$f \to {\left\{ {\int\limits_{ - 1}^{ + 1} {{{\left| {{f^{(m)}}(t)} \right|}^q}dt} } \right\}^{1/q}}resp.\quad f \to \mathop {ess\;\sup }\limits_{ - 1 \leqslant t \leqslant + 1} \left| {{f^{(m)}}(t)} \right|$$

.