Lacunary Quadrature Formulae and Interpolation Singularity

✍ Scribed by D.K. Dimitrov

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 299 KB
• Volume: 75
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1993.1101

✦ Synopsis

Birkhoff quadrature formulae (q.f.), which have algebraic degree of precision (ADP) greater than the number of values used, are studied. In particular, we construct a class of quadrature rules of (\mathrm{ADP}=2 n+2 r+1) which are based on the information (\left{f^{(j)}(-1), f^{(1 \prime}(1), j=0, \ldots, r-1 ; f\left(x_{i}\right), f^{(2 m)}\left(x_{i}\right), i=1, \ldots, n\right}), where (m) is a positive integer and (r=m), or (r=m-1). It is shown that the corresponding Birkhoff interpolation problems of the same type are not regular at the quadrature nodes. This means that the constructed quadrature formulae are not of interpolatory type. Finally, for each (m), we prove the existence of a quadrature formula based on the information (\left{f\left(x_{i}\right), f^{(2 m)}\left(x_{i}\right), i=1, \ldots, 2 m\right}), which has algebraic degree of precision (4 m+1). ' 1993 Academic Press. Inc.