Simultaneous Approximation and Quasi-Interpolants

✍ Scribed by Kang Zhao

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 748 KB
• Volume: 85
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.0037

✦ Synopsis

It is shown that the shift-invariant space S(.) generated by . # W m 2 (R s ) provides simultaneous approximation order k, with k>m, iff S(D : .) provides approximation order k&m in the L 2 (R s )-norm to all functions in D : W k 2 (R s ) for each |:| =m. Without appealing to the argument of polynomial reproduction, an explicit formula is presented for construction of quasi-interpolant of semi-discrete convolution type that achieves the approximation order provided by S(.). The traditional condition that the symbol : # Z s .(:) e : does not vanish on [ &? } } } ?] s is reduced to that it does not vanish on a neighborhood of the origin.

1996 Academic Press, Inc.

holds almost everywhere (a.e.) on a neighborhood of the origin, where [. ^, . ^] := : # Z s |. ^( }&2?:)| 2 . Here and below we adopt the convention that a fraction is zero when its numerator is zero, even if its denominator article no. 0037 201