Coefficients of Polynomials of Restricted Growth on the Real Line

✍ Scribed by Lawrence A Harris

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 285 KB
• Volume: 93
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1997.3138

✦ Synopsis

Let ,: (& , ) Ä (0, ) be a given continuous even function and let m be a positive integer. We show that, with some additional restrictions on ,, there exist decreasing sequences x 1 , ..., x m and y 1 , ..., y m&1 of symmetrically located points on (& , ) and corresponding polynomials P and Q of degrees m&1 and m, respectively, satisfying

where equality holds with alternating signs at the corresponding sequence of points (and also at \ for Q). Moreover, for any polynomial p of degree at most m, (a) if | p(x j )| ,(x j ) m for j=1, ..., m, then | p (k) (0)| |P (k) (0)| whenever k and m have opposite parity and 0 k<m; (b) if | p( y j )| ,( y j ) m for j=1, ..., m&1 and if lim sup y Ä | p( y)|Â,( y) m 1, then | p (k) (0)| |Q (k) (0)| whenever k and m have the same parity and 0 k m.

We give two computational methods for determining these sequences of points and thus P and Q.