Hyperdifferential Operators and Continuous Functions on Function Fields

✍ Scribed by Sangtae Jeong

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 142 KB
• Volume: 89
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2629

✦ Synopsis

We prove that a sequence of hyperdifferential operators is an orthonormal basis of the space of continuous F q -linear functions on F q [[T ]]. By Conrad's digit principle, the q-adic extensions of this sequence, called the digit derivatives, turn out to be an orthonormal basis of the whole space of continuous functions on F q [[T ]]. We then give the explicit derivation of the formula for digit derivative coefficients.

2001 Academic Press

And let LC(O, K ) be the closed subspace of continuous F q -linear functions f: O Ä K under the metric induced from C(O, K).

Recently, the close relationship between orthonornmal bases for the entire space C(O, K ) and those for the subspace LC(O, K ) is revealed by K. Conrad [Co]. He develops the digit principle, which says that any orthonormal basis of the subspace extends an orthonormal basis for the entire space via q-adic digit expansion of non-negative integers.