## A comparison of the Carlitz and digit derivative bases in function field arithmetic (2000)

Venue: | J. Number Theory |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Jeong00acomparison,

author = {Sangtae Jeong},

title = {A comparison of the Carlitz and digit derivative bases in function field arithmetic},

journal = {J. Number Theory},

year = {2000},

pages = {258--275}

}

### OpenURL

### Abstract

Abstract: We compare several properties and constructions of the Carlitz polynomials and digit derivatives for continuous functions on Fq[[T]]. In particular, we show a close relation between them as orthonormal bases. Moreover, parallel to Carlitz’s coefficient formula, we give the closed formula for the expansion coefficients in terms of the digit derivatives. 1

### Citations

150 |
Basic Structures of Function Field Arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3
- Goss
- 1996
(Show Context)
Citation Context ... x and for n ≥ 1 let Let F0 = L0 = 1 and for n ≥ 1 let en(x) = ∏ m ∈ Fq[T] deg(m) < n (x − m), Fn = [n][n − 1] q · · · [1] qn−1 , Ln = [n][n − 1] · · ·[1], where [n] = T qn − T. It is well-known [C1],=-=[Go2]-=- that en(x) is an Fq-linear polynomial of degree qn with coefficients in Fq[T], as it has an expansion: en(x) = n∑ (−1) n−i i=0 Fn FiLn−i qi x qi Another set of functions in LC(O, K) is the hyper-diff... |

63 |
complètement continus des espaces de Banach p-adiques
- Serre
- 1962
(Show Context)
Citation Context ...lary 2 Let f(x) = ∑ j≥0 AjGj(x) be a continuous function from O to K. Then f(x) ∈ C(O, O) if and only if {Aj}j≥0 ⊂ O. We close this section with another proof of Theorem 2 by using a useful criterion(=-=[Se]-=- Lemme I) as Conrad did with hyper-differential operators. This is really easy to see and does not depend on Wagner’s arguments. A necessary and sufficient condition for Carlitz Fq-linear polynomials,... |

49 |
On certain functions connected with polynomials in a Galois field
- Carlitz
- 1935
(Show Context)
Citation Context ... Fq-linear functions from O to K, denoted LC(O, K). It is also shown by them [W2], [Go1] that LC(O, K) has an orthonormal basis of Carlitz Fq-linear polynomials, {En(x)}n≥0. As it is shown by Carlitz =-=[C1]-=-, the Carlitz polynomials {Gj}j≥0 are formed out of the orthonormal basis of the subspace, {En(x)}n≥0 via the q-adic expansion of j. Recently, B. Snyder [Sn] and the author [J] showed with different a... |

44 |
Ultrametric calculus
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- 1984
(Show Context)
Citation Context ...s the inversion formula to the Voloch’s expansion stated in Proposition 1. In order to establish Theorem 2, Snyder follows the arguments of Wagner but the proof of the author is based on Theorem 50.7 =-=[S]-=- on orthonormal bases of Banach spaces over local fields. In fact, both proofs depend on Wagner’s result (Theorem 1) and use Voloch’s result. On the other hand, Conrad [Co2] proved the same result, in... |

20 |
A set of polynomials
- Carlitz
- 1940
(Show Context)
Citation Context ...vatives {Dj}j≥0 are formed from hyper-differential operators {Dn}n≥0 via the q-adic expansion of j in the same way that Carlitz did with the En(x). Then we see by the digit principle(due to K. Conrad =-=[C2]-=-) that the digit derivatives {Dj}j≥0 as well as Carlitz polynomials {Gj}j≥0 are orthonormal bases of C(O, K). Indeed, any orthonormal basis of LC(O, K) can extend to an orthonormal basis of C(O, K) vi... |

10 |
Linear operators in local fields of prime characteristic
- WAGNER
(Show Context)
Citation Context ...K), which is topologized by the sup-norm. Unlike the case of padic integers, the space C(O, K) has a subspace of continuous Fq-linear functions from O to K, denoted LC(O, K). It is also shown by them =-=[W2]-=-, [Go1] that LC(O, K) has an orthonormal basis of Carlitz Fq-linear polynomials, {En(x)}n≥0. As it is shown by Carlitz [C1], the Carlitz polynomials {Gj}j≥0 are formed out of the orthonormal basis of ... |

8 | The digit principle
- Conrad
(Show Context)
Citation Context ...hor is based on Theorem 50.7 [S] on orthonormal bases of Banach spaces over local fields. In fact, both proofs depend on Wagner’s result (Theorem 1) and use Voloch’s result. On the other hand, Conrad =-=[Co2]-=- proved the same result, independent of Wagner by simply using a criterion on orthonormal bases for Banach spaces over local fields and did not find the coefficient formulas. But the formula for the c... |

7 |
Interpolation series for continuous functions on π-adic completions of GF(q, x
- Wagner
- 1971
(Show Context)
Citation Context ...rlitz’s coefficient formula, we give the closed formula for the expansion coefficients in terms of the digit derivatives. 1 Introduction As an analogue of Mahler expansion for function fields, Wagner =-=[W1]-=- and Goss [Go1] proved that Carlitz polynomials {Gj}j≥0 are an orthonormal basis of the Banach space of continuous functions from O := Fq[[T]] to K := Fq((T)), denoted C(O, K), which is topologized by... |

3 |
Die Wronskische Determinante in beliebigen differenzierbaren Funktionenkörpern
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- 1939
(Show Context)
Citation Context ... ( ) n + m (3) Dn(Dm(x)) = Dn+m(x) for all x, y ∈ K. m The collection of additive operators satisfying the three properties above is called iterative derivations and for a detailed study on them, see =-=[Sch]-=-. Definition 1 (1) Put En(x) = en(x)/Fn for any integer n > 0 and E0(x) = x. (2) For the q-adic expansion of j ≥ 0, which is given by j = α0 + α1q + · · · + αsq s . with 0 ≤ αi < q, Put Gj(x) := s∏ n=... |

2 |
Diophantine Problems in Function Fields of Positive Characteristic
- Jeong
- 1999
(Show Context)
Citation Context ...it is shown by Carlitz [C1], the Carlitz polynomials {Gj}j≥0 are formed out of the orthonormal basis of the subspace, {En(x)}n≥0 via the q-adic expansion of j. Recently, B. Snyder [Sn] and the author =-=[J]-=- showed with different arguments that the certain Fq-linear hyper-differential operators {Dn}n≥0 are an orthonormal basis of LC(O, K). The digit derivatives {Dj}j≥0 are formed from hyper-differential ... |

2 |
Hyperdifferential Operators on Function Fields and Their Applications
- Snyder
- 1999
(Show Context)
Citation Context ...als, {En(x)}n≥0. As it is shown by Carlitz [C1], the Carlitz polynomials {Gj}j≥0 are formed out of the orthonormal basis of the subspace, {En(x)}n≥0 via the q-adic expansion of j. Recently, B. Snyder =-=[Sn]-=- and the author [J] showed with different arguments that the certain Fq-linear hyper-differential operators {Dn}n≥0 are an orthonormal basis of LC(O, K). The digit derivatives {Dj}j≥0 are formed from ... |

2 |
Differential operators and interpolation series in power series fields
- Voloch
- 1998
(Show Context)
Citation Context ...) n−i i=0 Fn FiLn−i qi x qi Another set of functions in LC(O, K) is the hyper-differential operators, namely Hasse derivations Dn, n ≥ 0, defined by Dn( ∑ aiT i ) = ∑ ( ) i aiT n i−n . It is known in =-=[V]-=- that Dn are continuous Fq-linear functions on K (hence on O). For later use, recall here the properties of hyper-differential operators Dn; (1) D0 = id n∑ (2) Dn(xy) = Di(x)Dn−i(y) i=0 ( ) n + m (3) ... |

1 |
A q-Analogue of Mahler Expansions I
- Conrad
(Show Context)
Citation Context ...mediately from Serre’s criterion ([Se] Lemme I) as we easily deduce from hypothesis that ||fn|| = 1 and the reduced functions for fn and en are identically same . Alternatively we refer the reader to =-=[Co1]-=- Lemma 3.2 for a concrete proof.Carlitz and Digit Derivatives Bases 6 Theorem 5 {En(x)}n≥0 is an orthonormal basis of LC(O, K) if and only if {Dn}n≥0 is an orthonormal basis of LC(O, K). Proof. We wi... |

1 |
Fourier series, Measures and Divided
- Goss
(Show Context)
Citation Context ...cient formula, we give the closed formula for the expansion coefficients in terms of the digit derivatives. 1 Introduction As an analogue of Mahler expansion for function fields, Wagner [W1] and Goss =-=[Go1]-=- proved that Carlitz polynomials {Gj}j≥0 are an orthonormal basis of the Banach space of continuous functions from O := Fq[[T]] to K := Fq((T)), denoted C(O, K), which is topologized by the sup-norm. ... |

1 |
Non-Archimedean Analysis Over Function Fields with positive characteristic. The Ohio State University(Coulmbus)(1999
- Yang
(Show Context)
Citation Context ...closed formula. Parallel to this formula, we will also obtain the closed formula for the expansion coefficients of functions in C(O, K) in terms of the digit derivatives by following Yang’s arguments =-=[Y]-=-. Indeed this is based on the close similarity between the orthogonality properties for two canonical orthonormal bases of C(O, K). Finally we show for any non-negative integer m that the two sets of ... |