## Abstract

For a real multiplicity sequence Λ={ λ_{n}, μ_{n} }_{n = 1}^{∞}, that is, a sequence where { λ_{n} } are distinct positive real numbers satisfying 0 < λ_{n} < λ_{n + 1} {mapping} ∞ as n {mapping} ∞ and where each λ_{n} appears μ_{n} times, we associate the exponential systemE_{Λ} = { t^{k} e^{λn t} : k = 0, 1, 2, ..., μ_{n} - 1 }_{n = 1}^{∞} .For a certain class of multiplicity sequences, we give necessary and sufficient conditions in order for E_{Λ} to be complete in some weighted Banach space of continuous functions on R, and in some weighted L^{p} (- ∞, ∞) spaces of measurable functions, with p ∈ [1, ∞). We also prove that if E_{Λ} is incomplete in the weighted spaces, then every function in the closure of the linear span of E_{Λ*}, where E_{Λ *} = { t^{μn - 1} e^{λn t} }_{n = 1}^{∞}, can be extended to an entire function represented by a Taylor-Dirichlet seriesg (z) = underover(∑, n = 1, ∞) c_{n} z^{μn - 1} e^{λn z}, c_{n} ∈ C .Furthermore, we prove that E_{Λ} is minimal in the weighted spaces if and only if it is incomplete.

Original language | British English |
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Pages (from-to) | 115-148 |

Number of pages | 34 |

Journal | Journal of Approximation Theory |

Volume | 146 |

Issue number | 1 |

DOIs | |

State | Published - May 2007 |

## Keywords

- Closure
- Completeness
- Minimality
- Taylor-Dirichlet series