A necessary and sufficient condition for the existence of a father wavelet

Gustaf Gripenberg

ArticleOriginal scientific text

Title

A necessary and sufficient condition for the existence of a father wavelet

Authors ¹

Affiliations

Department of Mathematics, University of Helsinki, P.O. BOX 4 (Regeringsgatan 15), 00014 University of Helsinki, Finland.

Abstract

It is proved that if

{2^{- \frac{m}{2}} ψ (2^{- m} • - k)}_{m, k \in ℤ}

is an orthonormal basis in

L^{2} (ℝ; ℂ)

, then the mother wavelet ψ is obtained from a multiresolution generated by a father wavelet if and only if

\sum_{p = 1}^{\infty | \sum_{k \in ℤ} | ψ ̂ (2^{p} (• + k)) ∣^{2} > 0}

a.e.

Keywords

ENG

wavelet, multiresolution, existence

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Title

A necessary and sufficient condition for the existence of a father wavelet

Affiliations

Abstract

Keywords

Bibliography