Rebricking Wavelets

Abstract The analytic signal combines a real function f with its Hilbert transform Hf to a complex function $$f + iH f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>H</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> . This idea goes back to Denis Gabor and leads to the question, under which conditions two real-valued bases $$\{f_{n}: n \in \mathbb {N}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> and $$\{g_{n}: n\in \mathbb {N}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> form a complex basis $$\{f_{n}+ig_{n}:n\in \mathbb {N}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . Moreover, for which bounded real linear operators A does $$\{ f_{n} + iA f_{n}: n \in \mathbb {N}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>i</mml:mi> <mml:mi>A</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> form a complex-valued basis, when $$\{f_{n}: n \in \mathbb {N}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is a real-valued basis? We have called this approach rebricking. In this article, we add additional structure in the form of wavelet systems. We characterize the rebricking of orthonormal wavelet bases and of multiresolution analyses. By simple application of the analytic signal to wavelet bases, the completeness property may be lost, which leads to a loss of information and of the perfect reconstruction property. We prove criteria under which the rebricking approach conserves the wavelet structure and insures complete information for signal processing.