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arXiv:0910.5354 · Chinese Physics B

Inversion formula and Parseval theorem for complex continuous wavelet transforms studied by entangled state representation

Liyun Hu, Fan Hong‐Yi

Published Jul 1, 2010• 8 citations
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Abstract

In a preceding Letter (Opt. Lett. 32, 554 (2007)) we have proposed complex continuous wavelet transforms (CCWTs) and found Laguerre--Gaussian mother wavelets family. In this work we present the inversion formula and Parsval theorem for CCWT by virtue of the entangled state representation, which makes the CCWT theory complete. A new orthogonal property of mother wavelet in parameter space is revealed.

DOI
10.1088/1674-1056/19/7/074205

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In a preceding Letter (Opt. Lett. 32, 554 (2007)) we have proposed complex continuous wavelet transforms (CCWTs) and found Laguerre--Gaussian mother wavelets family. In this work we present the inversion formula and Parsval theorem for CCWT by virtue of the entangled state representation, which makes the CCWT theory complete. A new orthogonal property of mother wavelet in parameter space is revealed.
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Based on the quantum mechanics representation, we propose a novel 3D entangled wavelet transformation. This kind of three entangled state representation can be generated theoretically by using a two-cascaded beam splitter, which has the completeness and orthogonal property. Some properties of 3D WTs are discussed, including the Parseval theory of 3D WTs, its inverse formula, and the orthogonality of the mother wavelets in parameter space. In addition, the mother wavelet condition for the 3D WTs is also derived.
In this paper, we propose the so-called continuous Fresnel-wavelet combinatorial transform which means that the mother wavelet undergoes the Fresnel transformation. This motivation can let the mother-wavelet-state itself vary from |〉 to F†r,s|〉, except for variation within the family of dilations and translations. The Parseval's equality, admissibility condition and inverse transform of this continuous Fresnel-wavelet combinatorial transform are analysed. By taking certain parameters and using the admissibility condition of this continuous Fresnel-wavelet combinatorial transform, we obtain some mother wavelets. A comparison between the newly found mother wavelets is presented.