Rocky Mountain Journal of Mathematics

Wavelet frames in $L^2(\mathbb{R}^d)$

Khole Timothy Poumai and Shiv Kumar Kaushik

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We obtain a necessary and sufficient condition for the existence of wavelet frames. We define and study the synthesis and analysis operators associated with wavelet frames. We discuss some applications of operator value (OPV) frames in the theory of wavelet frames. Also, we discuss the minimal property of wavelet frame coefficients and study the property of over completeness of wavelet frames. Various characterizations of wavelet frame, Riesz wavelet basis and orthonormal wavelet basis are given. Further, dual wavelet frames are discussed and a characterization of dual wavelet frames is given. Finally, we give a characterization of a pair of biorthogonal Riesz bases.

Article information

Rocky Mountain J. Math., Volume 50, Number 2 (2020), 677-692.

Received: 18 August 2018
Revised: 11 August 2019
Accepted: 30 September 2019
First available in Project Euclid: 29 May 2020

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Zentralblatt MATH identifier

Primary: 42C15: General harmonic expansions, frames 42C30: Completeness of sets of functions 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] 46B15: Summability and bases [See also 46A35]

wavelet frames Riesz wavelet bases orthonormal wavelet basis OPV frames


Poumai, Khole Timothy; Kaushik, Shiv Kumar. Wavelet frames in $L^2(\mathbb{R}^d)$. Rocky Mountain J. Math. 50 (2020), no. 2, 677--692. doi:10.1216/rmj.2020.50.677.

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