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Perturbations of the Haar wavelet


Authors: N. K. Govil and R. A. Zalik
Journal: Proc. Amer. Math. Soc. 125 (1997), 3363-3370
MSC (1991): Primary 42C99; Secondary 41A05, 46C99
DOI: https://doi.org/10.1090/S0002-9939-97-04002-1
MathSciNet review: 1416087
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Abstract: Let $m \in Z^+$ be given. For any $\varepsilon > 0$ we construct a function $f^{\{\varepsilon \}}$ having the following properties: (a) $f^{\{\varepsilon \}}$ has support in $[-\varepsilon , 1 + \varepsilon ]$. (b) $f^{\{\varepsilon \}} \in C^m(-\infty , \infty )$. (c) If $h$ denotes the Haar function and $0<\delta <\infty $, then $\Vert f^{\{\varepsilon \}} - h \Vert _{L^\delta (\mathcal R)} \le (1+2^\delta )^{1/\delta }(2\varepsilon )^{1/\delta }$. (d) $f^{\{\varepsilon \}}$ generates an affine Riesz basis whose frame bounds (which are given explicitly) converge to $1$ as $\varepsilon \rightarrow 0$.


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Additional Information

N. K. Govil
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849–5310
Email: govilnk@mail.auburn.edu

R. A. Zalik
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849–5310
Email: zalik@mail.auburn.edu

DOI: https://doi.org/10.1090/S0002-9939-97-04002-1
Keywords: Frames, affine frames, Riesz bases, Haar wavelet, basis perturbations, $\wedge$-bounded mean variation, cardinal splines
Received by editor(s): March 18, 1996
Received by editor(s) in revised form: June 21, 1996
Additional Notes: The authors are grateful to Ole Christensen, Sergio J. Favier, Christopher E. Heil, and Luis Miguel Pozo Coronado for their helpful comments.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society