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 | References | Similar Articles | Additional Information

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 $\varepsilon \rightarrow 0$.

- John J. Benedetto and David F. Walnut,
*Gabor frames for $L^2$ and related spaces*, Wavelets: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 97–162. MR**1247515** - Charles K. Chui,
*An introduction to wavelets*, Wavelet Analysis and its Applications, vol. 1, Academic Press, Inc., Boston, MA, 1992. MR**1150048** - Charles K. Chui and Xian Liang Shi,
*Bessel sequences and affine frames*, Appl. Comput. Harmon. Anal.**1**(1993), no. 1, 29–49. MR**1256525**, DOI https://doi.org/10.1006/acha.1993.1003 - Ingrid Daubechies,
*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107** - S. J. Favier and R. A. Zalik,
*On the stability of frames and Riesz bases*, Appl. Comput. Harmon. Anal.**2**(1995), no. 2, 160–173. MR**1325538**, DOI https://doi.org/10.1006/acha.1995.1012 - K. Gröchenig, Acceleration of the frame algorithm,
*IEEE Trans. Signal Proc.*41 (1993), 3331–3340. - Christian Houdré,
*Wavelets, probability, and statistics: some bridges*, Wavelets: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, 1994, pp. 365–398. MR**1247521** - Xian Liang Shi,
*On $\Lambda $BMV functions with some applications to theory of Fourier series*, Sci. Sinica Ser. A**28**(1985), no. 2, 147–158. MR**795170** - I. J. Schoenberg,
*Cardinal spline interpolation*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12. MR**0420078** - G. Strang and T. Nguyen, “Wavelets and Filter Banks”, Wellesley–Cambridge Press, Wellesley, Massachussetts, 1996.
- Robert M. Young,
*An introduction to nonharmonic Fourier series*, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**591684**

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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

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