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 be given. For any we construct a function having the following properties: (a) has support in . (b) . (c) If denotes the Haar function and , then . (d) generates an affine Riesz basis whose frame bounds (which are given explicitly) converge to as .

**1.**J. J. Benedetto and D. F. Walnut, Gabor frames for and related spaces, in ``Wavelets: Mathematics and Applications'' (J. J. Benedetto and M. W. Frazier, Eds.), pp. 97-162, CRC Press, Boca Raton, FL, 1994. MR**94i:42040****2.**C. K. Chui, ``An Introduction to Wavelets'', Academic Press, San Diego, 1992. MR**93f:42055****3.**C. K. Chui and X. L. Shi, Bessel sequences and affine frames,*Appl. Comput. Harm. Anal.*1 (1993), 29-49. MR**95b:42028****4.**I. Daubechies, ``Ten Lectures on Wavelets," SIAM, Philadelphia, 1992. MR**93e:42045****5.**S. J. Favier and R. A. Zalik, On the stability of frames and Riesz bases,*Appl. Comput. Harm. Anal.*2 (1995), 160-173. MR**96e:42030****6.**K. Gröchenig, Acceleration of the frame algorithm,*IEEE Trans. Signal Proc.*41 (1993), 3331-3340.**7.**C. Houdré, Wavelets, probability and statistics: some bridges in ``Wavelets: Mathematics and Applications'' (J. J. Benedetto and M. W. Frazier, Eds), pp. 365-398, CRC Press, Boca Raton, FL 1994. MR**95c:60046****8.**X. L. Shi, On BMV functions with some applications to the theory of Fourier series,*Sci. Sinica Ser. A*28 (1985), 147-158. MR**87a:42027****9.**I. J. Schoenberg, ``Cardinal Spline Interpolation'', SIAM, Philadelphia, 1973. MR**54:8095****10.**G. Strang and T. Nguyen, ``Wavelets and Filter Banks'', Wellesley-Cambridge Press, Wellesley, Massachussetts, 1996. CMP**97:02****11.**R. M. Young, ``An Introduction to Nonharmonic Fourier Series'', Academic Press, New York, 1980. MR**81m:42027**

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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