## Portraits of frames

HTML articles powered by AMS MathViewer

- by Akram Aldroubi
- Proc. Amer. Math. Soc.
**123**(1995), 1661-1668 - DOI: https://doi.org/10.1090/S0002-9939-1995-1242070-5
- PDF | Request permission

## Abstract:

We introduce two methods for generating frames of a Hilbert space $\mathcal {H}$. The first method uses bounded operators on $\mathcal {H}$. The other method uses bounded linear operators on ${l_2}$ to generate frames of $\mathcal {H}$. We characterize all the mappings that transform frames into other frames. We also show how to construct all frames of a given Hilbert space $\mathcal {H}$, starting from any given one. We illustrate the results by giving some examples from multiresolution and wavelet theory.## References

- Akram Aldroubi and Michael Unser,
*Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory*, Numer. Funct. Anal. Optim.**15**(1994), no. 1-2, 1–21. MR**1261594**, DOI 10.1080/01630569408816545
—, - Akram Aldroubi and Michael Unser,
*Families of multiresolution and wavelet spaces with optimal properties*, Numer. Funct. Anal. Optim.**14**(1993), no. 5-6, 417–446. MR**1248121**, DOI 10.1080/01630569308816532 - Akram Aldroubi, Murray Eden, and Michael Unser,
*Discrete spline filters for multiresolutions and wavelets of $l_2$*, SIAM J. Math. Anal.**25**(1994), no. 5, 1412–1432. MR**1289146**, DOI 10.1137/S0036141092234086 - John J. Benedetto,
*Gabor representations and wavelets*, Commutative harmonic analysis (Canton, NY, 1987) Contemp. Math., vol. 91, Amer. Math. Soc., Providence, RI, 1989, pp. 9–27. MR**1002584**, DOI 10.1090/conm/091/1002584 - John J. Benedetto,
*Irregular sampling and frames*, Wavelets, Wavelet Anal. Appl., vol. 2, Academic Press, Boston, MA, 1992, pp. 445–507. MR**1161260**
J. J. Benedetto and S. Li, - Ingrid Daubechies,
*The wavelet transform, time-frequency localization and signal analysis*, IEEE Trans. Inform. Theory**36**(1990), no. 5, 961–1005. MR**1066587**, DOI 10.1109/18.57199 - 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**, DOI 10.1137/1.9781611970104 - R. J. Duffin and A. C. Schaeffer,
*A class of nonharmonic Fourier series*, Trans. Amer. Math. Soc.**72**(1952), 341–366. MR**47179**, DOI 10.1090/S0002-9947-1952-0047179-6 - Christopher E. Heil and David F. Walnut,
*Continuous and discrete wavelet transforms*, SIAM Rev.**31**(1989), no. 4, 628–666. MR**1025485**, DOI 10.1137/1031129 - Stephane G. Mallat,
*Multiresolution approximations and wavelet orthonormal bases of $L^2(\textbf {R})$*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 69–87. MR**1008470**, DOI 10.1090/S0002-9947-1989-1008470-5
—, - Wim Sweldens and Robert Piessens,
*Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions*, SIAM J. Numer. Anal.**31**(1994), no. 4, 1240–1264. MR**1286226**, DOI 10.1137/0731065 - Akram Aldroubi and Michael Unser,
*Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory*, Numer. Funct. Anal. Optim.**15**(1994), no. 1-2, 1–21. MR**1261594**, DOI 10.1080/01630569408816545 - Michael Unser, Akram Aldroubi, and Murray Eden,
*On the asymptotic convergence of $B$-spline wavelets to Gabor functions*, IEEE Trans. Inform. Theory**38**(1992), no. 2, 864–872. MR**1162223**, DOI 10.1109/18.119742
—,

*Families of wavelet transforms in connection with Shannon’s sampling theory and the Gabor transform*, Wavelets—A Tutorial in Theory and Applications, 2 (C. K. Chui, ed.), Academic Press, New York, 1992, pp. 509-528.

*Multiresolution analysis frames with applications*, IEEE-ICASSP

**3**(1993), 304-307.

*A theory of multiresolution signal decomposition*:

*the wavelet representation*, IEEE Trans. Pattern Anal. Machine Intell. PAMI-11 (1989), 674-693.

*A family of polynomial spline wavelet transforms*, Signal Process.

**30**(1993), 141-162. M. Vetterli and C. Herley,

*Wavelets and filter banks*, IEEE Trans. Signal Proc.

**40**(1992), 2207-2231.

## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 1661-1668 - MSC: Primary 46C05; Secondary 42C15
- DOI: https://doi.org/10.1090/S0002-9939-1995-1242070-5
- MathSciNet review: 1242070