# GENERATING NEW FRAMES IN $L^2(\mathbb{R})$ BY CONVOLUTIONS

• Accepted : 2011.12.05
• Published : 2011.12.25

#### Abstract

Let $\mathbf{c}=\{c_n\}_{n{\in}\mathbb{Z}}\in{\ell}^1(\mathbb{Z})$ and $\{f_n\}_{n{\in}\mathbb{Z}}$ be a frame (Riesz basis, respectively) of $L^2(\mathbb{R})$. We obtain necessary and sufficient conditions of $\mathbf{c}$ under which $\{\mathbf{c}{\ast}_{\lambda}f_n\}_{n{\in}\mathbb{Z}}$ becomes a frame (Riesz basis, respectively) of $L^2(\mathbb{R})$, where ${\lambda}$ > 0 and $(\mathbf{c}{\ast}_{\lambda}f)(t)\;:=\;{\sum}_{n{\in}\mathbb{Z}}c_nf(t-n{\lambda})$. When $\{\mathbf{c}{\ast}_{\lambda}f_n\}_{n{\in}\mathbb{Z}}$ becomes a frame of $L^2(\mathbb{R})$, we present its frame operator and the canonical dual frame in a simple form. Some interesting examples are included.

#### Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

#### References

1. A. Aldroubi and M. Unser, Sampling procedure in function spaces and asymptotic equivalence with Shannon's sampling theory, Numer. Funct. Anal. and Optimiz., 15(1),(1994),1-21. https://doi.org/10.1080/01630569408816545
2. O. Christensen, Frames, Riesz bases, and discrete Gahor/wavelet expansions, Bull. Amer. Math. Soc., 38 (2001), 273-291. https://doi.org/10.1090/S0273-0979-01-00903-X
3. O. Christensen, An Introduction to Frame and Riesz Bases, Birkhauser, Boston, 2003.
4. O. Christensen, A Paley-Wtener theorem for frames, Proc. Amer. Math. Soc., 123 (1995), 2199-2202. https://doi.org/10.1090/S0002-9939-1995-1246520-X
5. C. E. Heil, A basis theory primer, Manuscript, 1997.
6. C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31(1989), 628-666. https://doi.org/10.1137/1031129
7. Y. M. Hong, J. M. Kim, Kil H. Kwon, and E. H. Lee, Channeled sampling in shift invariant spaces Intern, J. Wavelets, Multiresolution and Inform. Processing, 5(5), (2007), 753-767. https://doi.org/10.1142/S0219691307002038
8. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2001.
9. E. H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, 14, Amer. Math. Soc., Provi- dence, RI, 2001.
10. R. M. Young , An lntroduction to Nonharmonic Fourier Series, revised ed., Academic Press, San Diego, 2001.
11. X. Zhou and W. Sun, On the sampling theorem for wavelet subspaee, J. Fourier Anal. Appl., 5(4), (1999), 347-354. https://doi.org/10.1007/BF01259375