Bilinear Mappings and the Frame Operator

DOI: 10.4236/apm.2013.34062   PDF   HTML     3,354 Downloads   4,989 Views  


The theory of frames has been actively developed by many authors over the past two decades, both for its applications to signal processing, and for its deep connections to other areas of mathematics such as operator theory. Central to the study of frames is the frame operator. We initiate an investigation that extends the frame operator to the bilinear setting.

Share and Cite:

E. Au-Yeung, "Bilinear Mappings and the Frame Operator," Advances in Pure Mathematics, Vol. 3 No. 4, 2013, pp. 438-441. doi: 10.4236/apm.2013.34062.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] R. J. Dun and A. C. Schaeer, “A Class of Nonharmonic Fourier Series,” Transactions of the American Mathematical Society, Vol. 72, No. 2, 1952, pp. 341-366. doi:10.1090/S0002-9947-1952-0047179-6
[2] D. Han and David Larson, “Frames, Bases and Group Representations,” Memoirs of the American Mathematical Society, Vol. 147, 2000, p. 697.
[3] A. G. I. Daubechies and Y. Meyer, “Painless Nonorthogonal Expansions,” Journal of Mathematical Physics, Vol. 27, 1986, pp. 1271-1283.
[4] O. Christensen, “An Introduction to Frames and Riesz Bases,” Springer-Birkhauser, New York, 2003.
[5] I. Daubechies, “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992.
[6] C. Heil and D. Walnut, “Continuous and Discrete Wavelet Transforms,” SIAM Review, Vol. 31, No. 4, 1989, pp. 628-666. doi:10.1137/1031129
[7] R. R. Coifman and Y. Meyer, “On Commutators of Singular Integrals and Bilinear Singular Integrals,” Transactions of the American Mathematical Society, Vol. 212, 1975, pp. 315-331.
[8] L. Grafakos and N. J. Kalton, “The Marcinkiewicz Multiplier Condition for Bilinear Operators,” Studia Mathematica, Vol. 146, 2001, pp. 115-156.
[9] L. Grafakos and R. Torres, “Discrete Decompositions for Bilinear Operators and Almost Diagonal Conditions,” Transactions of the American Mathematical Society, Vol. 354, 2002, pp. 1153-1176.
[10] N. Tomita, “A Hormander Type Multiplier Theorem for Multilinear Operators,” Journal of Functional Analysis, Vol. 259, 2010, pp. 2028-2044.
[11] M. Lacey and C. Thiele, “On Calderon’s Conjecture,” Annals of Mathematics, Vol. 149, No. 2, 1999, pp. 475-496. doi:10.2307/120971
[12] A. Benyi, “Diego Maldonado, and Virginia Naibo. What Is a Paraproduct?” Notices of the American Mathematical Society, Vol. 57, No. 7, 2010, pp. 858-860.
[13] E. M. Stein, “Harmonic Analysis,” Princeton University Press, Princeton, 1993.
[14] C. Feerman and E. Stein, “Hp Spaces of Several Variables,” Acta Mathematica, Vol. 129, 1972, pp. 137-193.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.