Bilinear Mappings and the Frame Operator

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.


Introduction
The theory of frames was initiated by Duffin and Schaeffer [1] to study some deep problems in non-harmonic Fourier series.For more than three decades, their ideas did not seem to generate much interest outside of nonharmonic Fourier series.Finally in 1986, Daubechies, Grossman, and Meyer [2] in their groundbreaking paper observed that frames can be used for painless nonorthogonal expansions for functions.Since then, frames have been used in signal processing, image processing, and data compression, as well as being studied for their deep connections to operator theory [3].Frames are important in signal processing because they can be used to provide stable reconstruction of signals.For background in the theory of frames, see [4][5][6].Central to the study of frames is the frame operator.
We initiate an investigation that extends the frame operator to the bilinear setting.Bilinear operators in harmonic analysis have been studied by many authors, see, for example, [7][8][9][10].The conjecture that the bilinear Hilbert transform can be extended to a bounded operator has remained open for some 30 years before it was settled in the celebrated work of Lacey and Thiele [11].The results in our current work extend the results concerning a class of bilinear operators known as paraproducts; these operators are better behaved than the usual products of functions, see [12].The results in this article indicate that there is a rich underlying theory that awaits to be developed.The present work only touches on certain aspects of that theory.
Let H be a separable Hilbert space.A sequence


For the rest of this article, the Hilbert space H is taken to be   . Let be the Schwartz space of rapidly decreasing smooth functions on .The Fourier transform of a function is de-

Main Results
We begin with a useful lemma that will simplify our calculations later.
Lemma 2.1 (Convolution with a radial function is a self-adjoint operator) : .
Define an operator by    , , , , , , , , , , , , , , , By the above calculations on   , B f g , we see that: . This is our frame We have constructed a fram with a bilinear .Let us summarize all our calculations in the g lemma.Let ,

H H H
Then the above in ality holds for all , by 2 Hence,   , n B f g converges in H to an element in H , and we can define a bounded bilinear operator Here, Q f is the average of f over the cube.The integration is over the cube.T e smallest bound h A for which the above inequality is satisfied is taken to be the norm of f , and is denoted by .
In the last inequality, we used Theorem 2.5.Another m gives the following: application of Plancherel Theore  

Acknowledgments
The author is gratef detto from teaching him the subject of xten concl ds to a bo operator on f of the th ul to Professor John Bene [14] ces of Several Variables," Acta Mathematica, Vol. 129, 1972, pp. 137-193.

BMO
sFor background on BMO functions, Chapter 4 of[13], as well as the seminal paper by C Fefferman and St. n f see .ein[14]  The next theorem o BMO functions, together with Lemma 2.3, will allow us to establish the boundedness of the bilinear operator.be a bounded, integrable function that is positive, radial, and decreasing.Write

[ 1 ]
R. J. Dun and A. C. Schaeer, "A Class of Nonharmonic Series merican Mathematical Society, V 341-366.University of Maryland for harmonic analysis and the theory of frames.
and B is linear in each of the two variables separately.
d comp t support, and , for all g in H , and B extends to a bounded bilinear ope-