Advances in Pure Mathematics, 2013, 3, 438441 http://dx.doi.org/10.4236/apm.2013.34062 Published Online July 2013 (http://www.scirp.org/journal/apm) Bilinear Mappings and the Frame Operator Enrico AuYeung Pacific Institute for the Mathematical Sciences, Vancouver, Canada Email: enricoauy@math.ubc.ca Received March 28, 2013; revised April 30, 2013; accepted May 26, 2013 Copyright © 2013 Enrico AuYeung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT 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. Keywords: Harmonic Analysis; Frames 1. Introduction The theory of frames was initiated by Duffin and Scha effer [1] to study some deep problems in nonharmonic Fourier series. For more than three decades, their ideas did not seem to generate much interest outside of non harmonic Fourier series. Finally in 1986, Daubechies, Grossman, and Meyer [2] in their groundbreaking paper observed that frames can be used for painless nonor thogonal 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 im portant in signal processing because they can be used to provide stable reconstruction of signals. For background in the theory of frames, see [46]. Central to the study of frames is the frame operator. We initiate an investigation that extends the frame op erator to the bilinear setting. Bilinear operators in har monic analysis have been studied by many authors, see, for example, [710]. The conjecture that the bilinear Hil bert 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 ope rators are better behaved than the usual products of func tions, see [12]. The results in this article indicate that there is a rich underlying theory that awaits to be devel oped. The present work only touches on certain aspects of that theory. Let H be a separable Hilbert space. A sequence 1 nn x of elements in is a frame for if there exist positive constants and such that B 2 22 1 ,,. n n HAffx Bf For the rest of this article, the Hilbert space is taken to be 2d L d d 2d fL . Let be the Schwartz space of rapidly decreasing smooth functions on . The Fourier transform of a function is de fined by 2πi ˆed dx fx x . 2. Main Results We begin with a useful lemma that will simplify our calculations later. Lemma 2.1 (Convolution with a radial function is a selfadjoint operator ) Let d , d x be a radial function, i.e. x 22 :dd TL L . Define an operator by 2, d. d d fL Tfxf xyxyfx T Then is a selfadjoint operator. That means, 2 ,,,,. d gLTfg fTg 2 ,. d fg L 2d Tf L Proof. Let We first note that since 221 ff. C opyright © 2013 SciRes. APM
E. AUYEUNG 439 ,d dd dd, dd, dd dd, dd d ,. dd dd dd dd dd dd dd d Tfgfxyxy gxx fyx y ygxx fy yxygxx d x x yyxygxyxy fyxgx yxy fy xgyxxyx fy xgyxxy fyg yy fTg xxy x Consider a :.BH HH bilinear operator ,,, gHBfg H , and B is linear in each of the two variables separately. with ac Let , d compt support, and d0,d1.is radial,.. dd . xxx iexx alize We norm so that 2 0ˆt d1. t t Write 1. d x tt t Define ,Bfg by: 0tt Bfg xf d ,. tt g xt (1) 0 d Consider,,d , d ntttn t Bfg efgex t where . n e H 0 d ,, d d ,by d d, d d d d ntttn tt tn tt tn ttn t Bfg efgex t fgxexx fgxexx fxgex x Lemma 2.1 here the last line follows from w ,,, withtn . t ThTf hhe We are now re g nstruct the frame operator. Let ady to co e 1 nn be a sequence in . Fix H. Define the alysis operator :an 2 Hl by 3 , ,, , . H MfBfg eBfg eBfge The frame operator :SH H is given by S nt operator 2 12 ,,,,, , f .M M adjoiof M. :. First, calculate the Hl 2 : 2, nn l ,, , H McfcMfcB fge Using ,SMM we obtain n lH . 12 ,,,,cc be a sequence in Let 3 cc2 l. Then, 123 ,,, ,,, ,,, Sf Mf MBfg eBfgeBfg e By the above calculations on ,Bfg, we see that: M ,, n Bfg e 0 dd ,d , d d tt tn n t xgexx t fxBeg xx Copyright © 2013 SciRes. APM
E. AUYEUNG 440 where 0 ntt Be g xe d ,. nt t gxt Therefore, , n Beg H , , nn 2 1 1 ,, ,, ,,,where l H nn n nn n Mcf cMf cBegf cBeg f 1 So, n ccBeg 2 , :Ml H . 123 ,,cccc , Hence, 1 ,, , nn H n SfBfgeB eg ,:.S HH i.e. , , nn HBeg. This is our frame We have constructed a framwith a bilinear . Let us summarize all our calculations in the g lemma. Lemma 2.2 (Frame operator with bilinear mapping) 1, n SffB eg operator. e operator mapping followin Let be the 2d L compact support, and Hilbert space d with . Let , d0, dxx d1. dxx is radial, i.e. x We normalize so that 2dt 0t ˆ1.t Write 1. d t xt t Consider a BH HH defined by Equation (1). H H is given bilinear operator : Then the frame operator :S by ,, , ,. nn n SfBfgeB eg Be g tablished th To prove that this bilinear operator is bounded, we need some preparation. Let 1 1n Proof. Our calculations ese lemma. ,, nn fBeg Lemma 2.3 be a separable Hilbert space (or a seble Banacace). Let parah sp be a dense sub space of . Let :BM HH be a bilinear ope rator such that ,,,. H H Then the above inality holds for all fMgHB fgCf g (2) equ in , for all in , and B extends to a bounded bilinear ope rator from H t o . ach Proof. For e in , there exists a sequence 1 nnM such that 0, nH ff as .n Since for each H , ,, , y2 mn Cf f g mn H m BfgBf g f g nH HH Bf So, for each ,Bf uchy quence in 1 , nn gH g is a Case . Hence, , n Bf g converges in to an element in , and we can define a bounded bilinear operator ,:BfgH HH by ,lim ,. nn BfgBf g Definition 2.4 (BMO) If is a locally integrable functio d say that n in , we BMO if there exists a constant , suce d Q, h that for any cub 1 d. Q Q xfxA Q Here, Q is the average of over the cube. The integration is over the cube. Te smallest bound h for which the above inequality is satisfied is taken to be the norm of , and is denoted by . MO 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 oBMO functions, to gether with Lemma 2.3, will allow us to establish the boundedness of the bilinear operator. Theorem 2.5 Let d be such that d0. dxx Let be a bounded, integrable func tion that is positive, radial, and decreasing. Write 1. td xt t Let bBMO . Then 22 0 2 2 dd d. d d tt BMO t bxt Cbf xx fx Definition 2.6 BMOL then If 2d f, 2d BMO fL. Remark 2.7 The space 2d BMO L is a dense sub space of 2d L in the topology of 2d L Theorem 2.8 (Boundedness of the bilinear perator) Assume the hypothesis of Lemma 2.2. Define . o ,g by: Bf 2d 0tt ,, d ,. fgL g xg tt t Bf f x Then there exists 0C such that for each n eH , Copyright © 2013 SciRes. APM
E. AUYEUNG Copyright © 2013 SciRes. APM 441 with 1, n e 22 22 ,, ,, . dd BMO n fL gL Bfg eCfg 2 BMO 2. d Let Proof. Let fLgL n eL w . d Let 2d , ith21. n e 0 0 22 0 12 2 0 12 ,, dd d d d d n ttt 12 dd dd dd n t tn tt tn Bfg e t t gxex x tt gxexx t t fxgx x t t ex x t II Let 2d. t t Then 0t Gx gx 22 1 22 GCg by Plancherel Theorem. Hence, 22 dd d. t 2 10 2 2 1 d d tt BMO fx Cf gx gxx t x In the last inequality, we used Theorem 2.5. Another m gives the following: application of Plancherel Theore 2 2 2 0 d dtn t Iex t 2 2 d. n xCe Hence, 22 ,, . nn BMO Bfg eCfge Therefore B is a bounded operator on 22dd LL BMO . Recall that the space e 2d BMO L 2d La 2.3, B 22dd . This 3. Acknowledgments The author is gratefdetto from teaching him the subject of [1] R. J. Dun and A. C. Schaeer, “A Class of Nonharmonic Fourier Seriesmerican Mathema tical Society, V341366. University of Maryland for harmonic analysis and the theory of frames. REFERENCES ,” Transactions of the A ol. 72, No. 2, 1952, pp. doi:10.1090/S00029947195200471796 [2] A. G. I. Daubechies and Y. Meyer, “Painless Nonortho gonal Expansions,” Journal of Mathematical Physics 27, 1986, pp. 12711283. , Vol. 697. .1137/1031129 [3] D. Han and David Larson, “Frames, Bases and Group Re presentations,” Memoirs of the American Mathematical Society, Vol. 147, 2000, p. [4] O. Christensen, “An Introduction to Frames and Riesz Bases,” SpringerBirkhauser, New York, 2003. [5] I. Daubechies, “Ten Lectures on Wavelets,” SIAM, Phila delphia, 1992. [6] C. Heil and D. Walnut, “Continuous and Discrete Wave let Transforms,” SIAM Review, Vol. 31, No. 4, 1989, pp. 628666. doi:10 atical Society, Vol. 212, 01, pp. 115156. atical Society, Vol. 44. [7] R. R. Coifman and Y. Meyer, “On Commutators of Singu lar Integrals and Bilinear Singular Integrals,” Transac tions of the American Mathem 1975, pp. 315331. [8] L. Grafakos and N. J. Kalton, “The Marcinkiewicz Multi plier Condition for Bilinear Operators,” Studia Mathe matica, Vol. 146, 20 [9] L. Grafakos and R. Torres, “Discrete Decompositions for Bilinear Operators and Almost Diagonal Conditions,” Transactions of the American Mathem 354, 2002, pp. 11531176. [10] N. Tomita, “A Hormander Type Multiplier Theorem for Multilinear Operators,” Journal of Functional Analysis, Vol. 259, 2010, pp. 202820 [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, D. Maldonado and V. Naibo, “What Is a Para product?” Notices of the American Mathematical Society, Vol. 57, No. 7, 2010, pp. 858860. [13] E. M. Stein, “Harmonic Analysis,” Princeton University Press, Princeton, 1993. C. Feerman and E. Stein, “Hp Spa is a dense subs ude pace of unded s the proo . By Lemm LL eorem. x ten concl ds to a bo operator on f of the th ul to Professor John Bene [14] ces of Several Vari ables,” Acta Mathematica, Vol. 129, 1972, pp. 137 193.
