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Gabor, Wavelet and Chirplet Transforms in the Study of Pseudodi Erential

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Gabor, Wavelet and Chirplet Transforms in the Study of Pseudodi Erential Gab or wavelet and chirplet transforms in the study of pseudo dierential yz op erators Ryuichi ASHINO Michihiro NAGASE Rmi VAILLANCOURT CRM January In memoriam NobuhisaIwasaki y To app ear in RIMS Symp osium on the Structure of Solutons for Partial Dierential Equations z This work was partially supp orted through aGrantinAid of Japan NSERCofCanada and the Centre de recherches mathmatiques of the Universit de Montral Abstract Gab or transforms and Wilson bases are reviewed in the study of the b oundedness of pseudo dif ferential op erators in mo dulation spaces and of the decay of the singular values of compact pseudo dierential op erators Similarly the wavelet transform is reviewed in the study of the L b oundedness of pseudo dierential op erators The parameter Gab or transform and the parameter wavelet transforms are generalized to parameter chirplet transforms in the context of signal pro cessing Op en questions concerning chirplets are addressed for p ossible applications to pseudo dierential op erators and Fourier integral op erators Keywords and phrases Pseudo dierential op erators Gab or transform wavelet transform chirplet transform Rsum On fait la synthse de lutilisation des transformations de Gab or et des bases de Wilson dans ltude de la continuit des op rateurs pseudodirentiels sur les espaces de mo dulation et de la mme on dcroissance des valeurs singulires des op rateurs pseudodirentiels compacts De fait la synthse de lutilisation des transformations en ondelettes dans ltude de la continuit L des op rateurs pseudodirentiels Les transformations deux paramtres de Gab or et en ondelettes se gnralisent aux transformation en p pielettes huit paramtres dans le contexte du traitement du signal On p ose des questions sur lutilisation p ossible des p pielettes dans ltude des op rateurs pseudodirentiels et Fourier intgraux Intro duction n n Given a symbol S R R the Weyl corresp ondence denes a pseudo dierential op erator n n L D X S R S R by the formula ZZ ixy L f x e x y f y dy d The Weyl representation is common in harmonic analysis and its relation with quantum theory The repre sentation of pseudo dierential op erators as a generalization of partial dierential op erators is ZZ ixy x D f x e x f y dy d In fact the name pseudodierential operator comes from its use in the study of parametrices of elliptic op erators A further generalization to Fourier integral op erators ZZ ixy Af x e ax y f y dy d was needed to obtain an asymptotic representation of the solutions of hyp erb olic partial dierential equations see also To establish the continuity of the op erator L on certain spaces of functions with symbol in certain classes of functions one often attempts to express the op erator as a sum of simpler almost orthogonal op erators or use other metho ds see for example In the use of the es the b est p ossible result whichismuch more precise than the results obtained bythe PaleyLittlewo o d giv use Cotlars lemma under the hyp otheses of the following theorem Theorem Let n and Nn be two integers Suppose that the symbol x and al l its derivatives n n x for jjN and j jN arecontinuous on R R and verify x j jjj j x jC j j jjN j jN x n are two constants Then the operator x D is boundedonL R where and C In the context of op erator theoryinYves Meyer has used Littlewo o dPaley wavelets which provide p n unconditional bases for the spaces L R toprovethe T theorem of David and Journ and thus to derive the L continuity of nonclassic pseudo dierential op erators in a very simple way and study their symb olic p calculus For the L theory of pseudo dierential op erators see for example Wavelets have also found applications in new trends in microlo cal analysis In several microlo cal spaces are considered A windo wed Fourier analysis of hyp erb olic chirps is used as a digression to intro duce a more involved hyp erb olic paving of the timefrequency plane pro duced by the unfolding of an orthonormal wavelet basis called an adapted basis In the context of signal pro cessing in and Gab or transforms are used to establish the continuity of pseudo dierential op erators on mo dulation spaces and to study the decayofthesingularvalues of compact pseudo dierential op erators n In and KazuyaTachizawa has used Wilsons bases which are unconditional bases for L R p n but not for L R with p to establish the b oundedness of pseudo dierential op erators on mo dulation spaces and to study the sp ectrum of these op erators The Gab or transform or shorttime Fourier transform STFT pro duces a shift in time and in frequency in the timefrequency plane of a given analyzing compactlysupp orted window g On the other hand the wavelet transform WT pro duces a shift and a scaling in time in the time domain of a given analyzing mother wavelet In and the STFT and WT are generalized to the chirplet transform by considering an parameter transformation of the timefrequency plane By restricting this generalization to planar subspaces one obtains useful transformations such as bowtie disp ersion warblet and p ersp ective transforms It is an op en question whether results for Fourier integral op erators would follow simply bytheuseofchirplet expansions The mo dulation space approachis not new in the sense that it is a typ e of coherent state expansion and coherent states have b een around a long time The new asp ect is that Feichtinger and Gr chenig haveshown that coherent states are concrete bases or basislike systems for the mo dulation spaces and as a result one has convenient equivalent norms for these spaces in terms of the basis co ecients and this fact can be applied to derive results on pseudo dierential op erators In one sense this is similar to howwavelets have simplied some asp ects of the LittlewoodPaley theory Wavelets nowprovide concrete unconditional bases for a large class of function spaces In this sense Besov spaces are the wavelet analogue of the mo dulation spaces And Yves Meyer has used wavelets to go o d eect in the study of integral op er ators Wavelets are simpler and easier to use than say functiondep endent atomic decomp ositions The Gr chenigHeil approach is complementary it uses uniform decomp osition of phase space instead of dyadic decomp ositions but it is the basislike asp ect that makes things simple Preliminaries Weintro duce general notation Inner pro duct Z n fg f x g x dx f g L R Fourier and inverse Fourier transforms Z ix f F f e f x dx Z ix f xF f xf x e f d Translation T and mo dulation M ix n T f xf x y M f xe f x x R y Dilation n D f xjaj f xa a R nfg a We notice that iy F T f M f F M f T f M T e T M y y y y and F D f D f a a n n The shorttime Fourier transform STFT of a function f S R with resp ect to a window g SR Z iy S x fM T g e g y x f y dy g x n n The wavelet transform WT of a function f S R with resp ect to a window g SR is Z x y g y dy x ajaj S g g a n n n n The Schr dinger representation of the Heisenb erg group H R R T on L R ix iy x f y e e f y x If the toral variable is unimp ortant with x the reduced Schr dinger representation is f y x f y ix iy e e f y x ix T f y e M x The radar ambiguity function Af goff and g Af gx x f g ix e S f x g n n The Wigner distribution W f g of f and g is the R R Fourier transform of the ambiguity function W f gF Af g n The mappings A and W are sesquiunitary on L R Af g Af g f f g g Wf g Wf g The last equalityisMoyals identity n n Using the linear transformation M R R M where each blo c kisamultiple of the n n identity matrix wehave W f g M g W f g n n where and are the Schr dinger representations on L R and L R resp ectively The pseudo dierential op erator L can b e expressed in the following forms L f g b Ag f n Wg f f g SR Mo dulation spaces n n R Aweight function on R Mo dulation spaces measure the joint timefrequency distribution of f S n w R satises the inequality s w x y C jxj w y for some s n n Denition Given a weight function w on R a window g S R and two numbers p q such that w n p q we dene the mo dulation space M R to b e the Banach space of all distributions of slow pq n growth f S R for which the following norm is nite Z Z q pq p p w d kf k jS f x j w x dx M g pq with obvious mo dications if p or q n s s n If w we write M R and if w x jxj j j wewriteM R If w x w x pq pq w n then M R is invariant under Fourier transformation since pp S f x S f x g g The following wellknown function spaces are mo dulation spaces n n R L R M s The weighted L spaces with weight w x w x jxj s w n n s n M R L R ff f x jxj L R g s s The Sob olev spaces with weight w x w j j s s n s n w n R ff f j j L R g M R H s The space with weight w x jxj j j s n n s n M R L R H R s n The Feichtinger algebra M S R p is a very nice space For It is to b e noted that M is not equal to L The Feichtinger algebra M pp example it is an algebra under b oth p ointwise multiplication and convolution It is invariant under Fourier transform It is the minimal Banach space that is isometrically invariant using its own norm under b oth p translations and mo dulations It is contained in every L spaces in the Fourier algebra consisting of Fourier transforms of L functions and in the Wiener algebra C l of continuous functions
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