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The Use of P- Modules to Study Exponential TOPICS IN COMPLEX ANALYSIS n centerlineBANACH CENTERfTOPICS PUBLICATIONS IN COMPLEX ANALYSIS comma VOLUMEg 3 1 INSTITUTE OF MATHEMATICS nnoindentPOLISH ACADEMYBANACH CENTER OF SCIENCES PUBLICATIONS , VOLUME 3 1 WARSZAWA 1 9 9 5 TOPICS IN COMPLEX ANALYSIS n centerlineTHE USE OFfINSTITUTE D hyphen MODULES OF MATHEMATICS TO .. STUDYg BANACH CENTER PUBLICATIONS , VOLUME 3 1 EXPONENTIAL POLYNOMIALS INSTITUTE OF MATHEMATICS n centerlineC period ..f APOLISH period .. ACADEMY B ERENSTEIN OF SCIENCES g POLISH ACADEMY OF SCIENCES Mathematics Depa to the power of r-t ment ampersand Institute of Systems Research comma .. University of Maryland WARSZAWA 1 9 9 5 n centerlineCollege ParkfWARSZAWA comma Maryland 1 9 9 20742 5 g comma .. U period S period A period THE USE OF D− MODULES TO STUDY A period .. YGER EXPONENTIAL POLYNOMIALS n centerlineD acute-e pafTHE to the USE power OF of t-r$ D ement− de$ Math MODULES acute-e TO matiquesnquad commaSTUDY ..g Universit e-acute de Bordeaux I C . A . B ERENSTEIN 334 5 Talence comma France Mathematics Depar−t ment & Institute of Systems Research , University of Maryland n centerlineAbstract periodfEXPONENTIAL .. This is a summary POLYNOMIALS of recentg work where we introduced a class of D hyphen modules College Park , Maryland 20742 , U . S . A . adapted to study ideals generated by exponential polynomials period A . YGER n centerline0 period .. IntroductionfC. nquad periodA. ..nquad This isB an ERENSTEIN expanded versiong of the lecture given by the first D epa´ t−r ement de Math e´ matiques , Universit e´ de Bordeaux I author at the Banach Center comma during the workshop on residues comma November 1 992 period 334 5 Talence , France n centerlineWe introducefMathematics a new method t o $ study Depa ideals ^f r generated−t g$ mentby exponential $ n& $ polynomials Institute comma of Systems Research , nquad University of Maryland g Abstract . This is a summary of recent work where we introduced a class of D− modules inspired by the theory on D hyphen modules open square bracket 1 6 comma 1 7 comma 1 8 closing square bracket adapted to study ideals generated by exponential polynomials . periodn centerline .... Detailedf College proofs of Park some ,of Maryland the 20742 , nquad U.S.A. g 0 . Introduction . This is an expanded version of the lecture given by the first statements of this paper can b e found in open square bracket 14 closing square bracket period .... We take the author at the Banach Center , during the workshop on residues , November 1 992 . opportunityn centerline t of thankA. nquad YGER g We introduce a new method t o study ideals generated by exponential polynomials , Professors Jak o-acute b czak comma Ple acute-s niak comma and Aizenberg for their hospitality period inspired by the theory on D− modules [ 1 6 , 1 7 , 1 8 ] . Detailed proofs of some of the n centerlineLet us recallfD that $ ann exponentialacutefeg polynomialpa ^f t f− ofr ng complex$ ement variables de Math with $ nacutefeg $ matiques , nquad U n i v e r s i t statements of this paper can b e found in [ 14 ] . We take the opportunity t o thank $ nfrequenciesacutefeg in$ a finitely de Bordeaux generated I subgroupg Capital Gamma of C to the power of n i s a function of the form Professors Jako ´ b czak , Ples ´ niak , and Aizenberg for their hospitality . Line 1 f open parenthesis z sub 1 comma period period period comma z sub n closing parenthesis = f open parenthesis Let us recall that an exponential polynomial f of n complex variables with zn centerline closing parenthesisf334 = 5 sum Talence p sub gamma , France open parenthesisg z closing parenthesis exponent open parenthesis gamma times frequencies in a finitely generated subgroup Γ of n i s a function of the form z closing parenthesis comma Line 2 gamma in Capital Gamma C n hspace ∗fn f i l l g Abstract . nquad This is a summaryX of recent work where we introduced a class of where the sum is finite comma the p subf(z gamma; :::; z ) are= f polynomials(z) = p ( commaz) exp(γ ....· z) and; gamma times z = gamma 1 to the power$ D of− z 1$ plus modules period period period plus gamma1 n n to the power ofγ z n period Such a function belongs t o the algebra A sub phi open parenthesis C to theγ 2 powerΓ of n closing parenthesis .. of entire nnoindent adapted to study ideals generated by exponential polynomials . functions F satisfying z z where the sum is finite , the pγ are polynomials , and γ · z = γ1 1 + ::: + γn n: t h-line sub e growth condition : n n hspace ∗fn fSuch i l l g0 a function . nquad belongsIntroduction t o the algebra . nAquadφ(C )This of entire is an functions expandedF satisfying version t h − ofline thee lecture given by the first exists C greatergrowth 0 bar condition F open : parenthesis z closing parenthesis bar less or equal C exponent open parenthesis C phi open parenthesis z closing parenthesis closing parenthesis comma nnoindent author at the Banach Center , during the workshop on residues , November 1 992 . 1 99 1 Mathematics Subject Classification9C : 32 > A0 1j 5F comma(z) |≤ C 32exp( A 25Cφ comma(z)); 34 K 5 comma 34 K 35 comma 32 S 40 period nnoindentResearch ofWe the1 introduce 99first 1 Mathematics author apartially new Subject method supported Classification t by o the study: NSF 32 A ideals grants 1 5 , 32 DMS A generated 25hyphen , 34 K 5 9000619 , by34 K exponential 35 and , 32 CDR S 40 . hyphen polynomials , 880301 2 periodResearch of the first author partially supported by the NSF grants DMS - 9000619 and CDR - nnoindentResearch ofinspired880301 the second 2 . by author the partially theory supported on $ D by the− NSF$ modules grant DMS [ hyphen 16 ,17 9000619 ,18 period ] . n h f i l l Detailed proofs of some of the The paper is in finalResearch form ofand the no second version author of it partially will be published supported elsewhereby the NSF period grant DMS - 9000619 . nnoindentopen squarestatements bracket 77The closing of paper this square is in paper final bracket form can and b no e version found of it in will [ be 14 published ] . n h elsewhere f i l l We . take the opportunity t o thank nnoindent Professors Jak $ nacutefog $ bczak[77] , Ple $ nacutef s g $ niak , and Aizenberg for their hospitality . n hspace ∗fn f i l l g Let us recall that an exponential polynomial $ f $ of $ n $ complex variables with nnoindent frequencies in a finitely generated subgroup $ nGamma $ o f $ C ^f n g$ i s a function of the form n [ n begin f a l i g n e d g f ( z f 1 g , . , z f n g ) = f ( z ) = nsum p f ngamma g ( z ) nexp ( ngamma ncdot z ) , nn ngamma n in nGamma nendf a l i g n e d g n ] nnoindent where the sum is finite , the $ p f ngamma g$ are polynomials , n h f i l l and $ ngamma ncdot z = ngamma 1 ^f z g 1 + . + ngamma n ^f z g n . $ nnoindent Such a function belongs t o the algebra $ A f nphi g ( C ^f n g ) $ nquad of entire functions $ F $ satisfying t $ h−l i n e f e g$ growth condition : n [ n exists C > 0 nmid F ( z ) nmid n leq C nexp (C nphi ( z ) ) , n ] n centerline f1 99 1 Mathematics Subject Classification : 32 A 1 5 , 32 A 25 , 34 K 5 , 34 K 35 , 32 S 40 . g Research of the first author partially supported by the NSF grants DMS − 9000619 and CDR − 880301 2 . n centerline f Research of the second author partially supported by the NSF grant DMS − 9000619 . g n centerline fThe paper is in final form and no version of it will be published elsewhere . g n [ [ 77 ] n ] 78 .. C period A period BERENSTEIN AND A period YGER nnoindentwhere the78 weightnquad phi canC .be A taken . BERENSTEIN as bar z bar comma AND A the . YGER Euclidean norm of z comma or even better comma if we choose a system gamma to the power of 1 comma period period period comma gamma to the power of N comma ofnnoindent Q hyphen linearlywhere independent the weight generators $ nphi of Capital$ can Gamma be taken comma as as $ nmid z nmid , $ the Euclidean norm of $zphi open , $ parenthesis or even z closingbetter parenthesis , = maximum open parenthesis bar Re open parenthesis gamma to the power 78 C . A . BERENSTEIN AND A . YGER of j times z closing parenthesis bar : j = 1 comma period period period comma N closing parenthesis plus log open where the weight φ can be taken as j z j; the Euclidean norm of z; or even better , parenthesisnnoindent 1 plusif we bar choose z bar to the a system power of 2 $ closingngamma parenthesis^f 1 g comma,..., ngamma ^f N g , $ if we choose a system γ1; :::; γN ; of − linearly independent generators of Γ; as o fwhere $ Q Re− z denotes$ linearly the real part independent of the complex generatorsQ number z period of .. $ InnGamma the case that, $ Capital as Gamma subset i R to the power of n comma the exponential polynomials are just the Fourier transforms of distributions φ(z) = max(j <(γj · z) j: j = 1; :::; N) + log(1+ j z j2); n [ supportednphi ( by finitely z many) = pointsnmax in the lattice( n minusmid i CapitalnRe Gamma( ngamma comma and^f Aj subg phincdot i s a subalgebraz ) ofn themid :j=1,...,N)+ n log ( 1 + nmid zn nmid ^f 2 g Paley endashwhere Wiener<z denotes algebra the E to real the part power of the of prime complex hatwidest-parenleft number z: In the R ncase closing that Γparenthesis⊂ iR ; the .
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