Interpolating varieties for weighted spaces of entire functions in Cn

Publicacions Matemàtiques, Vol 38 (1994), 157-173 . INTERP OLATIN G VARIETIE S FOR WEIGHTED SPACE S OF ENTIRE FUNCTIONS IN C n CARLOS A . BERENSTEIN AND BAO QIN L I A bstract  We prove in this paper that a given discrete variety V in C n is a n interpolating variety for a weight p if and only if V is a subset of th e variety {e E C n : f l (1) = f 2 (1) = • • • = f n (e) = O} of m function s f 1, • • • , fm in the weighted space the sum of whose directiona l derivatives in absolute value is not less than E exp(—Cp(()), ( E V for some constants E, C > O . The necessary and sufficien t conditions will be also given in terms of the Jacobian matrix o f f , • • • , f,n . . As a corollary, we solve an open problem posed b y Berenstein and Taylor about interpolation for discrete varieties . 1 . Introduction . In this note, we shall study the problem of findin g necessary and sufficient conditions for a given discrete subset of C n to b e interpolating for spaces of entire functions in several complex variable s satisfying growth conditions . Let V = {Oc} be a discrete subset of C n and p(O, _, (, . .• , ~ n ) E C n , a plurisubharmonic weight function on C n . We consider the follow - ing interpolation problem: under what conditions is it true that far an y sequence {a k } of complexnumbers satisfying the growth conditio n lk E N :_ {1, 2, • • • } for some constants A, B > O, there exists an entire function f on C n suc h that f (( k ) = a k and f satisfies the same kind of estimate for E e n , namely, f E A(C) or equivalentl y .Í <A'exp ( B'p (1) ) This research is supported in part by NSF grants DMS 92-25043 and CDR 88-03012 . 158  C . A . BERENSTEIN, B . Q . L I for some constants A ' , B' ~ O . We will then say that V is an interpolatin g variety (for the weight por the space A p (C n } } . In the case when V is a complete intersection and defined by so -calle d slowly decreasing functions, some interpolation results have been know n in [BT2], where a vector function F = (F 1 , F 2 , • • • , F n } , F i E A p (C n ) , is called slowly decreasing if and only if there exist e, C, C 1 , C 2 > 0 suc h that (i) the connected components of the set S(F ; E, C) are bounded , where S(F ; E, C) = {e E C n : 1= ( = I1)I 2 ) h/2 < E exp(—Cp(e)} } ; and (u) if SZ is a component of S(F ; E, C), then p(l) Ç C1p(w) + C 2 , for allwESZ . Note that the definition depends on the weight function p . F ma y be slowly decreasing for one weight and not for another . We refer t o [BT2] for a class of examples of slowly decreasing functions . With thi s notion, the interpolation theorem in [BT2I about discrete varieties ca n be stated as follows : Let F = (F 1 , F 2 , • • • , F a ), F i E A p (C n ) , be slowly decreasing . Assum e further that the points {k} of F(1) = 0 are simple ; that is , det J F (( k ) 0 (J F W = Jacobian matrix of F) . Then V = {ÇO is interpolating for A p (C n ) if and only if there exis t constants c > 0, C > 0 such tha t 1  ? e exp(—Cp((k)), k E N . For n = 1, p(z) = p (1 z 1), this result is due to A . F . Leont'ev [L] . Fo r n = 1, p(z) = ~ Im z i + lo g (1 + lzD, this result is due to Ehrenpreis an d Malliavin [EMI . For n = 1 and general weights p, this result is given i n [BT1] . Note that the aboye result only applies to the case of discrete varietie s V = {( k} which are exactly complete intersections of ri slowly decreasin g functions F 1 , F 2 , • • • , E n , i .e . , V = Z(F 1 , Fa, . . . , F am ) := {1 E C n : FI(1) = F 2(1) = ... = F n (e) } with F = (F 1 , F 2 , • • • , F n } slowly decreasing . It was pointed out i n [BT2, p . 210] that the case when the variety is nat such a complet e intersection seems to be quite different . Therefore it is a natural goal t o try to find conditions without the aboye hypotheses on the varieties, i n particular, to find conditions necessary and sufficient for interpolation INTERPOLATING VARIETIES FOR WEIGHTED SPACES IN e n  15 9 which apply to any given discrete variety . W e shall salve this problem i n the present note . It turns out, roughly speaking, that a discrete variet y V is interpolating for p if and only if V is a subset of the analytic variet y Z~ f ~ , • • • , f ,~ } for m(> n) functions f 1 , • • • , f,, . t E Ap the sum of whos e directional derivatives in absolute value is not "too small" (see Theore m 2 .5) . As a corollary, we salve an open problern posed by Berenstein an d Taylor in [BT3, p . 10] about interpolation far discrete varieties (se e Remark 2 .8) . 2 . Definitions and results. First of all, let us fix the notation s which we shall use throughout this paper . A plurisubharmonic function p : C n --*[0, oo) is called a weight functio n if it satisfies the following conditions (c .f . [BT2]) : (2 .1)  log(l + lel 2 ) = O(p(0 ) and there exist constants C l and C2 such that — wi < 1 implie s (2 .2)  p () c i p ( w ) + C 2 . Definition 2 .1 . Let A(C n ) be the ring of all entire functions o n C n .The n A p = A p (C n ) = = {f E A(C n ) : lAexp(Bp()) for some A, B > 0} . It is not the specific conditions on p which are important, but rathe r their consequences for the ring A p . It follows from (2 .1) that A p (C n ) contains the polynomials and, from (2 .2), that f E A p (C n ) implie s á~ E A p (C n ) (see e .g . [HoJ) . One can replace (2 .2) by the following Hórmander's condition ([Ho]) : there exist four positive constants c 1 , • • • , c 4 such that — wi Ç exp(—c i p(w) — C2) implies tha t pW Ç c3p(w) + c 4 . We use (2 .2) only for the sake of convenience . Remark 2 .2 . The two basic examples of such weight functions ar e p(O = lCV( p > 0) and (C) = ~ Im el + log ( 1 + 11 2 ) corresponding to th e space A p of all entire functions of order Ç p and finite type and the spac e E' (Rn ) of Fourier transforms of distributions with compact support i n R n (see e .g . [ E l) . Definition 2 .3 . Let V = {(k} be a discrete variety on C n , i .e ., a discrete sequence in C n with 1( k ~ fi oo as k ~ oo . The n A p (V) = {a = {ak}kEN : 2A, B > 0, la k l Aexp(Bp(( k )), Vk E N} . 160  C . A . BERENSTEIN, B . Q . L I With . the aboye definitions, the interpolation problem is simply t o determine when the following restriction map p : A p ---} A(V) define d by p( f) = { f ((O}, is onto from . A p to Ap (V) . Definition 2 .4 . A discrete varietyV = {( k } is an interpolating variety for A p -= A p (C n } if the restriction map p is onto from A p to A p (V ) . Now we can state our main theorem and its corollaries, whose proof s will be given in the next section . Theorem 2 .5 . Let V = {( k} be a discrete variety on C n and m(> n ) be an integer . Then V is an interpolating variety for Ap(C n ) if and onl y if there exist m functions f l , f 2i • • • ,f m E A p (C n ) such tha t (2 .3)  .V C 7i lfl~ .f2~ . . . ~ .fm l and for some c, C ~ o I? Ee)¿p(—CP((k)), Vk E N, u E S 2n— 1 i = l where D u f (1) := j_u i + • • • + jun is the directional derivative of f along the direction u E S 2n -1 . H er e S 2n—1 '_ {S =(51, . . .  ' 111 '— (iI2 + . . . -}- e = l J . Corollary 2 .6 . Let V = {(k} be a discrete subset of C n and m(> n ) be an integer . . Then V is an interpolating variety for A p (C n ) if and onl y if there exist m functions f 1 , f 2 , • • • , f, ., . E A p such tha t V C Z( ,f2, . . . J m ) and for each k E N, there exists an n x n miñor ~ of the Jacobianmatri z of f i ,••• , f m such tha t ~? Eexp(—Cp((k)) , where €, C are two positive constants independent of k . Corollary 2 .7 . Let V = {(k} be a discrete subset o f C n . Then V i s an interpolating variety for A p (C n ) if and only if there exist n function s f2,  f9 ' L E Ap such tha t V C Z ffi , f 2y  frzl INTERPOLATING VARIETIES FOR WEIGHTED SPACES IN C n  16 1 and f or some €, C > 0 det J~ x, . . .  ((k) f ~ E exp(—Cp(( k ) } a k E N , where Jf1, . . . , fn is the Jacobian matriz of f I , • • • , f n . Let us mention here that the sufficiency of Corollary 2 .6 or 2 .7 ca n not follow from [BT3] where the variety was again restricted to be th e complete intersection of some functions in A p . In the case n = 1, a stronger version of Corollary 2 .7 which allows arbitrary multiplicities ca n be given . We refer the reader to our papers [BL1], [BL2] and [BLV] fa r related results in this direction . Remark 2 .8 . Observe that when the conditions (2 .3) and (2 .4) ar e satisfied for a weight p, they are automatically satisfied for any weigh t q > p . This gives an affirmative answer to an open problem in [BT3, p . 10] for discrete varieties : Let V be an interpolating variety for A p (C n ) and q another weight satisfying q > p . Is V interpolating forA q (C n ) ? We conclude this section by providing an interpolation example usin g Theorem 2 .5 . Example 2 .9 . Let pi : C - -} [0, oo) be weights in C (1 Ç j Ç n) an d V i = { z k,i }11 1 be an interpolating variety for A pa (C) . Then we clai m that V := «k = (Ck,1, • ,(k,n)  E V .? }7 :1 1 is an interpolating variety for Ap (c n ) , where p(e) = p i (e) + • • • + p n (In ) for ~ -- ( e I, . . . ' en) . In fact, since V i is an interpolating variety for A p i , we know that b y Corollary 2 .7, there exists an entire function f i (z) in .r4p i (C } such tha t V~ C Z(f i ) and far some Ea , c i > 0 , .f . ;((k,i) ~ Ei exp ( —c i p i (g - k,j) } . Let F i(1)  fi(1 .i) ( 1 ~  n ) . Then clearly, F i E A p (C n ) and V c Z(F l , • • • , F n ) . Moreover , det JF t(k) ? E exp(--cp((k)} , with c = max c i , e = E l • • • En . Now weconclude by Corollary 2 .7 that V is an interpolatingvariety for Ap (C n ) . Furthermore, V 1 x • • • x V n is als o an interpolation variety . As a concrete example, we see that the lattic e V_,?2x . . .xz2 = =  ,(n .) C C n :  E 2'2, Ç . 7  n}, 162  C . A . BERENSTEIN, B . Q . L I where Z 2 = {m + in : m, n E Z}, is an interpolating variety for A P (C n ) with p(e) _  =  + • • • + 11n1 2 for = (Ci,•• • , fin) E C om , since i t is known that Z 2 is an interpolating variety for A P (C) with p(z) = lzl 2 for z E C . 3 . Proofs of Theorem 2 .5 and its corollaries . Let us first prov e the following lemmas . Lemma 3 .1 . Let {( k } be a discrete subset of C n and 6 k ç k l} . If for some constants e, B > 0 (3 .1)  S k > eexp(—Bp((k)), k E C then there exists a M~ 0 such tha t E exp(—Mp((k)) < oo . k= 1 Proof : Denote nk = min {l,S k } and ,= B ((k nk ) := {e E Cn : le – (k ~ 7 10 - Let dV(1) be the Euclidean volume element in C nand 113k1 = f~ k dV(1) . Then (3 .1) implies that I B kl ? e l ex P( —B i p (( k) ) for some positive numbers e l , B l . We then conclude that for large M , E eXP(—Mp((k)) = E TT L k exP(—Mp(Srç))dV (1) ~ 1 E f exp((B i — M ) p (( k)) dV ~~) ~ _ El k–1 k < cz(Bi — M) É f eX p (Ci( B i — M ) p (C)) dV (1) ~ k= 1 C , a( B 1 — M) ¡ n  J eXP(C i (B i — M)p(())dV(I) < o 0 by virtue of the property (2 .2) and (2 .1) of the weight p . ■ k=1  k=1 INTERPQLATING VARIETIES FOR WEIGHTED SPACES IN C n  16 3 Lemma 3 .2 . Let f 1, • • • , f, n be m(> n) entire functions in C n . The n m (3 .2)  ~ ~? eexp(—Cp((k)), Vk E N, u E S 2n1 j= 1 for some constants E, C > o if and only if far each k E N, there exists a n x n minor J'C of the Jacobian matrix J fl . . . fm of f 1 , • • • , f r , L such tha t (3 .3)  ~  ? E l exP(—cip((k)) , where Ei, C 1 are two positive constants independent of k . Proof : For any matrix A = (aj á ), we define I = E i ,i  . The n it is easy to verify that II AB II < II A IIXII B I I far any matrices A and B provided that the left hand side of the aboy e inequality makes sense . Suppose that there exist some constants E, C > o such that (3 .2) holds . Denote J := J fx , . . . , fm , which is a m x n matrix - valued function . The n for any u = (u 1 . • - u n ) E S2n-- 1, if Ju' = v, the n (3 .4)  Il v ((k)II ? eexp(—CP((k)) , where u t denotes the transpose of u . This shows that for each k, th e kernel of the mapping J(( k ) : C n —> C m , defined by J(u) = J • u t for u = (u i , . . . , u n ), is zero and thus the dimension of the image 3(C ) is n . Up to an isomorphism, we can identify J (Cn ) with the space C n . It is then clear that there exists an operator T : C m —} C n , given by a n x m matrix such that Tv t = v' for v E,7 (C n ) and HTII Ç L for som e constant .L > 0 . Let Q = Tj . Then it is easy to check that the imag e Q (C n ) is the whole space C n . Thus, the matrix Q is invertible . No w set P = Q —1 T . Then PJ = (T3)T3 = E n , the n x n unit matrix . By the well-known Binet - Cauchy theorem (seo e .g . [A]}, any r -rowe d determinant of P~ is equal to a sum of terms each of which is a produc t of an r -rowed determinant of P and an r -rowed determinant of J . I n particular, we have tha t N (3 .5)  E(det P l )(det ) = det E me, = 1, 164  C . A . BERENSTEIN, B . Q . L 1 where N is an integer only dependingon n and m, and ~z and a ar e n x n minors of P and ,T, respectively . On the other hand, far an y u E S 2n-1 , we have that Qu t = Tju t = j u t and so that by (3 .4) , llQu t il > eexp(—cp ((k )) • Notice that det Q = A l • • • an, where a i s are eigenvalues of Q . Let  E S 2nl be the unit eigenvector corresponding to a i . The n Q u S ,a = à i va j . Hence n P ■ il  = 1IQ uII ? Eex P( —cp ((k) ) l A il ? ~eexP(—Cp ((k )) • W e then have that n 1  ? ~ exp(—Cnp((k)) • It is obvious tha t IIQ((k)II = I~ I x I Ç LII .7((k)II  LAex P( Bp ( (k ) ) for some constants A, B > 0 since the f l , • • • , f m are in the space A p and A p is closed under differentiation . Thus, Q * (( k ) < A l exp(B l p(0 ) for some constants A l , B 1 > O, where Q * denotes the adjoint matrix o f Q . It now follows tha t  1   Q * ((k) c A 2 exp ( B 2 p ~Sk) ) ilQ((k)I1 = Il detQ((k) I I and thus tha t  IIPII =  L I1Q -1 11 ~ LA2e x p( B 2p((k)) • Therefore, each n x n minor P I (1 = 1, • • • , N) of P satisfies tha t ~ det  < A 3 exp(B 3 p(( k )) . It follows from (3 .5) that there exists at least one of 1 (1 = 1, • • • , N ) such that 1 1 det ~ ~ > NA3 exp(—B3p((k)) . or 1NTERPOLATING VARIETIES FOR WEIGHTED SPACES IN C n  16 5 This concludes the proof of the necessity . Conversely, if (3 .3) holds far some e l , C 1 > O . For any u E S 2nT1 , w e let 1Cu' = v . Then u t = 1C'v and thu s (3 .$)  II~II <_ Ix lvii,  or  Il v ll ? I ' Notice that JC —1 = d l * lc , where JC* denotes the adjoint matrix of 1C . Since A p is closed under differentiation, we deduce tha t II K* (OII 5 A 4 eXp ( B 4 p O, E C n and thus that by (3 .3) , I~ A 4 eX P(( B 4 ~- C l) p ( e )) , for some constants A 4 , B 4 > O . We then obtain that, by (3 .5) and takin g into account that ii u il ~ ~ for any u E S"' , ~i vll ? E l for some constants e, C > O . Henc e m E I D u .fi((k)I = 113'41 ? 11 Kut II ? eexp(—CP((k)) • j= 1 The proof of the lemma is thus complete . ■ We are now going to prove Theorem 2 .5 . In the sequel, we shall use A and B to denote positive constants the actual values of which may var y from one occurrence to the next . . Proof of Theorem 2 .5 : Sufficiency : For any fixed k E N, consider th e entire functio n fi,u (z) := ,f i (( k + uz) : C , c , Then 1 Ç j Ç m, u = (u i , . . . , un ) E S 2 ' 1 , f, u( o ) = af~~(k) u ~ + af ~~ (k) u2 + . + a a~~sk) un =D u f i (( k ) Therefore, by (2 .4), E lf 'j,u(o)l ~ E exp(-Cp((k)) . J=1 172  C . A . BERENSTEIN, B . Q . L I It now follows from Lemma 3 .1 tha t E exp((C — 2M)p(( k )) := D < oo , k= 1 provided that M is large . Combining this result with (3 .17), we kno w that the series (3 .13) is uniformly convergent in compact sets of C n and so f i is an entire function on e' . Furthermore , !MI) Ç A exP(Bpn x D , that is, f i E AP (Cn ) . The proof of Theorem 2 .5 is thus complete . ■ Proof of Corollary 2 .6 : The corollary follows directly from Theore m 2 .5 and Lemma 3 .2 . ■ Proof of Corollary 2 .7 : The corollary is obtained from Corollary 2 . 6 bytakingm — n . ■ Reference s [A] A . C . AITKEN, "Determinants and matrices, " Interscience Publishers, Inc ., New York, 1962 . [BG] C . A . BERENSTEIN AND G . GAY, "Complex variables, an introduct2on," Springer - Verlag, New York, 1991 . [BL1] C . A . BERENSTEIN AND B . Q . LI, Interpolating varieties for spaces of meromorphic functions, to appear in J . Geometric Analys2s . [BL2] C . A . BERENSTEIN AND B . Q . LI, Interpolation problems wit h growth conditions for entire functions in one and several comple x variables, preprint, 1993 . [BLV] C . A . 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