On interpolation and sampling in Hilbert spaces of analytic functions

ON INTERPOLATION AND SAMPLING IN HILBERT SPACES OF ANALYTIC FUNCTIONS Bo Berndtsson and Joaquim Ortega Abstract. In this paper wegive new pro ofs of some theorems due to Seip, Seip-Wallsten and Lyubarskii-Seip on sequences of interp olation and sampling for spaces of analytic functions that are square integrable with resp ect to certain weights. The results are also given in a somewhat more general setting. 1. Introduction In a series of recent pap ers Seip S], Seip-Wallsten S-W] and Lyuabarskii-Seip L-S] have studied sets of interp olation and sampling for various spaces of analytic functions of one variable. Part of these results concern Hilb ert spaces of functions that are square integrable against certain weights, and another, closely related part deals with similar spaces with uniform norms. The metho ds used in the cited pap ers are based on classical-typ e but intricate constructions of one-variable nature that to some extentgoback to Beurling B]. In O] Ohsawa has suggested the use of L 2 -techniques for  @ to prove results of the ab ove typ e. In particular, Ohsawa gives a pro of of the suciency part of the theorem of Seip-Wallsten concerning interp olation in the space of entire functions in C satisfying Z j f j 2 e ;j z j 2 < 1 : As in the approach initiated byBombieri, Hormander and Sko da (see H]), the main di- culty in such a pro of is the construction of a (pluri)-subharmonic function with prescrib ed singularities at the points where one wishes to interp olate. For this Ohsawa uses part of the constructions of Seip-Wallsten, ultimately going back to Beurling, and he p oses as a problem to give a more elementary pro of. One purp ose of this note is to show how that can be done. As it turns out, the metho d we use also works for more general weights and therefore also implies the suciency part of the theorem by Lyubarskii-Seip. Furthermore, we shall show how the p ositive direction of the sampling theorem can be obtained in a similar manner, and we shall also permit somewhat more general growth conditions than Lyubarskii-Seip (see Theorem 2). In O] Ohsawa also gives a new pro of of the theorem of Seip ab out interp olation in Bergman spaces of the disk. This part of Ohsawa's work contains two essential ingredients. The rst one is, as in the case of entire space, the construction of a subharmonic function First author supported by the NFR Second author was partially supp orted by the DGICYT grant PB92-08084-C02-01. The research necessary to conduct this work has b een partially supported by the Comissionat p er Universitats i Recerca de la Generalitat de Catalunya Typ eset by A M S -T E X 1 2 BO BERNDTSSON AND JOAQUIM ORTEGA with prescrib ed singularities. In O] no details of this construction is given, so in section 3 of this pap er we shall showhow the construction in entire space can b e adapted to the disk case. For this we use a so called invariant convolution in the disk, intro duced by Ulrich U]. The other ingredient in Ohsawa's pro of is a generalized version of Hormander's L 2 - estimates for  @ , inspired by a theorem of Donelly and Feerman D-F]. Ohsawa deduces this L 2 -estimate from a more general version involving vector bundles over Kahler manifolds. Since this terminology probably is not so well known among sp ecialists in one complex variable, we shall also take this opp ortunityto give a direct pro of for the disk (see also Be1] for a related argument). In section 3 we shall also prove the p ositive part of the sampling theorem, and show that both the interp olation and the sampling theorems of Seip S2] hold for more general weights. The theorems of Seip et al also concern spaces that are dened by other norms than L 2 , notably uniform norms. In section 4 we show that the metho ds of this pap er also give results of this typ e, even in our somewhat more general setting. The general lines of the pro ofs are the same as in the L 2 case, with the dierence that the L 2 -estimates of Hormander have to be replaced by similar estimates in uniform norms from Be2] and Be3]. One comment is in order. The results of Seip et al are attractive partly b ecause they are so precise and give necessary and sucient conditions for interp olation and sampling. This pap er gives dierent pro ofs of the positive directions of these theorems, but we have no new ideas ab out pro ofs of the converse directions. In particular, we do not knowifthe conditions in Theorem 2 and 4 are also necessary for sampling and interp olation, although this seems likely, and p erhaps can be proved along the lines of Beurling B], and Seip S], S2]. Acknoledgement. Part of this work was done while the second author was visiting the Math. Department at Goteb org. He wants to express his heartfelt gratitude to the institution for its invitation and to its memb ers for their hospitality. 2. Interpolation and sampling in C . Let  be a subharmonic function in C , and let F 2  be dened by F 2  = f h 2 H ( C ) k f k 2  =: Z j f j 2 e ;  < 1g : By denition, a sequence ;= f z j g C is sampling for F 2  if there are constants A and B such that for any h 2 F 2  A k h k 2   X j h ( z j ) j 2 e ;  ( z j )  B k h k 2  : The sequence ; is called interpolating if for any sequence f c j g such that X j c j j 2 e ;  ( z j ) < 1 we can nd an h 2 F 2  such that h ( z j ) = c j . Intro ducing the notation l 2  for the space of space of all sequences  = f c j g such that k  k 2 = P j c j j 2 e ;  ( z j ) < 1 we see that ; is ON INTERPOLATION AND SAMPLING 3 interp olating i the natural restriction map from F 2  to l 2  is surjective, and sampling i it is b ounded and injective, with closed range. In S] is dened the notion of upp er and lower density of a sequence in the following way: D + (;) = lim sup r !1 sup z 2 C n (; \ ( z r )) = r 2 and D ; (;) = lim inf r !1 inf z 2 C n (; \ ( z r )) = r 2 : (If E is a set n ( E ) denotes the number of elements in E .) Finally a sequence is called uniformly separated if the inmum of the distances between distinct p oints is strictly p ositive. The main results of Seip S] and Seip-Wallsten S-W] are as follows. Theorem A. Asequence ; is interpolating for F 2  with  =  j z j 2 if and only if it is uniformly separated and D + (;) < = : Theorem B. A sequence ; is sampling for F 2  with  =  j z j 2 if and only if it can be written as a nite union of uniformly separatedsequences and moreover contains a uniformly separated subsequence ; 0 satisfying D ; (; 0 ) > = : In Lyubarskii-Seip L-S], an analogous notion of density dep ending on an angle is intro duced and shown to characterize interp olating and sampling sequences for F 2  when  is subharmonic, of class C 2 and p ositively homogenous of degree 2 (i.e.  ( tz )= t 2  ( z ) for t> 0). Unwinding the denitions of D + and D ; we see that D + <  if and only if for some > 0and all suciently large r and all z it holds that (1) n (; \ ( z r )) = r 2 < ;  and that D ; > if and only if for some > 0 and all suciently large r and all z it holds that (2) n (; \ ( z r )) = r 2 > + : It is not hard to see that both these conditions are equivalent to saying that (1) and (2) hold for some large r . For a general subharmonic function  we now dene a similar notion. Here,aswell as in the rest of this pap er we let the Laplace op erator  b e dened as = @ 2 =@ z @  z , a convention which diers from the standard one by a factor 4. Denition. The sequence ; is dense with respect to   if for some r < 1 and  > 0 it holds that n (; \ ( z r )) =r 2 >   ( z )+  for al l z . ; is thin with respect to   if for some r< 1 and > 0 it holds that n (; \ ( z r )) =r 2 <   ( z ) ;  for al l z We are now ready to formulate the rst version of our main result. 4 BO BERNDTSSON AND JOAQUIM ORTEGA Theorem 1. Suppose  is subharmonic in C and that   is uniformly bounded. Then a uniformly separated sequence ; is (a) interpolating for F 2  if ; is thin with respect to   , and (b) sampling for F 2  if ; is dense with respect to   . In the light of what wejustsaid it is clear that when  =  j z j 2 this is just a rephrasing of the suciency part of the theorem of Seip-Wallsten. Moreover, one can check that when  is p ositively homogenous of degree 2 we get the result of Lyubarskii-Seip (note that such a function always has a uniformly bounded laplacian if it is of class C 2 ). Theorem 1 has as a consequence the following, p erhaps more natural, theorem . Theorem 2. Suppose  is subharmonic in C and that   is uniformly bounded. Then a uniformly separated sequence ; is (a) interpolating for F 2  if for some r< 1 and > 0 it holds that n (; \ ( z r )) =r 2 < 1 r 2 Z j  ; z j <r   (  ) ;  for al l z and (b) sampling for F 2  if for some r< 1 and > 0 it holds that n (; \ ( z r )) =r 2 > 1 r 2 Z j  ; z j <r   (  )+  for al l z . Assume that the condition in Theorem 2a (or b) holds for a certain value of r .Let  r = 1 r 2  (0 r ) and  r =    r  be the averages of  over disks with radius r .  r is again subharmonic, and the conditions mean precisely that ; is thin (or dense) with resp ect to   r . From Theorem 1 we conclude that ; is interp olating (sampling) for F 2  r .But it is easily seen that, if the Laplacian of  is uniformly b ounded, then  ;  r = O ( r 2 ). Since r is a xed numb er, this proves Theorem 2, given Theorem 1. To prepare for the pro of of Theorem 1 let  = P  z j be the measure consisting of apointmass at each pointin our sequence, which we assume from now on is uniformly separated. Let E =1 = log j z j 2 b e the fundamental solution of the Laplace op erator (with our convention  E =  0 ). We now dene an auxiliary function v =(  ;    r )  E: This function is certainly well dened if ;is nite. Notice that the value of v at z then dep ends only on the p oints in ; with j z j ; z j <r , since for the other p oints the two terms in the denition of v cancel by the mean value prop erty for harmonic functions. Therefore ON INTERPOLATION AND SAMPLING 5 we can dene v for arbitrary sequences by a limiting pro cedure. An alternative way to dene v is to rst let u r =(  r ;  0 )  E: This function satises (and of course is characterized by) the prop erties  u r =1 = r 2 ;  0 in j z j <r , and u r = @u r =@ n =0 on j z j = r . Moreover u r =0 in j z j >r . Explicitly, u r is given by u r = 1   j z j 2 r 2 ; 1+log  r 2 j z j 2  if j z j <r and u r =0 otherwise. In particular u r has compact supp ort so we can dene v by v = ; u r  : Since E is subharmonic it follows from the submeanvalue prop erty that u r  0 so we always have v  0. Of course it holds that  v =  ;    r : Assume now that ; is thin with resp ect to   .This means precisely that for some large enough r and some p ositive     r <   ; : Let  = v + : Then     +   :  satises the estimates (i)    in C and (ii) j  ; log j z ; z j j 2 ;  j C r in j z ; z j j < 0 , if  0 is chosen so that j z j ; z k j > 2  0 for j 6 = k .(ii) just says that j v ; log j z ; z j j 2 j C r which is clear since v = ; u r   in view of our explicit formula for u r . 6 BO BERNDTSSON AND JOAQUIM ORTEGA The pro of of Theorem 1 nowfollows standard lines. Let f c j g be a sequence of values such that X j c j j 2 e ;  ( z j )  A< 1 : The rst step in the construction of an interp olating function is to interp olate lo cally near each p oint in the sequence f z j g . For each j we apply the Riesz decomp osition formula in adisk  j of radius  0 and center z j . Write  = h j + G   ] where h j is harmonic and G   ]is a Green p otential. Write h j = 2 < H j ,where H j is holomorphic. Let G j = H j ; H j ( z j ). Then G j is a holomorphic function satisfying G j ( z j )= 0 and j  ;  ( z j ) ; 2 < G j j C in  j . This means that f j = c j e G j solves f j ( z j )= c j and j f j j 2 e ;   C j c j j 2 e ;  ( z j ) in  j . We next combine the f j :s using a partition of unity. Let g 2 C 1 c ( C ) b e suchthat g = 1 for j z j < 0 = 2, g = 0 for j z j > 0 and j @g j C  0 . Put f ( z )= X g ( z ; z j ) f j : Then f ( z j ) = c j so f interp olates the right values. Finally we shall mo dify f to get a holomorphic interp olating function by solving a  @ -equation. Note rst that j  @f j 2 e ;   C X j c j j 2 e ;  ( z j ) j  @g ( z ; z j ) j 2 : We then apply Hormanders  @ -theorem H], which implies that we can nd a solution U to  @U =  @f satisfying Z j U j 2 e ;   Z j  @f j 2 e ;    : Since     and  @f vanishes when z 2 ( z j  0 = 2) for some z j the right hand side is dominated by some constant times Z j  @f j 2 e ;   C 0 X j c j j 2 e ;  ( z j ) j  @g ( z ; z j ) j 2  C 00 A: Consequently Z j U j 2 e ;   Z j U j 2 e ;   C 00 A: ON INTERPOLATION AND SAMPLING 7 Moreover U ( z j )=0 for each z j since e ;   1 = j z ; z j j 2 near z j . Let h = f ; U: Then h ( z j )= c j , and R j h j 2 e ;  < 1 since b oth f and U satisfy this estimate. The pro of of Theorem 1 a is therefore complete. We now turn to the sampling part of Theorem 1. For a moment, let  be an arbitrary p ositive measure on C . Assume    r  C and that    r >   +  which is the analog of the density condition for general measures. Put v =(  ;    r )  E and  =( v +  ) : As b efore    but this time we have     ; : Let h 2 F 2  ,and let U = j h j 2 e ;  . Since log U ;   it follows that  U ; U  : Moreover, U 2 L 1 ( C ). Let g be a cut-o function such that g  0, g = 1 for j z j < 1 and g =0 for j z j > 2. Then lim R !1 Z g ( z=R ) U =lim R !1 Z 1 =R 2 ( g )( z=R ) U =0 : By the dierential inequality for U Z U    0 so it follows that (3)  Z j h j 2 e ;   Z j h j 2 e ;  d: This is already an inequality of sampling typ e. Wewould liketocho ose as b efore  = P  j but we cannot do that directly since  would then be identically equal to ;1 on the supp ort of  ,so the inequality would be of no value. 8 BO BERNDTSSON AND JOAQUIM ORTEGA Instead we shall take  to b e a smo othed version ofasumof Dirac measures. Let ; b e a sequence which is dense with resp ect to   . Let  be dened as  = t X 1  2  (0  ) ( z ; z j ) where 0 <t< 1 ( we make no distinction between an absolutely continuous measure and its density with resp ect to Leb esgue measure). Since ; is dense we can cho ose t so close to 1that    r >   + = 2  for some large r . Then  = v +  will satisfy the estimates (i')  ; C      and (ii') j  ; t log  2 ;  j C in j z ; z j j  . Take h 2 F 2  . Then h 2 F 2  so (3) holds. This in conjunction with (i') and (ii') gives (4) = 2 Z j h j 2 e ;   C X  ; 2 t  2 Z j z ; z j j < j h j 2 e ;  : Now we have to use that h is holomorphic (so far we have used only that log j h j 2 is subharmonic.) Fix j for the moment and write (5) Z j z ; z j j < j h j 2 e ;  = Z j z ; z j j < j he ; G j j 2 e ;  e 2 < G j  Z j z ; z j j < j g j j 2 e ;  ( z j )  where g j = he ; G j is holomorphic in j z ; z j j < 1 and g j ( z j )= h ( z j ). Clearly 1  2 Z j z ; z j j < j g j j 2 e ;  ( z j )  2 j h ( z j ) j 2 e ;  ( z j ) + C 2 e ;  ( z j ) sup j z ; z j j < j g 0 j j 2 : But, by Cauchy's estimate sup j z ; z j j < j g 0 j j 2 e ;  ( z j )  C Z j z ; z j j < 1 j g j j 2 e ;  ( z j )  Z j z ; z j j < 1 j h j 2 e ;  : Summing over j we nd = 2 Z j h j 2 e ;   C ; 2 t X j h ( z j ) j 2 e ;  ( z j ) + C 2 ; 2 t Z C j h j 2 e ;  : Cho osing  small enough we can absorb the second term on the right in the left hand side. This proves that ; satises the left of the inequalities in the sampling condition. The other inequality is trivial since we have assumed that ; is separated, so Theorem 1b is now completely proved. ON INTERPOLATION AND SAMPLING 9 3. Interpolation and sampling in D . Let  be a subharmonic function in D , and let F 2  ( D )be dened by F 2  ( D ) = f h 2 H ( D ) k f k 2  =: Z D j f j 2 e ;  1 ;j z j 2 < 1g : A sequence ;= f z j g D is sampling for F 2  if there are constants A and B such that for any h 2 F 2  ( D ) A k h k 2   X j h ( z j ) j 2 e ;  ( z j ) (1 ;j z j j 2 )  B k h k 2  : The sequence ; is called interpolating if for any sequence f c j g such that X j c j j 2 e ;  ( z j ) (1 ;j z j j 2 ) < 1 we can nd an h 2 F 2  ( D ) such that h ( z j )= c j . A sequence ; is called uniformly discrete or separated if inf j 6 = k  ( z j z k ) > 0  where  is the pseudo-hyp erb olic distance in D ,  ( z  )=     z ;  1 ;  z     : Following S2] we shall dene the notion of upp er and lower density of a uniformly separated sequence in D . Notice however that our denition diers from the one in S2] by a factor of two. D + (;) = lim sup r ! 1 sup ' 2 Aut( D ) X z n 2 ' (;) 1 = 2 < j z n j <r log 1 j z n j 2 log 1 1 ; r and D ; (;) = lim inf r ! 1 inf ' 2 Aut( D ) X z n 2 ' (;) 1 = 2 < j z n j <r log 1 j z n j 2 log 1 1 ; r The main results of S2] concerning weighted Bergman spaces in the disk are the following Theorem C. A sequence ; is interpolating for F 2  ( D ) with  =  log 1 = (1 ;j z j 2 ) if and only if it is uniformly separatedand D + (;) <: 16 BO BERNDTSSON AND JOAQUIM ORTEGA Theorem F. Let  be a (pluri)subharmonic function in C n such that i@  @ >  > 0 . Then if f is a  @ -closed (0  1) - form in C n and u is the solution to  @u = f which is of minimal norm in L 2 ( C n e ;  ) , u satises (*) sup j u j e ; e = 2  C sup j f j e ; = 2 : This Theorem has as a consequence a more precise statement that we will need. Let h be any harmonic function in C and write h = < H where H is entire. Then v = ue ; H= 2 is the canonical solution to  @v = e ; H= 2 f in L 2 ( e ;  + h ). Applying Theorem F to this situation instead we see that e  ( z ) can be replaced by ^ (  ; h )( z )+ h ( z )  where h is any harmonic function in C . It is even enough to assume that h is harmonic in  z =: ( z  0 = 2), since any such h can be approximated by functions that are globally dened. Cho osing h to be the harmonic extension of  from @  z to the interior of  z , we see that e  ( z ) can be replaced by  ( z ) ; G ( z ) where G is the Green's p otential of   over  z . In particular, if the Laplacian of  is uniformly b ounded, the Green's potential will be b ounded, so Theorem F holds with e  replaced by  . In our case where  =  + v one sees in a similar waythatwe can replace e  by  in (*). Let us now for a moment supp ose that our sequence ; is nite. Then  @f lies in L 2 so Theorem F applies and we see that the canonical solution to  @u =  @f satises sup j u j 2 e ;   C sup j  @f j 2 e ; = 2  C sup j c j j 2 e ;  : Moreover the canonical solution also satises the L 2 estimate from section 2, so it follows that u ( z j ) = 0. Letting h = f ; u we get an interp olating function in BF  .Since the norm do es not dep end on the numb er of p oints in the sequence, we can also p ermit innite sequences by a normal family argument. We can also treat the disk case in asimilar manner. Let  b e subharmonic in the disk, and supp ose that the invariant Laplacian of  is uniformly b ounded. Put BF  ( D ) = f f 2 H ( D ) sup z 2 D j f ( z ) j 2 e ;   C g  : A subsequence ; of the disk is interp olating for BF  if for any sequence f c j g such that sup j c j j 2 e ;  ( z j ) (1 ;j z j j 2 )  C there is a function f in BF  such that f ( z j )= c j . We then have Theorem 6. Let ; beasequence in the disk which satises the hypothesis of Theorem 4 (a). Then ; is interpolating for BF  ( D ) . The pro of again follows the same pattern as in the L 2 case the only dierence b eing that weneed to replace the L 2 -estimates for  @ by uniform estimates. In Be3] there is also a theorem analogous to Theorem F for the case of the disk (or ball in C n )but it requires that e  > 4. In one dimension one can avoid this restriction by instead app ealing to the following theorem from Be2]. ON INTERPOLATION AND SAMPLING 17 Theorem G. Let  be a subharmonic function and  =: min ((1 ;j z j )  1 = (1 ;j z j )) : Let f be a function in D such that sup j f j  e ; = 2 < + 1 : Let u be the canonical solution to  @u = f in L 2  .Then sup j u j e ; e = 2  C sup j f j  e ; = 2  where e  ( z )= sup  ( z ) < 1 = 2  (  ) . In our construction from section 3 it holds (after subtraction of a harmonic function like in the case of entire space), that e  ;  is uniformly b ounded, and moreover the corresp onding  will be of size 1 = (1 ;j z j 2 ). Therefore Theorem 6 follows in the same way as Theorem 5. Finally, it may be worth remarking that since we use the canonical solution of the  @ -equation, we can interp olate between L 2 and L 1 and get similar results in L p for p between 2and innity. Appendix: A proof of Theorem E The one dimensional case of Hormander's theorem that we used in section 2 says that if  is subharmonic in a domain in C we can solvethe equation  @u = g with the estimate (8) Z j u j 2 e ;   Z j g j 2   e ;  in . That  @u = g in the sense of distributions means that if  is any function in C 1 c ( ) it holds that Z g    = ; Z u @ @z : Taking the supremum of the norm of the right hand side over all g suchthat R j g j 2   e ;   1, we get from (8) (9) Z   j  j 2 e   Z     @ @z     2 e   for all  2 C 1 c ( ). Conversely (9) implies Hormander's theorem, and the usual pro of consists precisely in establishing (9) using integration by parts. 18 BO BERNDTSSON AND JOAQUIM ORTEGA Toprove Theorem E we let = D b e the unit disk and put as b efore  0 = log 1 = (1 ;j z j 2 ). Put  =  +  0 in (9). The crucial p oint is that  0 satises (10)   0 j @ 0 j 2 : Substitute  = e ;  0 in (9), and assume       0 .Then we get (  +1) Z   0 j  j 2 e  ;  0  Z     @ @z ;  @ 0 @z     2 e  ;  0   (1 + 2 = ) Z     @ @z     2 e  ;  0 +(1+ = 2) Z j  j 2     @ 0 @z     2 e  ;  0 : Rearranging and using (10) we obtain ( = 2) Z   0 j  j 2 e  ;  0  (1 + 2 = ) Z     @ @z     2 e  ;  0  or more explicitly Z j  j 2 e  = (1 ;j z j 2 )  C= 2 Z     @ @z     2 (1 ;j z j 2 ) e  : A standard functional analysis argument then shows that for a given g wemay nd u such that Z g    = ; Z u  @ @z for all  2 C 1 c ( D ), and Z j u j 2 e ;  1 ;j z j 2  C  Z j g j 2 e ;  (1 ;j z j 2 ) : This completes the pro of of Theorem E. ON INTERPOLATION AND SAMPLING 19 References Be1] Berndtsson B., A simple proof of an L 2 -estimate for  @ on complete Kahler manifolds , Rep ort (1992). Be2] , Weighted estimates for  @ in C , Duke Math. Journal 66 (1992), 239{255. Be3] , Uniform estimates with weights for the  @ -equation , preprint (1994). B] Beurling A., The col lected works of Arne Beurling Vol 2 , Birkhauser, Boston, 1989, pp. 341{365. D-F] Donelly H. and Feerman C., L 2 -cohomology and index theorem for the Bergman metric ,Ann.of Math. 118 (1983). H] Hormander L., An introduction to complex analysis in several variables, 3rdedition ,Van Nostrand, 1990, p. 96. L-S] Lyubarskii Y. and Seip K., Sampling and interpolation of entire functions and exponential systems in convex domains , Arkiv for Matematik (1994). O] Ohsawa T., On the extension of L 2 -holomorphic functions IV: a new density concept ,Preprint Nagoya University (1993). S] Seip K., Density theorems for sampling and interpolation in the Bargmann-Fock space I , J.reine angew Math. 429 (1992), 91{106. S2] , Beurling type density theorems in the unit disk ,Invent. math. 113 (1993), 21{39. S-W] Seip K. and Wallsten R., Density theorems for sampling and interpolation in the Bargmann-Fock spaceII , J. reine angew Math. 429 (1992), 107{113. Sto] Stoll M., Invariant potential theory in the unit bal l of C n ,Cambridge Unviversity Press, Cambridge, 1994, pp. 31{40. U] UlrichD., Radial limits of M -subharmonic functions ,Trans. Amer. Math. So c. 292 (1985), 501{ 518. Bo Berndtsson: Department of Mathematics CTH S-412 96 G  oteborg Sweden. e-mail: [email protected] Joaquim Ortega: Department of Mathematics, UPC (ETSEIB) 08028 Barcelona, Spain. e-mail: jor[email protected]