Geometric conditions for the reconstruction of a holomorphic function by an interpolation formula

GEOMETRIC CONDITIONS FOR THE RECONSTR UCTION OF A HOLOMORPHIC FUNCTION BY AN INTERPOLA TION F ORMULA AMADEO IRIGO YEN Abstract. W e giv e here some precisions and improv emen ts ab out the v alid- ity of the explicit reconstruction of an y holomor phic function on a ball of C 2 from its restrictions on a family of complex lines. Suc h v alidity depends on the mutual distribution of the lines. This condition can b e geometrically de- scrib ed and is equiv alen t to a stronger stabili t y of the reconstruction f or m ula in terms of p erm utations and subfami lies of these li nes. The motiv ation of this problem comes from possibl e applications in mathematical economics and medical imaging. Contents 1. Int ro duction 1 2. On the non-equiv alence of the geometric cr iterion 6 3. An essential result o n the extrac tio n o f subs e quences 11 4. Pro of of the main theor em 17 References 21 1. Introduction 1.1. Setting of the probl em and so me remi nders. In this pap er w e give some answers and impro vemen ts of the res ults from [7], where w e dea l with a specia l case of the gener a l problem o f r econstruction of a holomorphic function from its restrictions on a family o f a nalytic submanifo lds . Here the s etting is the following: on the o ne hand, we consider for the analytic submanifolds any family o f complex lines in C 2 that cr o ss the orig in. Such a family can b e written as  z ∈ C 2 , z 1 − η j z 2 = 0  j ≥ 1 , (1.1) where the directions η j ∈ C are all different (we omit the sp ecial line { z 2 = 0 } ). On the other hand, let b e f ∈ O  C 2  (resp. f ∈ O ( B 2 (0 , r 0 )) where for any fixed r 0 > 0, B 2 (0 , r 0 ) ⊂ C 2 is the complex ball defined a s B 2 (0 , r 0 ) = n z ∈ C 2 , | z 1 | 2 + | z 2 | 2 < r 2 0 o ) . 1 2 AMADEO IRIGOYEN W e then wan t to give a n effective r econstruction o f f from its restrictions on these complex lines. An application of so me metho ds fro m [1] yields the following in ter- po lation formula, that we remind from [6 ] and [7]: E N ( f ; η )( z ) := N X p =1   N Y j = p +1 ( z 1 − η j z 2 )   N X q = p 1 + η p η q 1 + | η q | 2 1 Q N j = p,j 6 = q ( η q − η j ) × (1.2) × X m ≥ N − p  z 2 + η q z 1 1 + | η q | 2  m − N + p 1 m ! ∂ m ∂ v m | v =0 [ f ( η q v , v )] , where N ≥ 1 and z = ( z 1 , z 2 ) ∈ C 2 . W e know (see Pr opos ition 3 from [7]) that for all N ≥ 1 and f ∈ O  C 2  (resp. f ∈ O ( B 2 (0 , r 0 ))), E N ( f ; η ) is well-defined and satisfies the fo llo wing pro perties: • E N ( f ; η ) ∈ O  C 2  (resp. E N ( f ; η ) ∈ O ( B 2 (0 , r 0 ))); • E N ( f ; η ) is an e x plicit formula that is constructed with the data  f |{ z 1 = η j z 2 }  1 ≤ j ≤ N ; • ∀ j = 1 , . . . , N , E N ( f ; η ) |{ z 1 = η j z 2 } = f |{ z 1 = η j z 2 } ; • ∀ P ∈ C [ z 1 , z 2 ] with deg P ≤ N − 1, E N ( P ; η ) ≡ P . One reason for the choice of a family of lines (1 .1) is that it is well suited fo r the metho ds in [1 ], which rea dily pr oduce formula (1.2). But the e s sen tial r eason comes from p ossible a pplications to the r eal Rado n tr ansform theo ry , tha t may hav e consequences in mathematical ec o nomics and medica l imaging. Indeed, let µ b e a measure with c ompact supp ort K ⊂ R 2 (w.l.o.g. one can a ssume that 0 ∈ K ). W e wan t to rec o nstruct it from its Radon transfo rms on a finite num be r o f directions, i.e. from ( R µ )  θ ( j ) , s  with  θ ( j ) , s  ∈ S 1 × R and j = 1 , . . . , N , where S 1 is the unit sphere of R 2 and ( R µ )  θ ( j ) , s  := ∂ ∂ s Z { x ∈ R 2 , θ ( j ) 1 x 1 + θ ( j ) 2 x 2 ≤ s } µ ( dx ) . (1.3) As it was explained at the Introduction of [7], we consider the F antappie transform Φ µ of µ , that is defined on the dua l space K ⋆ :=  ξ = [ ξ 0 : ξ 1 : ξ 2 ] ∈ C P 2 , < ξ , x > 6 = 0 , ∀ x ∈ K  and is holomorphic there. Explicitly , Φ µ : ξ ∈ K ⋆ 7→ < µ, ξ 0 < ξ , x > > := Z x ∈ K ξ 0 < ξ , x > µ ( dx ) (see [9]). In addition, we know that there is r K > 0 such that fo r all θ ∈ S 1 and all u ∈ C with | u | < r K (so that [1 : uθ 1 : uθ 2 ] ∈ K ⋆ ), Φ µ ([1 : uθ 1 : u θ 2 ]) = Z + ∞ −∞ ( R µ )( θ, s ) 1 + s u ds , i.e. the kno wledge o f ( R µ )( θ ( j ) , s ) , j = 1 , . . . , N , s ∈ R , allows to know the re- striction of Φ µ ∈ O ( B 2 (0 , r K )) on every line L θ ( j ) = { ( u θ 1 , uθ 2 ) , u ∈ C } = { z ∈ C 2 , z 1 = η j z 2 } where η j = θ ( j ) 1 /θ ( j ) 2 ∈ R , j = 1 , . . . , N (1.4) (w.l.o.g. one can ass ume that θ ( j ) 2 6 = 0 for all j = 1 , . . . , N ). GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 3 The family of measures defined for N ≥ 1 by µ N := Φ − 1 [ E N (Φ µ ; η )] (where E N ( · ; η ) is the ab ov e fo m ula (1.2), and Φ − 1 is the r ecipro cal isomorphism, whose existence is g ua ranteed by [9]), is interpolating in the meaning that < µ N , x k 1 x l 2 > = < µ, x k 1 x l 2 > fo r all N ≥ 1 and k , l ≥ 0 with k + l ≤ N . Since by (1.4) the set { η j } j ≥ 1 is a subset of R , by an applica tion o f Theor em 1 below, we will conclude tha t the family { µ N } N ≥ 1 will approximate µ in an appro priate top ology . In addition, a n application of some results of Henkin and Shananin from [5 ] will allow to c ompute the rec o nstruction with go o d estimates. These exp ected r esults are ha ndled in [8], that is currently in progres s. 1.2. Essential results. The es sen tial problem is tha t there is no g uarantee that, as N → ∞ , E N ( f ; η ) will con verge to f (although it coincide s with f on an in- creasing n um b er of lines). W e know f rom [7] that in g eneral it is not the case, i.e. there are families of lines with (at least) an asso ciated holomor phic function f such that E N ( f ; η ) will no t converge. Since we are interested in a reconstructio n formula whose conv ergence is guaranted for every holo morphic function f , w e wan t to determine a ll the go o d families of lines η = ( η j ) j ≥ 1 for which the conv ergence o f the a sso ciated interp olation for mula E N ( · ; η ) is gua ranteed for every holomor phic function. Theorems 1 and 2 from [7] give e q uiv alent criteria for the v alidity of such a reconstruction: roug hly spe aking, the sequence of the directions ( η j ) j ≥ 1 of the lines (1.1) must s a tisfy an expo nen tial estimate of their divided differ ences (an op erator of succe s siv e discrete der iv atives, see for example [2 ], [4] a nd [10] for the definition a nd essential results). Nev ertheless, the difficult y to ch eck this condition on divided differences g iv es us the motiv ation to find a c r iterion that is easier to understand. This leads to the following definition: Definition 1. The set { η j } j ≥ 1 is lo cally interpolable by rea l- analytic curves if, for al l ζ ∈ { η j } j ≥ 1 (the top ol o gic al closur e of { η j } j ≥ 1 in C P 1 ), ther e exist a neighb or- ho o d V of ζ and a s mo oth r e al-analytic curve C s uch that ζ ∈ C and V \ { η j } j ≥ 1 ⊂ C . (1.5) This new geometric condition is a sufficient criterion for the conv ergence of the int erp olation formula E N ( · ; η ) a nd y ields the following result, given as Theo rem 3 from [7]. Theorem 1 . If { η j } j ≥ 1 is lo c al ly interp olable by r e al-analytic curves, then the interp ola tion formula E N ( f , η ) c onver ges to f un ifo rmly on any c omp act K ⊂ C 2 and for al l f ∈ O  C 2  . Similarly, r 0 b eing given, ther e is ε η > 0 such that, for al l f ∈ O ( B 2 (0 , r 0 )) , E N ( f ; η ) c onver ges to f un iformly on any c omp ac t subset K ⊂ B 2 (0 , ε η r 0 ) . Nevertheless, this new criterion is not equiv alen t. Indeed, as it has been sus - pec ted in the In tro duction of [7], there a re sequences of lines that are not lo cally in- terp olable by real-a nalytic curves and whose as s ocia ted for mula E N ( · ; η ) conv erges. Prop osition 1 b elo w is the firs t r esult of this pap er and gives a n explicit example o f such a family: it consists on constructing a sequence ( η j ) j ≥ 1 as the increasing union of 1 / 2 r -nets, r ≥ 0 , of the sq ua re [0 , 1 ] + i [0 , 1] = { z ∈ C , 0 ≤ ℜ ( z ) , ℑ ( z ) ≤ 1 } (so that fo r all N ≥ 1, the first N p oints η j ’s are the m ost separ ated p ossible from e ac h other). 4 AMADEO IRIGOYEN Prop osition 1 . Ther e exists (at le ast) one se quenc e ( η j ) j ≥ 1 that is not lo c al ly inter- p ol able by r e al-analytic curves but whose asso ciate d interp olation formula E N ( · ; η ) c onver ges, i.e. for al l f ∈ O  C 2  , E N ( · ; f ) c onver ges to f uniformly on any c om- p ac t subset K ⊂ C 2 (similarly, r 0 > 0 b eing fixe d, ther e is ε η > 0 such that for al l f ∈ O ( B 2 (0 , r 0 )) , E N ( f ; η ) c onver ges to f uniformly on any c omp act subset K ⊂ B 2 (0 , ε η r 0 ) ). This first conclusion leads to the following question: wh y is this geometric cr i- terion not (a lw ays) necessa r y? On the other hand, the express io n (1.2) of E N ( · ; η ) clearly involv es the e n umeration of the lines η j ’s. Since Definition 1 is a conditio n ab out sets that do es no t depend on a n y of its enum eratio ns, one is tempted into considerating the actio n of the group o f per m utations S N and chec k the v alidity of the conv ergence of E N ( · ; σ ( η )), where the sequence σ ( η ) is defined from η = ( η j ) j ≥ 1 by σ ( η ) =  η σ ( j )  j ≥ 1 (1.6) (in order to simplify the no tation, S N will mean S N \{ 0 } since all the co nsidered sequences in the pap er star t by j = 1 ). Now σ ∈ S N being given, one could first think that E N ( f ; σ ( η )) and E N ( f ; η ) are essentially the same . Indeed, if M N := max { N , σ (1 ) , . . . , σ ( N ) } , then E M N ( f ; η ) and E M N ( f ; σ ( η )) b oth in terp olate f on the M N first lines η 1 , . . . , η N and η σ (1) , . . . , η σ ( N ) . Nevertheless, if we change the o rder of the sequence of the sq uare from P rop osi- tion 1 ab ov e, the ass ocia ted in terp olation formula may not conv erge anymore. This leads to the following q uestion: a given sequence ( η j ) j ≥ 1 whose a sso c ia ted in terp o- lation formula E N ( · ; σ ( η )) alwa ys converges under the a ction of any p erm utation σ , should b e lo cally in terpola ble by rea l-analytical cur v es? W e will see that the answer is a ffirmative as claimed by Theo rem 2 b elow. In o rder to deal with this problem, we need to consider the following question: the s equence η = ( η j ) j ≥ 1 being fixed, if the for mula E N ( · ; η ) conv erges, wha t a bout E N ( · ; η ′ ), where η ′ := ( η j k ) k ≥ 1 is any given (infinite) subsequence o f η ? On a first sight, the answer lo oks p ositive b ecause o f the following intu itive arg ument: if E N ( f ; η ) can interpola te f on mor e lines than E N ′ ( f ; η ′ ) do es (where N ′ is the nu mber of k ≥ 1 such tha t j k ≤ N ) and E N ( f ; η ) co n verges, then why should not E N ( f ; η ′ ) to o ? The true answer is that this heuris tic ar g umen t is false. Indeed, it is als o a strong co ndition that is equiv alen t to the geometric cr iterion (1.5). This claim and the abov e one are sp ecified by the following result, that is the main theorem of this pap er. Theorem 2 . L et η = ( η j ) j ≥ 1 b e any se quenc e define d as in (1.1). The fol lowing c ondi tions ar e e quivalent: (1) the set { η j } j ≥ 1 is lo c al ly interp olable by r e al-analytic curves; (2) for al l f ∈ O  C 2  and al l σ ∈ S N , E N ( f ; σ ( η )) c onver ges to f uniformly on any c omp act subset K ⊂ C 2 ; (3) for al l f ∈ O  C 2  and al l subse quenc e η ′ = ( η j k ) k ≥ 1 , E N ( f ; η ′ ) c onver ges to f u n iformly on any c omp ac t subset K ⊂ C 2 . First, this res ult fina lly gives an equiv alence, for a given sequence ( η j ) j ≥ 1 , b e- t ween the strong geometric hypo thesis (1.5) and sharp er conditions in ter ms of GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 5 the v alidit y o f the conv ergence of the asso ciated int erp olation formula E N ( · ; η ). In particular, it clarifies in which s ense this ge ometric co ndition is sufficient. Next, this r esults only deals with the conv ergenc e of E N ( f ; η ) for any f ∈ O  C 2  and we would like to know what happ ens if we consider the same a ssertion with any f ∈ O ( B 2 (0 , r 0 )) for any fixed r 0 > 0. One o f the applications of Theorem 2 is its generaliza tion to the ca se of every complex ba ll B 2 (0 , r 0 ), as sp ecified by the following result. Corollary 1. L et η = ( η j ) j ≥ 1 b e any se quenc e define d as in (1.1) and let b e r 0 > 0 . The fol lowing c onditions ar e e quivalent: (1) the set { η j } j ≥ 1 is lo c al ly interp olable by r e al-analytic curves; (2) ther e is ε η > 0 such that for al l f ∈ O ( B 2 (0 , r 0 )) and al l σ ∈ S N , E N ( f ; σ ( η )) c onver ges to f uniformly on any c omp act subset K ⊂ B 2 (0 , ε η r 0 ) ; (3) ther e is ε η > 0 such that for al l f ∈ O ( B 2 (0 , r 0 )) and al l subse quenc e η ′ = ( η j k ) k ≥ 1 , E N ( f ; η ′ ) c onver ges to f u niformly on any c omp act subset K ⊂ B 2 (0 , ε η r 0 ) . As it can b e noticed, these results are equiv a lences b et ween a geometric condition and the v alidit y of the conv ergence of its asso ciated in terp olation formula E N ( · ; η ) (i.e. in terms of functional a ppr o ximation theory). Moreov er, we hav e ano ther conseq ue nc e that gives s ome pr ecision o n the sp eed of conv ergence of E N ( f ; η ) to f . Corollary 2. When any of the e quiva lent c onditions fr om The or em 2 or Cor ol lary 1 is fulfil le d, one has in addition t he fo l lowing estimate: for al l K ⊂ O  C 2  (r esp. K ⊂ O ( B 2 (0 , r 0 )) ) and K ⊂ C 2 (r esp. K ⊂ B 2 (0 , ε η r 0 ) ) c omp act subset s, ther e ar e C K ,K and ε K > 0 such that for al l σ ∈ S N , for al l η ′ = ( η j k ) k ≥ 1 and al l N ≥ 1 , sup f ∈K sup z ∈ K | f ( z ) − E N ( f ; σ ( η ))( z ) | ≤ C K ,K (1 − ε K ) N and sup f ∈K sup z ∈ K | f ( z ) − E N ( f ; η ′ ) ( z ) | ≤ C K ,K (1 − ε K ) N . In particular , as it has b een p oin ted o ut in [7], a simple conv ergence of E N ( · ; η ) (i.e. conv ergence of E N ( f ; η ) for every fixed holomorphic function f ) implies a uniform one. This can be interpreted as a Banach-Steinhaus prop erty for the family of op erators { E N ( · ; η ) } N ≥ 1 in the ca nonical top olog y for the holomorphic functions (i.e. the top ology of uniform conv ergence on any compa ct s ubs et). Finally , the essen tial argument for the pro of of Theor em 2 follows fro m the following result, whose pro of is given in Section 3. Prop osition 2. L et ( η j ) j ≥ 1 b e any se quenc e such that t he set { η j } j ≥ 1 is not lo c al ly interp ola ble by re al-analytic curves. Then ther e exist s a s ubse qu enc e ( η j k ) k ≥ 1 of ( η j ) j ≥ 1 that satisfies the fol lowing c onditio ns: • the se quenc e ( η j k ) k ≥ 1 is c onver gent in C P 1 ; • the set { η j k } k ≥ 1 is not lo c al ly interp ola ble by r e al-analy tic curves. W e know that if { η j } j ≥ 1 (coming from any sequence ( η j ) j ≥ 1 ) is lo cally interpo- lable by r eal-analytic curves, then so will be any of its subse ts (finite or infinite), in particular if it comes fro m a conv ergent subsequence ( η j k ) k ≥ 1 . Conv ersely , we 6 AMADEO IRIGOYEN may ask what happens if { η j } j ≥ 1 is not loca lly interpo lable by real-analy tic curves. Prop osition 2 gives an affirma tive answer, in the sense that one can ex tract conv er- gent subsequenc e s that are s till not lo cally in terp olable by rea l-analytic curves. I would lik e to thank G. Henkin for having in tro duced me this interesting problem and J. Orteg a-Cerd` a for all his guidance, rewarding ideas and discussions to pr ogress in. 2. On the non-equiv alence of the geometric criterion In this sectio n w e deal with the pro of of Pr opos ition 1. W e first need some reminders and preliminary res ults. 2.1. Some rem inders. First, the following r esult is given as Lemma 20 fro m [7] and is a necessa ry c o ndition for a s et to satis fy the geo metric condition (1.5 ). Lemma 1. The top olo gic al closur e of a set that is lo c al ly interp ol able by r e al- analytic curves, has empty interior. Next, we remind Theor em 1 as one o f the essential res ults fro m [7] and that gives an equiv a len t criter ion for a b ounded sequence ( η j ) j ≥ 1 to make converge its asso ciated interpolation formula E N ( · ; η ). Theorem 3. L et ( η j ) j ≥ 1 b e b ounde d and fix any r 0 > 0 . The fol lowing c onditio ns ar e e quivalent: (1) ther e is ε η > 0 such t hat, for al l f ∈ O ( B 2 (0 , r 0 )) , the int erp olation formula E N ( f ; η ) c onver ges to f , uniformly on any c omp ac t subset of B 2 (0 , ε η r 0 ) ; (2) for al l g ∈ O  C 2  , the interp ola tion formula E N ( g ; η ) c onver ges to g , u ni- formly on any c omp act subset of C 2 ; (3) ∃ R η ≥ 1 , ∀ p, q ≥ 0 ,      ∆ p, ( η p ,...,η 1 ) "  ζ 1 + | ζ | 2  q # ( η p +1 )      ≤ R p + q η . (2.1) The op erator ∆ p is called divide d di ffer enc es and is defined as follows (for any application h that is defined at the η j ’s):              ∆ 0 ( h )( η 1 ) = h ( η 1 ) ; for all p ≥ 1 , ∆ p, ( η p ,...,η 1 ) ( h )( η p +1 ) = = ∆ p − 1 , ( η p − 1 ,...,η 1 ) ( h )( η p +1 ) − ∆ p − 1 , ( η p − 1 ,...,η 1 ) ( h )( η p ) η p +1 − η p . (2.2) ∆ p ( h ) can b e seen a s the discrete deriv ativ e of order p of the function h . A lot its prop erties can b e found in the references , in par ticular the following one (see for example [3], Chapter 4, 7 (7.7 )). Lemma 2. L et { η j } j ≥ 1 b e any set of differ ent p oints and h any funct ion define d on them. One has for al l p ≥ 0 , ∆ p, ( η p ,...,η 1 ) [ h ] ( η p +1 ) = p +1 X q =1 h ( η q ) Q p +1 j =1 ,j 6 = q ( η q − η j ) . GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 7 W e also deduce a s an applicatio n the following result that will b e useful for the pro of of Theor em 2. Corollary 3 . F or al l p, q ≥ 0 ,      ∆ p, ( η p ,...,η 1 ) " ζ 7→  ζ 1 + | ζ | 2  q # ( η p +1 )      ≤ p +1 X l =1 1 Q p +1 j =1 ,j 6 = l ( η l − η j ) . In p articular, the b ound do es not dep end on q ≥ 0 . Pr o of. The proo f immediately follo ws by Lemma 2 with the particular choice o f h ( ζ ) =  ζ 1 + | ζ | 2  q since fo r all ζ ∈ C , one ha s that       ζ 1 + | ζ | 2  q      =  | ζ | 1 + | ζ | 2  q = s | ζ | 2 1 + | ζ | 2 ! q × 1  p 1 + | ζ | 2  q ≤ 1 . √ 2.2. Construction of a coun terexample. In this subsection, we construct the explicit sequence ( η j ) j ≥ 1 ⊂ Q , where Q is the clos ed squar e Q = [0 , 1 ] + i [0 , 1] = { z ∈ C , 0 ≤ ℜ ( z ) , ℑ ( z ) ≤ 1 } . (2.3) Its e s sen tial required prop erty is that the η j ’s m ust b e the m ost s eparated p ossible from each other. W e start b y setting η 1 = 0 , η 2 = 1 , η 3 = 1 + i, η 4 = i . W e find the maximal num b er of po in ts of Q whos e mut ual distance is not smaller than 1. When it is not possible anymore, we add the maxima l num b er of p oin ts who se mutual distance is a t least 1 / 2, then η 5 = 1 / 2 , η 6 = i/ 2 , η 7 = (1 + i ) / 2 , η 8 = 1 + i/ 2 , η 9 = 1 / 2 + i . More ge nerally , we will cho ose by induction o n r ≥ 0 the maximal num b er of p oin ts whose mutual dis tance is a t least 1 / 2 r . Let fix r ≥ 0 and let A r be an 1 / 2 r -net of Q , i.e. a set of p oin ts that are at least at a dista nce of 1 / 2 r from each other. One can choose A r =  s + it 2 r , ( s, t ) ∈ N 2 , 0 ≤ s, t ≤ 2 r  , (2.4) whose cardinal is (1 + 2 r ) 2 (one can chec k tha t A 0 = { 0 , 1 , i, 1 + i } = { η 1 , η 2 , η 3 , η 4 } ). In addition, one has the sequence of inclusio ns A 0 ⊂ A 1 ⊂ · · · ⊂ A r ⊂ · · · . (2.5) The seque nc e η = ( η j ) j ≥ 1 will be defined by induction on r ≥ 0 as follo ws: we first choos e η 1 , η 2 , η 3 and η 4 for the first set A 0 (notice that we do not s pecify any enum eratio n fo r these first η j ’s); nex t, if we assume having co nstructed η 1 , . . . , η N r with N r = (1 + 2 r ) 2 , (2.6) we define η N r +1 , . . . , η N r +1 so that  η N r +1 , . . . , η N r +1  = A r +1 \ A r . (2.7) Again, the enum eratio n for thes e η j ’s do es no t matter . The only imp ortant fact is that η j ∈ A r j for all j ≥ 1, where r j is the first r ≥ 0 such that j ≤ N r . 8 AMADEO IRIGOYEN The seq uence ( η j ) j ≥ 1 can b e defined by induction o n r ≥ 0 a nd o ne ha s { η j } j ≥ 1 = A ∞ := [ r ≥ 0 A r . As it has b een specified, the en umeration of ( η j ) j ≥ 1 do es not matter as long as one ha s the following imp ortant c o ndition: for a ll r ≥ 0 and all j, k ≥ 1 s uc h that η j ∈ A r and η k ∈ A r +1 \ A r , then o ne neces sarily has j < k . Equiv alen tly , for all r ≥ 0, the fir st N r po in ts η j ’s b elong to A r . W e can de duce the following preliminar result. Lemma 3. The se quenc e ( η j ) j ≥ 1 is wel l-define d and is dense in Q . In additio n, ( η j ) j ≥ 1 satisfies the fol lowing c ondition: ∀ r ≥ 0 , ∀ j ≤ N r = (1 + 2 r ) 2 , η j ∈ A r . (2.8) Pr o of. The la st assertion immediately follows from (2.6) a nd (2.7). In order to pr ove the density , let consider z ∈ Q , ε > 0 and let b e r ≥ 0 such that 1 / 2 r ≤ ε . There is η j z ∈ A r +1 such that |ℜ ( η j z ) − ℜ ( z ) | ≤ 1 / 2 r +1 and |ℑ ( η j z ) − ℑ ( z ) | ≤ 1 / 2 r +1 , then | η j z − z | ≤ √ 2 / 2 r +1 < 1 / 2 r ≤ ε . √ In or der to prove Pro position 1, we first need to give a n estimate of the divided differences { ∆ p } p ≥ 1 asso ciated with ( η j ) j ≥ 1 . 2.3. A b o u nd for the asso ciated divide d diffe rences. W e start by the follow- ing pr e liminar result that is a lower b ound for the products that app ear on the expression of the ∆ p ’s given by Lemma 2. Lemma 4. L et c onsider the se quenc e ( η j ) j ≥ 1 fr om L emma 3. Ther e is P η ≥ 2 such that, for al l p ≥ P η and al l q = 1 , . . . , p + 1 , one has p +1 Y j =1 ,j 6 = q | η q − η j | ≥ exp( − 9 p ) . Pr o of. Let fix any p ≥ 2 and let consider the unique r ≥ 0 such that  1 + 2 r − 1  2 < p + 1 ≤ (1 + 2 r ) 2 . (2.9) Now let fix η q with q = 1 , . . . , p + 1, i.e. η q = s q + it q 2 r where 0 ≤ s q , t q ≤ 2 r . Similarly , fo r all η j 6 = η q with j = 1 , . . . , p + 1 , one has η j = s j + it j 2 r with 0 ≤ s j , t j ≤ 2 r and ( s j , t j ) 6 = ( s q , t q ). Since | s j − s q | ≤ 2 r (resp. | t j − t q | ≤ 2 r ), then k q,j := ma x {| s j − s q | , | t j − t q |} ∈ { 1 , . . . , 2 r } . (2.10) It follows that η j ∈ D k q,j ( η q ), where D k ( η q ) is defined for all k ∈ N by D k ( η q ) :=  z = s + it 2 r , ( s, t ) ∈ Z 2 , max {| s − s q | , | t − t q |} = k  . W e firs t wan t to estima te card [ D k ( η q ) ∩ { η j , 1 ≤ j ≤ p + 1 , j 6 = q } ] for all k = 1 , . . . , 2 r . W e start by noticing that D k ( η q ) = S k ( η q ) \ S k − 1 ( η q ) , (2.11) GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 9 where S k ( η q ) is de fined for all k ∈ N by S k ( η q ) :=  z = s + it 2 r , ( s, t ) ∈ Z 2 , max {| s − s q | , | t − t q |} ≤ k  . Since on the one hand, one has for all k ≥ 0, that card [ S k ( η q )] = card [ S k (0)] = card  z = s + it 2 r , ( s, t ) ∈ Z 2 , − k ≤ s, t ≤ k  = (2 k + 1) 2 , and on the other ha nd, S k − 1 ( η q ) ⊂ S k ( η q ) for all k ≥ 1, it follows by (2.11) that card [ D k ( η q ) ∩ { η j , 1 ≤ j ≤ p + 1 , j 6 = q } ] ≤ card [ D k ( η q )] = card [ S k ( η q )] − card [ S k − 1 ( η q )] = (2 k + 1) 2 − (2 k − 1) 2 = 8 k . (2.12) Next, one has for all k = 1 , . . . , 2 r , and all η j ∈ D k ( η q ) (with η j = s j + it j 2 r ), | η j − η q | ≥ max  | s j − s q | 2 r , | t j − t q | 2 r  = k 2 r . (2.13) Finally , the estimates (2 .1 3) and (2.12) together yie ld for all k = 1 , . . . , 2 r , Y D k ( η q ) ∩{ η j , 1 ≤ j ≤ p +1 , j 6 = q } | η q − η j | ≥  k 2 r  card [ D k ( η q ) ∩{ η j , 1 ≤ j ≤ p +1 , j 6 = q } ] ≥  k 2 r  8 k , the seco nd inequality b e ing v alid s ince 0 < k / 2 r ≤ 1. By applying the following partition (justified b y (2.9) and Lemma 3), { η j , j = 1 , . . . , p + 1 , j 6 = q } = 2 r [ k =1 [ D k ( η q ) ∩ { η j , 1 ≤ j ≤ p + 1 , j 6 = q } ] (one indeed has 1 ≤ k ≤ 2 r by (2.10)), we can deduce that p +1 Y j =1 ,j 6 = q | η q − η j | = 2 r Y k =1   Y D k ( η q ) ∩{ η j , 1 ≤ j ≤ p +1 , j 6 = q } | η q − η j |   ≥ 2 r Y k =1  k 2 r  8 k = exp " 2 r X k =1 8 k ln ( k / 2 r ) # = ex p " 2 2 r + 3 × 1 2 r 2 r X k =1 k 2 r ln  k 2 r  # . (2.14) The last expressio n inv olves the Riemann’s sum of the co n tin uous function t ∈ ]0 , 1] 7→ t ln t , 0 7→ 0, whose integral is R 1 0 t ln tdt =  t 2 2 ln t  1 0 − R 1 0 t 2 dt = − 1 4 . Then 2 2 r + 3 2 r 2 r X k =1 k 2 r ln  k 2 r  = 2 2 r + 3 ( − 1 / 4 + ε (1 /r )) = 2 2 r 4 ( − 8 + ε (1 /r )) , (2.15) where ε (1 / r ) → 0 as 1 / r → 0. O n the o ther hand, one has by (2.9) that p ≥  1 + 2 r − 1  2 − 1 = 2 2 r − 2 + 2 r ≥ 2 2 r 4 . (2.16) 10 AMADEO IRIGOYEN In addition, (2.9) also gives that ε (1 / r ) = ε (1 /p ). It follows b y applying (2.1 4), (2.15) and (2.16) that p +1 Y j =1 ,j 6 = q | η q − η j | ≥ exp [ p × ( − 8 + ε (1 / p )) ] ≥ exp ( − 9 p ) , for all p ≥ P η ( ≥ 2) so that | ε (1 /p ) | ≤ 1 (notice tha t P η do es not dep end on q = 1 , . . . , p + 1). The estimate b eing true for all q = 1 , . . . , p + 1 , the pr oo f o f the lemma is achiev ed. √ This allows us to prov e the following r esult that will b e useful for the pro of of Prop osition 1. Lemma 5. L et h b e any function define d on t he set { η j } j ≥ 1 (c oming fr om the se quenc e ( η j ) j ≥ 1 of L emma 3) and that is b ounde d: k h k ∞ := sup j ≥ 1 | h ( η j ) | < + ∞ . Then ther e is R η ≥ 1 s uch that for al l p ≥ 0 ,   ∆ p, ( η p ,...,η 1 ) [ h ] ( η p +1 )   ≤ k h k ∞ R p η . Pr o of. Let b e p ≥ 0. One has by L e mma 2 that   ∆ p, ( η p ,...,η 1 ) [ h ] ( η p +1 )   ≤ p +1 X q =1 | h ( η q ) | Q p +1 j =1 ,j 6 = q | η q − η j | ≤ ( p + 1) k h k ∞ min 1 ≤ q ≤ p +1  Q p +1 j =1 ,j 6 = q | η q − η j |  . If p ≤ P η − 1, then   ∆ p, ( η p ,...,η 1 ) [ h ] ( η p +1 )   ≤ C η k h k ∞ , where C η := P η min 1 ≤ p ≤ P η − 1  min 1 ≤ q ≤ p +1 Q p +1 j =1 ,j 6 = q | η q − η j |  . Otherwise p ≥ P η ( ≥ 2) then one has by Lemma 4 that min 1 ≤ q ≤ p +1   p +1 Y j =1 ,j 6 = q | η q − η j |   ≥ 1 / exp( 9 p ) , th us   ∆ p, ( η p ,...,η 1 ) [ h ] ( η p +1 )   ≤ ( p + 1) k h k ∞ exp(9 p ) ≤ exp( p ) k h k ∞ exp(9 p ) = k h k ∞ exp(10 p ) (the seco nd es tima te be ing justified by the clas s ical one : 1 + t ≤ exp( t ) , ∀ t ∈ R ). It follows that for all p ≥ 1, one has   ∆ p, ( η p ,...,η 1 ) [ h ] ( η p +1 )   ≤ k h k ∞ × max [ C η , exp (10 p )] ≤ k h k ∞ (1 + C η ) exp(10 p ) ≤ k h k ∞ R p η , where R η := (1 + C η ) e 10 (for p = 0, one just has tha t | ∆ 0 ( h ) ( η 1 ) | = | h ( η 1 ) | ≤ k h k ∞ × R 0 η ) and the pro of is achiev ed. √ GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 11 R emark 2.1 . Notice that w e do not need to assume any kind of regular it y for the function h , e ls e that it is b ounded on the set { η j } j ≥ 1 . 2.4. Pro of of Prop ositio n 1. Now we can g iv e the pro of of the prop osition. Pr o of. Firs t, the set { η j } j ≥ 1 is dense in the squar e Q by Lemma 3 then its top o- logical clos ure { η j } j ≥ 1 has nonempty interior. It follows by Lemma 1 tha t { η j } j ≥ 1 cannot b e lo cally in terp olable by real-ana lytic curves. Next, the sequence ( η j ) j ≥ 1 being b ounded, in order to prov e that E N ( · ; η ) con- verges (i.e. for entire functions as well as for holomor phic functions on a n y fixed ba ll B 2 (0 , r 0 )), it suffices to show that ( η j ) j ≥ 1 satisfies the es timate (2.1) in Theorem 3. F o r all q ≥ 0, one ha s with the choice o f h ( ζ ) =  ζ 1 + | ζ | 2  q that       ζ 1 + | ζ | 2  q      ∞ ≤         s   ζ   2 1 + | ζ | 2   q       ∞ ×       1  p 1 + | ζ | 2  q       ∞ ≤      p 1 + | ζ | 2 1 + | ζ | 2      q ∞ × 1 ≤ 1 (in par ticula r h is b ounded on C ). It follows by Lemma 5 that for all p, q ≥ 0,      ∆ p, ( η p ,...,η 1 ) "  ζ 1 + | ζ | 2  q # ( η p +1 )      ≤       ζ 1 + | ζ | 2  q      ∞ × R p η ≤ R p η ≤ R p + q η , i.e. ( η j ) j ≥ 1 satisfies condition (2.1) from Theorem 3 and this completes the pro of of Pro position 1. √ 3. An essential resul t on the extraction o f subsequences In this par t we give the proo f of P rop osition 2 that will b e useful in o rder to prov e Theorem 2. W e first ne e d a couple of pr e liminar results. 3.1. Some rem inders and preliminar resul ts . Let fix η = ( η j ) j ≥ 1 . W e first remind the following identit y that is justified b y Prop osition 3 from [7] a nd that inv olv es another analog ous for m ula R N ( · ; η ), that is the essential remainder pa rt of f − E N ( f ; η ): f ∈ O ( B 2 (0 , r 0 )) (resp. f ∈ O  C 2  ) being given, one ha s for all N ≥ 1 and a ll z ∈ B 2 (0 , r 0 ) (resp. z ∈ C 2 ), f ( z ) = E N ( f ; η )( z ) − R N ( f ; η )( z ) + X k + l ≥ N a k,l z k 1 z l 2 ; (3.1) here, f ( z ) = X k,l ≥ 0 a k,l z k 1 z l 2 12 AMADEO IRIGOYEN is the T aylor expans io n of f , and R N ( f ; η )( z ) := N X p =1   N Y j =1 ,j 6 = p z 1 − η j z 2 η p − η j   X k + l ≥ N a k,l η k p  z 2 + η p z 1 1 + | η p | 2  k + l − N +1 (3.2) is well-defined and b elongs to O ( B 2 (0 , r 0 )) (r esp. O  C 2  ). Since the T aylor expansion of f ∈ O ( B 2 (0 , r 0 )) (re sp. f ∈ O  C 2  ) a lw ays conv erges to 0 uniformly on any compact subset K ⊂ B 2 (0 , r 0 ) (resp. K ⊂ C 2 ), this gives an equiv alence b etw een the conv ergence of E N ( · ; η ) a nd the one o f R N ( · ; η ). More precisely , w e have the following result that is Lemma 7 from [7]. Lemma 6. r 0 > 0 b eing fix e d, let c onsider f ∈ O ( B 2 (0 , r 0 )) (r esp. f ∈ O  C 2  ) and K a ny c omp act subset of B 2 (0 , r 0 ) (r esp. C 2 ). Then for al l N ≥ 1 , one has sup z ∈ K | f ( z ) − E N ( f ; η )( z ) | ≤ sup z ∈ K | R N ( f ; η )( z ) | + C K ( N + 2) sup k z k≤ r K | f ( z ) | (1 − ε K ) N , wher e k z k = p | z 1 | 2 + | z 2 | 2 is the usu al norm on C 2 and C K , r K dep end only on K . In p articular, E N ( f ; η ) c onver ges t o f (uniformly on any c omp ac t subset) if and only if so do es R N ( f ; η ) t o 0 . On the other hand, we will a lso dea l with the a ction of so me homog raphic trans- formations on ( η j ) j ≥ 1 . W e remind some nota tions and re s ults fro m [7] (b eginning of Section 4): let fix a n y η c / ∈ { η j } j ≥ 1 ∪ {∞} and let consider the unitar y matrix U η c ∈ U (2 , C ) defined b y U η c := 1 p 1 + | η c | 2  η c 1 1 − η c  ; let als o cons ide r the fo llowing homogr aphic applica tion h η c : C P 1 → C P 1 (3.3) ζ 7→ 1 + η c ζ ζ − η c (where C P 1 = C ∪ {∞} ) and the new seq uence θ = ( θ j ) j ≥ 1 := ( h η c ( η j )) j ≥ 1 . (3.4) Then the set { θ j } j ≥ 1 is w ell-defined as a subset of C and o ne has the following result that is Lemma 16 from [7]. Lemma 7. L et b e f ∈ O ( B 2 (0 , r 0 )) (re sp. f ∈ O  C 2  ). F or al l N ≥ 1 and z ∈ B 2 (0 , r 0 ) (r esp. z ∈ C 2 ), R N ( f ; η )( z ) = R N  f ◦ U − 1 η c ; θ  ( U η c z ) . These le mma s yield the fo llowing co ns equence. Corollary 4. η = ( η j ) j ≥ 1 b eing any se quenc e, η c / ∈ { η j } j ≥ 1 ∪ {∞} b eing fixe d and h η c (r esp. θ ) b ei ng define d by (3.3) (r esp. (3.4)), the formula E N ( f ; η ) c onver ges to f u n iformly on any c omp act subset K ⊂ C 2 and for every function f ∈ O  C 2  , if and only if so do es E N ( f ; θ ) . GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 13 Pr o of. f ∈ O  C 2  being giv en, one ha s b y Lemma 6 th at the for m ula E N ( f ; η ) conv erges to f if and only if R N ( f ; η ) converges to 0 (uniformly on any compact subset). U η c being an isometr y , it follows by Lemma 7 that R N ( f ; η ) conv erges to 0 uniformly o n a n y co mpact subset K ⊂ C 2 and for every function f ∈ O  C 2  , if a nd only if so do es R N  f ◦ U − 1 η c ; θ  , thus if and only if so do es R N ( f ; θ ) fo r all f ∈ O  C 2  . Finally , b y applying Le mma 6 again, it is true if and only if E N ( f ; θ ) conv erges to f (uniformly on a n y compact s ubset) for a ll f ∈ O  C 2  . √ W e also prove the following prelimina r result ab out the homo graphic tr a nsfor- mations defined by (3.3). Lemma 8. F or al l η c / ∈ { η j } j ≥ 1 ∪ {∞} , one has h − 1 η c = h η c wher e h η c : C P 1 → C P 1 ζ 7→ 1 + η c ζ ζ − η c . In addition, one also has that η c / ∈ { h η c ( η j ) } j ≥ 1 S {∞} , i.e. h − 1 η c = h η c is of t he same kind (3.3) for the asso ciate d set { h η c ( η j ) } j ≥ 1 = { θ j } j ≥ 1 . Pr o of. Indeed, for a ll ζ ∈ C \ { η c } , one ha s that  h η c ◦ h η c  ( ζ ) = 1 + η c 1 + η c ζ ζ − η c 1 + η c ζ ζ − η c − η c = ζ + η c η c ζ 1 + η c η c = ζ , then the equa lity holds for all ζ ∈ C P 1 . The second assertion follows b y (3.3) since h η c ( ∞ ) = η c , then h η c ( η j ) 6 = η c for all j ≥ 1. √ W e finish the subsection with the following result reminded as Lemma 18 from [7], and that gives an equiv alen t definition for the geometric criterio n (1 .5). Lemma 9. The set { η j } j ≥ 1 is lo c al ly interp olable by r e al-analytic curves if and only if it c an lo c al ly holomorphi c al ly int erp olate the c onjugate function, i.e. for al l ζ ∈ { η j } j ≥ 1 (the top olo gic al closur e of { η j } j ≥ 1 in C P 1 ), ther e ar e a neighb o rho o d V of ζ and g ∈ O ( V ) such that η j = g ( η j ) , ∀ η j ∈ V . (3.5) 3.2. On the extraction of certain su bs equences. No w we can give the pr oo f of Pro position 2. Pr o of. Since the set { η j } j ≥ 1 is not lo cally interpola ble b y r eal-analytic curves, it follows by Lemma 9 that there is ζ 0 ∈ { η j } j ≥ 1 without any neighborho o d V ∈ V ( ζ 0 ) and holomor phic function g ∈ O ( V ζ 0 ) that ca n interpo late the conjugate function on { η j } j ≥ 1 T V , i.e. ∀ V ∈ V ( ζ 0 ) , ∀ g ∈ O ( V ) , ∃ η j ∈ V , g ( η j ) 6 = η j . (3.6) 14 AMADEO IRIGOYEN In particula r, ζ 0 cannot b e isolated in { η j } j ≥ 1 . Other wise, if ζ 0 6 = ∞ (resp. ζ 0 = ∞ ), then by taking V ζ 0 ⊂ C s uc h tha t { η j } j ≥ 1 T V ζ 0 = { ζ 0 } (resp. V ∞ = C P 1 \ K , where the compact subset K is big enough so that { η j } j ≥ 1 \ K ⊂ {∞} ) a nd g ζ 0 : V ζ 0 → C ζ 7→ g ζ 0 ( ζ ) ≡ ζ 0 (resp. g ∞ : V ∞ → C P 1 ζ 7→ g ∞ ( ζ ) ≡ ∞ ) , we would get a contradiction with (3.6). As a consequence, ther e is a subsequence ( η j k ) k ≥ 1 ⊂ ( η j ) j ≥ 1 that satisfies: ( ( η j k ) k ≥ 1 conv erges to ζ 0 ; η j k 6 = ζ 0 for all k ≥ 1. (3.7) W e will then deal with the cases ζ 0 ∈ C and ζ 0 = ∞ resp ectiv ely . ζ 0 ∈ C . W e star t by setting S 0 := ( η j k ) k ≥ 1 and S 1 := S 0 \ D ( ζ 0 , 1) , (3.8) where D ( ζ 0 , 1) = { ζ ∈ C , | ζ − ζ 0 | < 1 } . By construction, S 1 gives a (nonempty a nd infinite) sequence that conv erges to ζ 0 . If S 1 (as a se t) is not lo c ally interpola ble by rea l-analytic curves, the pro position is prov ed. Other w is e (beca use ζ 0 is a limit po in t of S 1 ), there are V 1 ∈ V ( ζ 0 ) and g 1 ∈ O ( V 1 ) such that η j = g 1 ( η j ) for all η j ∈ S 1 ∩ V 1 . By re ducing V 1 if necessary , w e can assume that V 1 ⊂ D ( ζ 0 , 1) and V 1 is connected. Since ζ 0 satisfies (3.6), it follows that g 1 | V 1 cannot interpolate the conjugate function on V 1 T { η j } j ≥ 1 , i.e. there is η s 1 ∈ V 1 T { η j } j ≥ 1 such that g 1 ( η s 1 ) 6 = η s 1 . W e set S 2 := S 1 [ { η s 1 } and S 2 (with any enumeration) s till gives a sequence that conv erges to ζ 0 . Let fix m ≥ 1 and let a ssume having constructed η s 1 , . . . , η s m , S 1 , . . . , S m , V 1 , . . . , V m and g 1 , . . . , g m such that for all q = 1 , . . . , m , one has the following prop erties: V q ∈ V ( ζ 0 ) and V q is connected; (3.9) η s q ∈ V q ⊂ D  ζ 0 , 1 / 2 q − 1  ; (3.10) g q ∈ O ( V q ) and g q  η s q  6 = η s q ; (3.11) g q ( η j ) = η j for all η j ∈ S q ∩ V q , (3.12) where S q = S 1 [  η s 1 , . . . , η s q − 1  for all q = 2 , . . . , m . (3.13) W e first consider the set S m +1 := S 1 [ { η s 1 , . . . , η s m } and this sa tisfies (3.13) fo r a ll q = 2 , . . . , m + 1 . Next, S m +1 (with any enumeration) will g ive a s e quence that still co n verges to ζ 0 as the union of S 1 (that conv erges GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 15 to ζ 0 by (3.8) and (3.7)) and the finite set { η s 1 , . . . , η s m } . If S m +1 is no t lo cally int erp olable by real-a nalytic curves, the pro position is proved. O therwise (b ecause ζ 0 is a limit po in t of S m +1 ), there are V m +1 ∈ V ( ζ 0 ) and g m +1 ∈ O ( V m +1 ) such that η j = g m +1 ( η j ) for a ll η j ∈ S m +1 ∩ V m +1 . By reducing V m +1 if necessary , we can assume that V m +1 ⊂ D ( ζ 0 , 1 / 2 m ) and V m +1 is connected. On the o ther hand, since ζ 0 satisfies (3.6), it f ollows that there is η s m +1 ∈ V m +1 such that g m +1  η s m +1  6 = η s m +1 . This proves (3.9), (3.10), (3 .1 1) and (3.12) for q = m + 1, and completes the induction. Now if there is m ≥ 1 such that the set S m defined by (3.13) is not lo cally in terp o- lable b y real-analytic cur v es, then the propo sition is proved (since a n y enumeration of S m will give a sequence that converges to ζ 0 ). Otherwise, we can construct for a ll m ≥ 1, such η s m , S m , V m and g m that fulfill (3.9), (3.10), (3.11) and (3.12) for all q = 1 , . . . , m , and co nsider the following set S ∞ := [ m ≥ 1 S m = S 1 [ { η s m , m ≥ 1 } . Then an y en umeration of S ∞ will give a sequence that conv erges to ζ 0 as the union of S 1 (that conv erges to ζ 0 by (3.8) and (3.7)) and the convergen t sequence ( η s m ) m ≥ 1 (since by (3.10), one has η s m ∈ D  ζ 0 , 1 / 2 m − 1  ). If we prov e that S ∞ is not lo cally interp olable by real-ana lytic cur ves, the pro of o f the pr opo s ition will b e achiev ed in the c ase for which ζ 0 ∈ C . Let a ssume on the contrary that S ∞ is, i.e. (since ζ 0 is a limit point of S ∞ ) ther e are V ∞ ∈ V ( ζ 0 ) and g ∞ ∈ O ( V ∞ ) such that g ∞ ( η j ) = η j for all η j ∈ S ∞ T V ∞ . In particular, this yields for all m ≥ 1 (since S m ⊂ S ∞ ), g ∞ ( η j ) = η j , ∀ η j ∈ S m \ V ∞ . (3.14) On the other hand, by (3.10) there is m 0 ≥ 1 s uc h that V m 0 ⊂ D  ζ 0 , 1 / 2 m 0 − 1  ⊂ V ∞ . In addition, one has by (3.12) for q = m 0 , that g m 0 ( η j ) = η j for all η j ∈ S m 0 ∩ V m 0 . Hence g m 0 and g ∞ | V m 0 are b oth holo morphic functions on the doma in V m 0 , that coincide o n the set S m 0 ∩ V m 0 . Since by (3 .13), S m 0 ∩ V m 0 ⊃ S 1 ∩ V m 0 that is infinite with limit po in t ζ 0 ∈ V m 0 by (3 .8), (3.7) and (3.9), it follows tha t g ∞ | V m 0 ≡ g m 0 . (3.15) But an application of (3.14) for m = m 0 + 1 yields g ∞ ( η j ) = η j for a ll η j ∈ S m 0 +1 ∩ V ∞ . In particula r, since η s m 0 ∈ V m 0 ⊂ D  ζ 0 , 1 / 2 m 0 − 1  ⊂ V ∞ (b y (3 .10) for q = m 0 ) and η s m 0 ∈ S m 0 +1 by (3.13) for m = m 0 + 1, one has η s m 0 ∈ S m 0 +1 ∩ V ∞ then g ∞  η s m 0  = η s m 0 . (3.16) Moreov er, an application of (3.11) for q = m 0 , als o yields g m 0  η s m 0  6 = η s m 0 . (3.17) Finally , (3.15), (3.16) and (3.17) toge ther lead to (since η s m 0 ∈ V m 0 ) η s m 0 = g ∞  η s m 0  = g ∞ | V m 0  η s m 0  = g m 0  η s m 0  6 = η s m 0 , 16 AMADEO IRIGOYEN and this is impo ssible. Necessarily , S ∞ cannot be lo cally in terp olable b y real- analytic cur ves and the prop osition is pr o ved in the case for which ζ 0 ∈ C . ζ 0 = ∞ . First, by removing 0 from { η j } j ≥ 1 if necessary , we can assume that η j 6 = 0 , ∀ j ≥ 1 (as well a s η j k 6 = 0 , ∀ k ≥ 1). Indeed, since the sequence ( η j k ) k ≥ 1 conv erges to ∞ by (3.7 ), it follows that the subset { η j k } k ≥ 1 \ { 0 } is infinite, then so is the s et { η j } j ≥ 1 \ { 0 } ⊃ { η j k } k ≥ 1 \ { 0 } . In a ddition, the new s ubs et { η j k } k ≥ 1 \ { 0 } gives a new sequence that still satisfies (3.7). Now let consider the sequence ( θ j ) j ≥ 1 where θ j := 1 η j for all j ≥ 1 . First, ( θ j ) j ≥ 1 is well-defined. Next, since ( η j k ) k ≥ 1 satisfies (3.7) with ζ 0 = ∞ , it follows that so do es the subsequence ( θ j k ) k ≥ 1 with the choice o f ζ ′ 0 := 0, i.e. ( ( θ j k ) k ≥ 1 conv erges to 0 ; θ j k 6 = 0 for all k ≥ 1. (3.18) Lastly , w e claim that ζ ′ 0 = 0 satisfies (3.6 ) as well. Indeed, let be V ∈ V (0) and g ∈ O ( V ). W e wan t to prov e that there exists θ j ∈ V such that g ( θ j ) 6 = θ j . If g (0 ) 6 = 0, then by (3.18), θ j k − → 0 and g ( θ j k ) − → g (0) 6 = 0 as k → + ∞ . It follows that g ( θ j k ) 6 = θ j k for all k lar ge enough and the claim is prov ed in this case. Otherwise, g (0) = 0. Let co ns ider W :=  1 ζ , ζ ∈ V \ { 0 }  [ {∞} and h : W − → C P 1 ∞ 7− → ∞ , ζ ∈ C ∩ W 7− →    1 g (1 /ζ ) if g (1 /ζ ) 6 = 0 , ∞ otherwise. Then W ∈ V ( ∞ ), h is well-defined and h ∈ O ( W ). It fo llo ws by (3.6) that there is η j ∈ W such that h ( η j ) 6 = η j . If g (1 /η j ) = 0, i.e. g ( θ j ) = 0, then θ j = 1 /η j 6 = 0 = g ( θ j ), and this pro ves the claim in that case . Otherwise, g (1 / η j ) 6 = 0, i.e. g ( θ j ) 6 = 0 then 1 g ( θ j ) = 1 g (1 /η j ) = h ( η j ) 6 = η j = 1 θ j , hence g ( θ j ) 6 = θ j and the claim is proved in this last ca se. W e ca n now apply the pr evious cas e of the propo sition with the choice of ( θ j ) j ≥ 1 and ζ ′ 0 = 0 to get a subsequence  θ j ′ k  k ≥ 1 (maybe different from ( θ j k ) k ≥ 1 ) that conv erges to 0 and that is not lo cally in terp olable by r eal-analytic curves. It follows that the seq uence  η j ′ k  k ≥ 1 =  1 /θ j ′ k  k ≥ 1 conv erges to ∞ . On the other ha nd, the inv erse function ζ 7→ 1 /ζ being a homogra phic trans fo rmation, it is in particula r a biholomor phic application o f C P 1 . Hence the subset n η j ′ k o k ≥ 1 = n 1 /θ j ′ k o k ≥ 1 GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 17 cannot b e either lo cally int erp olable b y rea l- analytic cur v es. Finally , one a lso has that η j ′ k 6 = 0 for all k ≥ 1. This proves the pro positio n in this second case a nd completes its whole pro of. √ R emark 3.1 . As we hav e seen in the ab ov e pro of, we know in a ddition that in the case for which ζ 0 = ∞ , we a ls o hav e that η j k 6 = 0 for all k ≥ 1. 4. Proof o f the main theorem In the first subsection, we deal with the proo f of the equiv alence bet ween (1) and (3) in the statement of Theorem 2. 4.1. On the s tabilit y by extraction of subsequences . Befo re giving the pro of of this pa rt, we remind the following result as Prop osition 2 fro m [7], and that is a sp ecial cas e of equiv alence for the g eometric c r iterion (1.5), i.e. in the par ticular case when ( η j ) j ≥ 1 is a co n vergen t sequence, co ndition (1.5) also b ecomes necessary . Prop osition 3. L et ( η j ) j ≥ 1 b e any c onver gent se quenc e (in C ). If the interp ola tion formula E N ( f ; η ) c onver ges t o f (uniformly on any c omp act su bset) for al l f ∈ O  C 2  , then { η j } j ≥ 1 is lo c al ly interp olable by r e al-analytic curves. R emark 4.1 . T o be rigor ous, in order to apply Propos ition 3, we sho uld also as- sume that E N ( f ; η ) conv erges to f for all f ∈ O ( B 2 (0 , r 0 )). But as sp ecified by Remark 5.2 from [7], it is sufficient to assume the co n vergence of E N ( f ; η ) for all f ∈ O  C 2  . Pr o of. Firs t, if { η j } j ≥ 1 is loca lly interpolable b y r eal-analytic curves, then so is any (infinite) subset { η j k } k ≥ 1 . The implication (1) = ⇒ (3) then follows by The o rem 1. Conv ersely , let assume that { η j } j ≥ 1 is not lo cally interpo lable by r eal-analytic curves. By Prop osition 2, ther e is a subsequence ( η j k ) k ≥ 1 that is not lo cally in ter- po lable by real-ana lytic curves and that is con v ergent (in C P 1 ). In order to get the conv erse implication (3) = ⇒ (1), we wan t to prov e that η ′ := ( η j k ) k ≥ 1 do es not make co n verge its a sso ciated interpolation formula E N ( · ; η ′ ) for entire functions, i.e. there exists (at leas t) one function f ∈ O  C 2  such that E N ( f ; η ′ ) do es not conv erge to f (uniformly on any compact subset K ⊂ C 2 ). Let b e ζ 0 = lim k → + ∞ η j k . If ζ 0 is finite, the required asser tio n follows b y Pro po- sition 3. Otherwise, ζ 0 = ∞ and b y Remark 3.1, o ne also ha s that η j k 6 = 0 for all k ≥ 1. It follows that the sequence θ ′ := ( θ j k ) k ≥ 1 where θ j k := 1 η j k for all k ≥ 1 , is w ell-defined (a s a subset of C ), b ounded a nd conv erges to 0 . On the other hand, θ j k = h 0 ( η j k ) for all k ≥ 1 , where h 0 is the homog raphic trans formation defined as h 0 ( ζ ) = 1 / ζ (see (3.3) with the choice of η c = 0). Th us { θ j k } k ≥ 1 is not lo cally int erp olable by real-analytic curves (becaus e any homographic transforma tion is in particular bih olomor phic). Again, by Prop osition 3, the seq uence θ ′ = ( θ j k ) k ≥ 1 do es not mak e conv erge its assoc ia ted interpo lation form ula E N ( · ; θ ′ ) for en tire functions, i.e. there exis ts f ∈ O  C 2  such that E N ( f ; θ ′ ) do es not conv erge to 18 AMADEO IRIGOYEN f (uniformly on an y co mpact subse t). Finally , since E N ( · ; θ ′ ) = E N ( · ; h 0 ( η ′ )), it follows by Cor ollary 4 that neither c an do the sequence η ′ = ( η j k ) k ≥ 1 for the formula E N ( · ; η ′ ) for ent ire functions , and this pr o ves the implicatio n (3) = ⇒ (1 ). √ 4.2. On the action by p ermut ations. Now we can give the pro of of the second part of Theorem 2 that is the eq uiv alence b etw een (1) and (2), and a c hieve its whole pro of. W e fir s t need a spec ific result that is a part of the pro of for The o rem 1 from [7] (reminded ab ov e as Theor em 3). Lemma 10. L et b e ( η j ) j ≥ 1 such that, for al l f ∈ O  C 2  , R N ( f ; η ) is uniformly b ounde d on any c omp act subset of C 2 . Then the estimate (2.1) fr om The or em 3 is satisfie d. This result is Lemma 11 from [7 ] and yields the part (2) = ⇒ (3) in the s tatemen t of Theorem 3. In pa rticular, the impo rtan t fa c t is that no one condition is needed for the set { η j } j ≥ 1 (like boundedness, see Remark 3 .1 from [7]). This will be useful in order to prove the implication (2) = ⇒ (1) in the statement of Theorem 2. Pr o of. The implication (1) = ⇒ (2) immediately follo ws by Theorem 1 since t he prop erty of b eing lo cally interp o lable by rea l-analytic curves is a condition ab out sets, then it do es not dep end on any enumeration of { η j } j ≥ 1 . Conv ersely , let assume that { η j } j ≥ 1 (coming fro m the sequence η = ( η j ) j ≥ 1 ) is not locally in terp o lable b y re a l-analytic curves. W e wan t to find a p ermutation σ o f N \ { 0 } such that E N ( · ; σ ( η )) do es not c on verge for entire functions (where σ ( η ) :=  η σ ( j )  j ≥ 1 ). W e kno w by Prop osition 2 that there are ζ 0 ∈ C P 1 and a subsequence η ′ = ( η j k ) k ≥ 1 of ( η j ) j ≥ 1 that sa tisfy the following conditions: ( the sequenc e ( η j k ) k ≥ 1 conv erges to ζ 0 ; the set { η j k } k ≥ 1 is no t lo cally in terp olable by real-ana lytic curves. (4.1) First, we claim that can w.l.o .g. assume that ζ 0 is finite. Indeed, if ζ 0 = ∞ , let consider η c / ∈ { η j } j ≥ 1 S {∞} , h η c ∈ O  C P 1  defined by (3 .3) and the a sso ciated sequence θ = ( θ j ) j ≥ 1 = ( h η c ( η j )) j ≥ 1 (that is well-defined by (3.4)) . Then the subsequence ( θ j k ) k ≥ 1 = ( h η c ( η j k )) k ≥ 1 satisfies (4 .1 ) with ζ ′ 0 = h η c ( ∞ ) = η c ∈ C (beca use ( η j k ) k ≥ 1 do es and h η c is biholomorphic). It will fo llow that there will b e a per m utation σ such that the sequence σ ( θ ) =  θ σ ( j )  j ≥ 1 do es not make converge its asso ciated interp olation fo rm ula E N ( · ; σ ( θ )) for entire functions. Since h − 1 η c ( σ ( θ )) = h − 1 η c h  θ σ ( j )  j ≥ 1 i =  h − 1 η c  θ σ ( j )  j ≥ 1 =  h − 1 η c  h η c  η σ ( j )  j ≥ 1 =  η σ ( j )  j ≥ 1 = σ ( η ) (notice that n  h − 1 η c ( σ ( θ ))  j o j ≥ 1 =  h − 1 η c ( θ j )  j ≥ 1 is well-defined as a subse t of C by Lemma 8), a n a pplication of Co rollary 4 (which is p ossible bec ause h − 1 η c = h η c by Lemma 8) will allow us to deduce that neither will do the sequence h − 1 η c ( σ ( θ )) = σ ( η ) for E N ( · ; σ ( η )) for entire functions, i.e. there will exist (a t least) o ne function f ∈ O  C 2  such that E N ( f ; σ ( η )) will not conv erge to f (uniformly o n any compact subset K ⊂ C 2 ). This will prove the required implication (2) = ⇒ (1) of the theorem GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 19 for the ca se ζ 0 = ∞ a nd co mplete the who le pr oof of the equiv alence b etw een (1) and (2) in the general cas e. W e can then assume that ζ 0 ∈ C in (4 .1). Let fix the enumeration o f the ass oci- ated subsequenc e η ′ = ( η j k ) k ≥ 1 as well as the c a nonical one for the complementary subsequence η ′′ = ( η r m ) m ≥ 1 := ( η j ) j ≥ 1 \ ( η j k ) k ≥ 1 . (4.2) Since η ′ = ( η j k ) k ≥ 1 is a (b ounded) conv ergent seque nce that is not lo cally inter- po lable by real-a nalytic cur v es, it follows by Pro positio n 3 that E N ( · ; η ′ ) canno t conv erge f or entire functions. By an application of (2) ⇐ ⇒ (3) in Theorem 3, it follows that the sequence of the asso ciated divided differences is not exp onentially bo unded, i.e. ∀ R ≥ 1, ∃ p R , q R ≥ 0 such that      ∆ p R , ( η j p R ,...,η j 1 ) "  ζ 1 + | ζ | 2  q R #  η j p R +1       > R p R + q R . (4.3) In particular , ther e a re p 1 , q 1 ≥ 0 such that      ∆ p 1 , ( η j p 1 ,...,η j 1 ) "  ζ 1 + | ζ | 2  q 1 #  η j p 1 +1       ≥ 1 . (4.4) W e set σ ( k ) = j k for all k = 1 , . . . , p 1 + 1 and σ ( p 1 + 2) = r 1 . (4.5) Then σ is injective on the first ( p 1 + 2) indices, 1 is attained s ince 1 ∈ { j 1 , r 1 } ⊂ σ ( { 1 , . . . , p 1 + 1 , p 1 + 2 } ) and (4 .4) can b e rewritten as      ∆ p 1 , ( η σ ( p 1 ) ,...,η σ (1) ) "  ζ 1 + | ζ | 2  q 1 #  η σ ( p 1 +1)       ≥ 1 . (4.6) The p ermutation σ will b e constructed by inductio n o n m ≥ 1. W e fir s t se t p 0 := − 2 , (4.7) and we a ssume having defined σ on { 1 , . . . , p m + 2 } wher e p l − 1 + 2 ≤ p l for all l = 1 , . . . , m , (4.8) as follows: for all l = 1 , . . . , m , σ ( k ) = ( j k − l +1 for all k = p l − 1 + 3 , . . . , p l + 1 , r l if k = p l + 2 . (4.9) W e also a ssume that for all l = 1 , . . . , m ,      ∆ p l , ( η σ ( p l ) ,...,η σ (1) ) "  ζ 1 + | ζ | 2  q l #  η σ ( p l +1)       ≥ l p l + q l . (4.10) W e indeed chec k that (4.8) is fulfilled for m = 1 sinc e p 1 ≥ 0 and p 0 = − 2 by (4.7). Similarly , (4.9) (resp. (4.10)) is satisfied for m = 1 b y (4.7) and (4.5) (resp. by (4.6 )). Now let consider the sequence η ( m ) =  η ( m ) k  k ≥ 1 defined as follows: η ( m ) k := ( η σ ( k ) for all k = 1 , . . . , p m + 2 , η j k − m for all k ≥ p m + 3 . (4.11) 20 AMADEO IRIGOYEN Since  η ( m )  (as a set) is the union of { η j k } k ≥ 1 and the finite set { η r 1 , . . . , η r m } by (4.9), the sequence η ( m ) is b ounded and satisfies (4.1) as well (with the same limit p oint ζ 0 ). Again, by successive applications of Prop osition 3 and (2) ⇐ ⇒ (3) from Theorem 3, it follows that η ( m ) satisfies (4.3). In particular, with the choice of R = m + 1, there are p m +1 , q m +1 ≥ 0 such that      ∆ p m +1 ,  η ( m ) p m +1 ,...,η ( m ) 1  "  ζ 1 + | ζ | 2  q m +1 #  η ( m ) p m +1 +1       ≥ ( m + 1) p m +1 + q m +1 . (4.12) In addition, we ca n c ho ose p m +1 ≥ p m + 2 (this will satisfy (4.8) for all l = 1 , . . . , m + 1). Indee d, if it w ere not p ossible, this would mea n that for all R ≥ m + 1, the asso ciated p R should b e b ounded. B y Corollary 3, so w ould b e all the terms      ∆ p R ,  η ( m ) p R ,...,η ( m ) 1  "  ζ 1 + | ζ | 2  q #  η ( m ) p R +1       for all q ≥ 0 , and this would contradict (4.3) for η ( m ) . W e can then extend σ to { 1 , . . . , p m +1 + 2 } as follows: σ ( k ) = ( j k − m for all k = p m + 3 , . . . , p m +1 + 1 , r m +1 if k = p m +1 + 2 . (4.13) The induction hypothesis (4.9) and (4.13) show that σ is well-defined on { 1 , . . . , p m +1 + 2 } . Moreov er, one has by (4.11) and (4.1 3) that η σ ( k ) = η ( m ) k for all k = 1 , . . . , p m +1 + 1 , then it follows by (4.12) that (4.1 0) is still satisfied fo r l = m + 1. This last a sser- tion with the induction hypotheses (4.9) and (4.1 0) co mplete the ca se for m + 1, i.e. (4.9) and (4.10) are s till satisfied for all l = 1 , . . . , m + 1. The sequence ( p m ) m ≥ 1 constructed above allows us to define σ for all k ≥ 1 by (4.9) s ince we have the following par tition fr om (4.7) a nd (4 .8), N \ { 0 } = [ m ≥ 1 { k , p m − 1 + 3 ≤ k ≤ p m + 2 } . Next, σ is a p erm utation of N \ { 0 } : indeed, it follows fr o m (4 .8) that every se t { k , p m − 1 + 3 ≤ k ≤ p m + 2 } co ntains a t least tw o elements, i.e. σ attains by (4.9) exactly one of the type r m and at least one of the type j k as well. On the other ha nd, one has by (4.9) again tha t for all m ≥ 1 , σ ( p m + 1) = j p m − m +2 and σ ( p m + 3) = j p m +3 − ( m +1)+1 = j p m − m +3 . This last a ssertion a nd (4.5) toge ther show that all the j k ’s (resp. r m ’s) ar e reached exac tly once. Finally , the estima te (4.1 0) b eing satisfied for all m ≥ 1 (i.e. this contradicts the es tima te (2.1) from Theo rem 3), it fo llo ws by a n application of L emma 10 tha t there is f ∈ O  C 2  such that R N ( f ; σ ( η )) cannot b e unifor mly b ounded (on any compact subset K ⊂ C 2 ). In pa rticular, R N ( f ; σ ( η )) cannot even co n verge to 0, then by Lemma 6, E N ( f ; σ ( η )) do es not c o n verge to f (uniformly on a n y c o mpact subset K ⊂ C 2 ). This achieves the implication (2) = ⇒ (1 ) from Theorem 2 a nd completes its whole pro of. √ 4.3. Pro of of Coroll aries 1 and 2. In order to prov e Corollar y 1, we first remind the following auxiliary res ult that is Lemma 8 fr om [7]. GEOMETRIC CONDITIONS FOR AN INTERPOLA TION FORMULA 21 Lemma 1 1. L et r 0 > 0 b e fi xe d. If ther e is ε η > 0 such that, ∀ f ∈ O ( B 2 (0 , r 0 )) , R N ( f ; η ) c onver ges to 0 uniformly on any c omp act subset of B 2 (0 , ε η r 0 ) , then ∀ g ∈ O  C 2  , R N ( g ; η ) c onver ges to 0 un iformly on any c omp ac t subset of C 2 . W e can then g iv e the pro of o f Corolla ry 1. Pr o of. Firs t, as for the pro of of Theorem 2, the implication (1) = ⇒ (2) (resp. (1) = ⇒ (3)) from the statement of the cor ollary immediately follows by Theorem 1 since, if { η j } j ≥ 1 is lo cally interpo lable by real- analytic curve, then so will b e the (same) set { σ ( η ) } =  η σ ( j )  j ≥ 1 for all σ ∈ S N (resp. the subset { η j k } k ≥ 1 coming from any subsequence ( η j k ) k ≥ 1 ). Conv ersely , in or der to prov e the implication (2) = ⇒ (1) (resp. (3) = ⇒ (1)), let fix σ ∈ S N (resp. η ′ = ( η j k ) k ≥ 1 ) and g ∈ O  C 2  . The hypothesis (2) (resp. (3)) and Lemma 6 imply that for all f ∈ O ( B 2 (0 , r 0 )), R N ( f ; σ ( η )) (resp. R N ( f ; η ′ )) conv erges to 0 uniformly on a n y co mpact subset K ⊂ B 2 (0 , ε η r 0 ). It follows by Lemma 11 that in particular R N ( g ; σ ( η )) (resp. R N ( g ; η ′ )) conv erge s to 0 uniformly on any compact subset K ⊂ C 2 . Aga in, by a n a pplication of Lemma 6 , o ne can deduce that E N ( g ; σ ( η )) (resp. E N ( g ; η ′ )) c on verges to g uniformly on a n y co mpact subset K ⊂ C 2 . Finally , σ ∈ S N (resp. η ′ = ( η j k ) k ≥ 1 ) and g ∈ O  C 2  being arbitrar y , the condi- tion (2) (resp. (3)) from the statement of Theorem 2 is satisfied, whose application yields the required a ssertion (1). √ Lastly , the pro of o f Co rollary 2 immediately follows by an application of The- orem 1 (or T he o rem 3 from [7 ]) and Corolla ry 1 from [7] since the set  η σ ( j )  j ≥ 1 (resp. { η j k } k ≥ 1 ) is still lo cally interp o lable by real-ana lytic curves. References [1] B. Berndtsson, A formula for in terpolation and divisi on in C n , Math. 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