Identifying Linear Combinations of Ridge Functions

Transcription

1 Advaces Appled Matheatcs 22, Ž Atcle ID aaa , avalable ole at o Idetfyg Lea Cobatos of Rdge Fuctos Mat D. Buha Matheatk, Lehstuhl 8, Uestat Dotud, Dotud, Geay ad Alla Pkus Depatet of Matheatcs, Techo-Isael Isttute of Techology, Hafa, 32000, Isael Receved July 1, 1997; accepted Septebe 19, 1997 Ths pape s about a vese poble. We assue we ae gve a fucto fž. x whch s soe su of dge fuctos of the fo Ý g Ža x. 1 ad we just kow a uppe boud o. We seek to detfy the fuctos g ad also to detfy the dectos a fo such lted foato. Seveal ways to solve ths olea poble ae dscussed ths wok Acadec Pess 1. INTRODUCTION A dge fucto s a ultvaate fucto of the sple fo h:, hž x 1,..., x. gž a1x1 ax. g Ž a x., whee g: ad a a,...,a I othe wods, t s a ultvaate fucto costat o the paallel hypeplaes a x c, c. The vecto a 04 s geeally called the decto. Rdge fuctos appea vaous aeas ad ude vaous guses. We fd the the aea of patal dffeetal equatos Žwhee they have E-al: db@ath.u-dotud.de. E-al: pkus@ath.techo.ac.l $30.00 Copyght 1999 by Acadec Pess All ghts of epoducto ay fo eseved.

2 104 BUHMANN AND PINKUS bee kow fo ay, ay yeas ude the ae of plae waes. 7. We also fd the used coputezed toogaphy 10, statstcs Žwhee they appea pojecto pusut algoths. 5, eual etwoks, ad of couse appoxato theoy. Moe about dge fuctos ay be foud Pkus 11, ad efeeces thee. Whe dealg wth dge fuctos, oe s geeally teested oe of thee possble sets of fuctos. The fst s gve by 1 ½ Ý 5 1 RŽ a,...,a. g Ž a x.: g CŽ., 1,...,. That s, we fx a fte ube of dectos ad we cosde lea cobatos of dge fuctos wth these dectos. The fuctos g ae the vaables. Ths s a lea space. The secod set s ½ Ý R g a x : a 0, g C, 1,...,. Hee, we fx ad we choose both the fuctos g ad the dectos a. Ths s ot a lea space. The thd set s otvated by a odel eual etwoks. It s a subset of the secod. We fx CŽ., called the tasfe fucto eual etwok lteatue, ad we let ½ Ý N c a x b : a 0, c, b, 1,...,. Hee we also fx ad choose both the dectos a Ž called the weghts., ad the shfts b Ž called the thesholds.. Ths s ot a lea space. Oe poble et wth whe dealg wth fucto sets such as the pecedg, s kowg f ad whe a gve fucto s the set. That s, do we have ay way of kowg whethe ay pescbed f s RŽa 1,...,a. fo soe gve dectos a 1,...,a,o R,o N? Whle the fst set s lea, the latte two ae ot, ad ths poble s theefoe fa fo tval. A secod uch elated poble s the followg. Assue you kow Ž 1. 1 o suppose that f s R a,...,a fo soe gve dectos a,...,a, o R,o N. How ca we detee the ukows, be they the fuctos, the dectos, o the shfts? ŽIf we kow a ethod fo deteg these ukows, we essetally have a ethod of fdg whethe we ae the appopate set: we assue that we ae the appopate set,

3 LINEAR COMBINATIONS OF RIDGE FUNCTIONS 105 we detfy the ukows, ad the we check whethe the esultg fucto s fact ou ogal fucto.. I ths pape we addess these questos ad we gve a athe geec ethod of asweg the latte questo. That s, we show that the poble ca be solved. A ajo dawback s that ths ethod s athe oe theoetcal tha pactcal, although pcple ou poofs ae costuctve. I a pevous pape 3, we cosdeed a sla ecovey poble. Thee we assued that we wee gve a fucto G: ad a fucto f:, the latte we kew to be of the fo f Ž x. Ý cg j Ž x t j., x, j1 fo soe ukow coeffcets c 4 4 j j1 ad shfts t j j1, whee Ž o a uppe boud o. s kow. The poble was to detfy the coeffcets ad shfts. The techques developed that pape ae used hee. ŽI that pape we also cosdeed N. The esult obtaed Ž Theoe 6 of. 3, whle ot techcally eo, was eagless. Thus ths pape also allows us to edess ths wog.. 2. UNIQUENESS AND SMOOTHNESS Whe epesetg a fucto as a su of dge fuctos, o whe seekg to detfy ts vaous copoets, t s of fudaetal potace to ty to udestad the extet to whch ay epesetato s uque. We theefoe ask, f k l f Ž x. Ý gž a x. Ý hž b x., 1 1 what ca be sad about the g 4 k 4 k 4 1 ad a 1 elatve to the h 1 l ad b 41 l? Fo leaty popetes, ths poble educes to the followg foulato: Assug Ý g Ž a x. 0, 1 o at least a uppe boud o t beg kow, what ca we say about the g? The followg esult s vald.

4 106 BUHMANN AND PINKUS PROPOSITION 1. If s fte, the a ae pawse lealy depedet, the g CŽ., 1,...,, ad Ý g Ž a x. 0, 1 fo all x, the each g s a polyoal of degee at ost 2. Reak. I fact the boud o the degee of the polyoal ca be futhe educed. Poof. The poof of the poposto s eleetay f each of the g les C 1 Žsee Lea 1 Dacos ad Shahshaha. 4. It goes as 4 4 j follows. Fx 1,...,. Fo each j 1,...,, j, let c satsfy c j a j 0 ad c j a 0. Ths s possble because the a ae pawse lealy depedet. Fo geeal c c,...,c, let Now 1 Dc Ý c s. x s s1 Dgax c a g c Ž a x.. Thus, because each g s suffcetly sooth, 0 ŁD c j Ý g a x j1 1 j ž / j Ž 1. Ý Ł c a g a x 1 j1 j j Ž 1. Ł Ž c a. g Ž a x.. j1 j Fo ou choce of the c j, t follows that g Ž 1. a x 0,

5 LINEAR COMBINATIONS OF RIDGE FUNCTIONS 107 fo all x. Ths ples that g Ž 1. Ž t. 0, fo all t, ad g s theefoe a polyoal of degee at ost 2. If the g ae oly CŽ., the the esult eas vald. I fact we ay 1 eve suppose g s just locally tegable,.e., g L Ž. loc. To pove ths, we use soe vey basc deas fo dstbuto theoy. Choose u such that a u b 0, 1,...,. The 0 Ý gž a Ž x tu.. Ý gž a x tb., 1 1 fo evey t ad x. Let D Co the ftely sooth Ž fuctos wth copact suppot whch we shall call test fuctos.. Thus, 1 s a x 0 g a x tb t dt g s ds. Ý Ý H b ž 1 1 b b H Fx 1,...,4 ad let the c j 4 whch s the sae as j1, j / be as the pevous text. The ž / / 1 1 s a x 0 ŁD j c Ý H gž s. ds, ž b b b j1 j ž / 0 g Ž s. ds Ž c a.. Ž s a x 1 Ž 1. H Ł b b b j1 j j Because s a abtay fucto D, t follows that H Ž 1. g Ž s. Ž s. ds 0, fo all D. It s well kow that ths ples that g s a polyoal of degee at ost 2. Fo copleteess, hee s a shot poof. Assue fst fo splcty that g g C ad that H Ž 1. g s s ds 0, 2.2 fo all D. Fo each atual ube k, let k D have suppot 1k,1k, 0, ad H Ž s. ds 1. Thus 4 s a sequece of k k k k1

6 108 BUHMANN AND PINKUS appoxate dettes, the eleets of such a sequece beg chaactezed by copact suppot that shks to 04 wth gowg dex k, oegatvty, ad ut tegal. Let H g Ž t. gž s. Ž t s. ds. k The g C, ad k H Ž 1. Ž1. g k Ž t. gž s. k Ž t s. ds 0, fo each t. Theefoe g k s a polyoal of degee at ost 2. I addto, g k coveges ufoly to g as k o evey fte teval. Theefoe g s also a polyoal of degee at ost 2. 1 If g s ot cotuous but oly L ad satsfes Ž 2.2. loc fo ay test fucto D, we ca take Foue tasfos ad we ca get by Placheel s detty, H ˆŽ g x. x Ž x. dx 0, 1ˆ whee D ad thus ˆ ae stll abtay. Theefoe, the sese of dstbutos, x 1 ˆgŽ x. 0 whch eas ˆgŽ x. 0 eveywhee except at zeo. Thus Theoe poves the esult, aely, that ˆg s a fte lea cobato of the delta fucto ad ts devatves ceteed at the og, the degees beg lted by 2. ŽWe ae usg hee, of couse, that the vese Foue tasfo of the kth devatve of the delta fucto ceteed at zeo s a algebac polyoal of degee k.. Theefoe we have poved that g ust be a polyoal of degee less tha 1, alost eveywhee. Retug to ou g, t thus follows that each of the g s alost eveywhee a polyoal. The pots whee ay of the g ght ot be a polyoal of degee less tha 1 exted, though the e poduct sde g Ža x., to a hypeplae othogoal to a. Because the su ove of these expessos s detcally zeo ad because the dectos a ae utually lealy depedet, t s a staghtfowad cosequece that each g ust fact be a polyoal eveywhee. Reak. Ths questo of uqueess has bee cosdeed by Albet, Sotag, Mallot 2, Sussa 12, ad Feffea 6 fo N, especally the case whee x tah x. If the dectos a ae pawse lealy depedet, the we ca apply Poposto 1. The codto of pawse lea depedece s ot, howeve, atual ths settg. Poposto 1 k

7 LINEAR COMBINATIONS OF RIDGE FUNCTIONS 109 does pet us to educe the poble to studyg whe Ý c Ž a x. b 1 s a polyoal of degee at ost 2. Hee a ozeo a s fxed; we have gouped all tes wth lealy depedet dectos. We ae teested codtos o,, ad b whch the ply that the c ae all zeo. Is Poposto 1 vald wthout ay estcto o the g? Is t tue that f g :, 1,...,, ad the su ove the g Ža x. vashes detcally fo pawse lealy depedet a, the each g s a polyoal of degee at ost 2? The aswe, ufotuately, s o. It s well kow, see Aczel 1, p. 35 that thee exst hghly ocotuous fuctos h: satsfyg hž x y. hž x. hž y., fo all x ad y. These h ae costucted usg Hael bases. Thus, fo exaple, the equato, 0 g1ž x1. g2ž x2. g3ž x1 x2. has hghly opolyoal, ocotuous solutos g g g h The pecedg text togethe wth Poposto 1, begs the followg teestg questo. Naely, f f x Ý g a x, k Ž. l ad f C, do thee the exst g C fo whch f Ž x. Ý g Ž a x.? 1 I addto, what s the elatoshp betwee k ad l? Fo stace, we would wsh the pevous asseto to hold wth l k. The case k 0sof specal teest. Howeve, hee we wll cosde oly k 1 ad we wll pove ude ld assuptos that deed l k ths case. k Ž PROPOSITION 2. Assue that f has the fo 2.3 ad that f C. 1 fo soe k 1. If, addto, g L fo each, the g C k Ž.. loc

8 110 BUHMANN AND PINKUS Poof. We use the c as the poof of Poposto 1. Let be a abtay test fucto. The the expesso ž / H ŁD j c Ý g Ž a Ž x tu.. Ž t. dt j1 1 j H ž / Ł j1 Ž. D j c f x tu t dt 2.4 j s detcal to the ght-had sde of Ž I othe wods, the dstbutoal devatves of f Ž f s a dstbuto as well as a fucto!. of total ode 1 alog the dectos gve by the c j ae detcal to soe dstbutoal devatve of ode 1of g Ž tes ceta costats.. It does ot atte hee that Ž 2.4. s a uvaate tegal although a weak foulato of a devatve of the ultvaate fucto f would eque a tegal agast a ultvaate test fucto. Ths s because f has cotuous devatves ayway. Now, the dstbutoal devatve s equal to the cotuous classcal devatve of f accodg to Theoe , p. 195 because f C 1 Ž. ad theefoe the sae ust be tue fo the ght-had sde. Thus g C 1 ad Ł j1 j Ž 1. Ł j j1 j D j c f x g a x c a, j Ž 1. kž 1. whee all a ad c ae kow. As such g C Ž., k 1. The esult ow follows. A fucto f x, y s of the fo, Ž R a,...,a 2 fž x, y. Ý g Ž a x by., Ž x, y., 1 fo gve Ž a, b., but ukow cotuous g, 1,...,, f ad oly f ž / b a fž x, y. 0, Ž 3.1. x y Ł 1 a dstbutoal sese. Ths athe sple esult s based o the sae esult the case 1 whch s staghtfowad. Note that ths povdes a

9 LINEAR COMBINATIONS OF RIDGE FUNCTIONS 111 ethod of detfyg RŽa 1,...,a. f 2. Ufotuately a sple chaactezato such as Ž 3.1. does ot hold the case of thee o oe vaables. How ca we detee f a fucto f Ždefed o. s of the fo fo soe gve a,...,a 0 4, but ukow cotuous g,..., g? That s, how do we chaacteze RŽa 1,...,a. 1? Oe aswe ay be foud L ad Pkus 9. Let PŽa 1,...,a. deote the set of polyoals whch vash o all the les a : 4, 1,...,. Ths s a deal. THEOREM 3 ŽL ad Pkus. 9. The cotuous fucto f RŽa 1,...,a. f ad oly f 4 1 f spa q: q polyoal, p D q 0 fo evey p P a,...,a. Ths theoe, ad of tself, does ot povde a sple ethod of checkg whethe a patcula fucto s RŽa 1,...,a.. Howeve, t follows fo the theoy of polyoal deals that oe eed ot check evey p PŽa 1,...,a.. It s a cosequece of Hlbet s bass theoe, c.f. 13, p. 18, that t suffces to cosde ay set of p PŽa 1,...,a. whch ae geeatos fo the polyoal deal PŽa 1,...,a.. Sets of geeatos ae hghly ouque. But t s ot dffcult to show that thee always exsts a faly sple set of geeatos of cadalty. We shall, howeve, ot futhe pusue these deas hee. I Dacos ad Shahshaha 4 ae to be foud two addtoal theoes whch chaacteze RŽa 1,...,a.. The fst s stated oly fo the case 3. But t s fact vald fo evey. 1 THEOREM 4 Dacos ad Shahshaha 4. Let a,...,a be pawse lealy depedet ectos. Let deote the hypeplae c : c a 0, 4 1,...,. A fucto f C Ž. has the fo, f x Ý g a x P x, fo soe polyoal P of degee less tha, f ad oly f Ł c 1 D f 0, fo all choces of c, 1,...,. The fee polyoal te Ž 3.2. s a cosequece of Poposto 1. Aothe uch oe coplcated chaactezato of RŽa 1,...,a. fo ceta patcula choces of Ž due to Royde. ay be foud at the ed of the pape by Dacos ad Shahshaha 4.

10 112 BUHMANN AND PINKUS Ou goal hee s oe odest. Assue we ae gve f RŽa 1,...,a. of the fo Ž We wsh to detfy the g. If f s of the fo 2.3 wth pawse lealy depedet a, the, fo Poposto 1, these g ae uque up to polyoals of degee at ost 2. How do we detee these g? Hee s a ecpe based o the deas we have aleady dscussed. Assue fst that f C 1 Ž. s of the fo Ž Fx 1,...,4 ad let the c j 4j1, j, be as the poof of Poposto 1. I patcula, c j a j 0 ad c j a 0. The we have ŁD jf x D j c Ł c Ý g a x j1 j1 1 j j ž / j Ž 1. Ý Ł c a g a x 1 j1 j ž / j Ž 1. Ł Ž c a. g Ž a x.. j1 j By assupto we kow f ad thus the left-had sde of the pevous equato. We also kow the costat, Ł j1 j j c a, ad the vecto a tself. Thus ths sple ethod gves us g Ž 1.. I othe wods we ca detee g up to a polyoal of degee 2, ageeet wth Poposto 1. 1 Ths aguet also woks f f ad the g ae just L Ž. loc ad 1 L Ž. loc, espectvely. Fo ths, we ague as the poof of Poposto 1;.e., we fd a u as that poof ad we tegate agast a test fucto 1 D. Let ths test fucto scaled by b,.e., Žb 1., be a eleet fo a sequece of appoxate dettes. I patcula, H 1 b. Now, we ca detfy fo the dsplay Ž 2.1. the Ž 1. st devatve of g tegated agast that. Deote the esult by g Ž 1.. So we have 1 Ž 1. t a x j Ž 1. Ł H Ž c a. g Ž t. dt b ž b b, / j1 j 1 Ž 1. Ł j, j1 j g Ž a x. Ž c a., b ž /

11 LINEAR COMBINATIONS OF RIDGE FUNCTIONS 113 Ž 1. whch defes the g. I othe wods, tegatg Ž 1., tes gves g,, aely, the g soothed by the test fucto the afoeetoed way,.e., also scalg by b 1, / t a x g t dt. H ž b b Lettg the suppot of the appoxate detty ted to 04 whle atag ut tegal gves g Ža x. odulo a polyoal of degee 2. The latte s a cosequece of ou Ž 1. -fold tegato of the devatve. Because we ca do ths fo each 1,..., 4, we kow that f s cotaed the set of fuctos, Ý Ž g Ž a x. p Ž a x.., 1 whee each p s a abtay uvaate polyoal of degee at ost 2 ad g s such that g Ž 1. g Ž 1., 1,...,. That s, fo soe choce of polyoals p p we have g g p. Alteatvely, we ay state that f x Ý g a x p x s a ultvaate polyoal of total degee at ost 2 whch ay be deteed ad ca be wtte the fo, pž x. Ý p Ž a x., 1 fo soe choce of p of degee at ost 2. These p Ž ad theefoe the g. ae ot uque. Thee see to be vaous ethods of deteg appopate p. Hee s oe such ethod. We kow that 2 j pž x. Ý p Ž a x. Ý Ý jž a x., 1 1 j0 fo soe choce of coeffcets j. If we ca fd appopate j, we have costucted sutable p, 1,...,. Ths ca be doe the followg Ž. j 4, 2 fasho: Aog the a x 1, j0, choose a bass, ad the wte p tes of ths bass. The coeffcets j fo the polyoal tes of ths bass ca be foud explctly because we kow p. As such we have elucdated oe ecpe fo deteg appopate fuctos g.

12 114 BUHMANN AND PINKUS 4. R Assue f R,.e., f x Ý g a x, fo soe set of cotuous, but ukow, ozeo fuctos g, ad ukow dectos a 0 4, 1,...,. Ca we detee the g ad a? To be oe pecse Ž because thee s a poble of uqueess. we wsh to fd soe g ad a such that Ž 4.1. holds. Fo sooth g, we ca do ths, ude ceta futhe ld assuptos. I ths secto we expla how to fd dectos a 41. We the efe the eade to the pevous secto fo a ecpe fo deteg the g 4 1 based o kowledge of the a 41. The case 1 s elatvely sple. Let us see how t ay be doe because t s stuctve fo the oe coplcated ssues to follow. Assue that f Ž x. g Ž a x., fo soe ukow a Ž a. ad g. Assue also that f Ž 1 ad theefoe g. s cotuously dffeetable. The f x Takg atos we have f x ag a x, 1,...,. Ž x. x f Ž x. x j a,, j 1,...,, a j so log as a ad g Ž a x. j do ot vash. The ght-had sde s depe- det of x fo evey choce of, j 1,..., 4. Note that a ad g ae ot uquely deteed. Specfcally, we ca always eplace a by ca fo ay costat c 0, ad appopately alte g. Thus, kowg all the atos aaj effectvely detees a. Oce gve a, we obta g fo Secto 3. The geealzato to 1 s oe coplcated. We wll use the followg esult, the poof of whch ay be foud Secto 2 of Buha ad Pkus 3, see Theoe 1 ad Coollay 2 thee.

13 LINEAR COMBINATIONS OF RIDGE FUNCTIONS 115 THEOREM 5. Assue that we ae ge ubes b whch satsfy k Ý k 1 k k0 c Ž d. b, k 0,...,21, Ž 4.2. fo ukow ozeo c ad ukow dstct d 1. The d ae uquely deteed as follows. The fucto 0 b0 b-1 b BŽ x. det Ž 4.3. b-1 b2 2 b x x s a polyoal of exact degee. The d, 1,...,, ae ts dstct zeos. The c, 1,...,, ae easly calculated fo the lea equatos Ž 4.2. oce we kow the d, 1,...,. Reak. Note that we ae ot sayg that fo evey choce of b k k0 the syste Ž 4.2. has a soluto. A oe coplete ad detaled exaato of ths poble ay be foud 3. We ow gve a ecpe fo deteg the a 4 Ž , based o vaous Ž ot ueasoable. assuptos. The basc assuptos whch ae used thoughout ae that the a 41 ae pawse dstct, that each g s 2 1 Ž2 1. C soe eghbouhood of 0, ad that g Ž. 0 0, 1,...,. Let c 0 4. Assue that c has bee chose such a way that the values 2 1 Ž2 1. ½ 51 Ž c a. g Ž 0. Ž 4.4. ae ozeo ad dstct. Ths s always possble wth a sutable c, because of the lea depedece of the a. Fo each d 04 ad k 0, 1,..., 2 1, 2 1k k 2 1k k Ž2 1. c d 1 Ý D D f Ž 0. Ž c a. Ž d a. g Ž Ž d a. Ž2 1. Ý Ž c a. g Ž 0.. Ž c a. 1 k

14 116 BUHMANN AND PINKUS If the d a ½ Ž c a. 5 1 Ž 4.5. ae dstct, the t follows fo Theoe 5 that they ay be uquely deteed. Takg lealy depedet d d j, j 1,...,, whch satsfy the pecedg, we obta fo each 1,...,4 the values, d j a ½, j 1,...,. Ž Ž c a. Ths detees the a, 1,...,; Ž c a. Ž Ž. 1. ths case c a. Thus the a ae totally deteed Žsee the eak the case 1.. Ths s ou geeal ecpe. Let us ow cosde ceta of ou equeets futhe detal. ŽSee also the dscusso the poof of Theoe 3 of Buha ad Pkus. 3. We eed to be able to tell whethe, fo a gve d 0 4, the values Ž 4.5. ae dstct. Ths s staghtfowad, because ude ou othe assuptos these values ae dstct f ad oly f the assocated polyoal BŽ x. Ž Ž s of exact degee ad has ths ay dstct zeos. Why do we eed the assupto that the coeffcets Ž 4.4. be dstct, as well as beg ozeo? The dstctess s ot used Theoe 5. It s, howeve, used fo labelg. If two coeffcets have the sae values the we do ot kow how to assg the assocated expessos Ž 4.6. ad thus we detee the a. If the values Ž 4.4. ae ot dstct, we ca alte the c. Fo Theoe 5 we get the values of these coeffcets, ad so t s easly deteed as to whethe these coeffcets ae dstct. ŽThe othe opto s to ty all the dffeet possbltes of assgg the tes Ž 4.6. to the appopate coeffcets, ad the atchg the esults obtaed wth the. Ž2 1. ogal f. Futhe, we ay weake the dead that g Ž. 0 0, 1,...,, by adttg ay shfts away fo the og. Fally, the soothess codto o g ay be weakeed by covolvg g wth a test.e., t detees a up to ultplcato by a ozeo, fte costat fucto ad by cosdeg g, stead of g, as the pevous Ž2 1. secto. Thus the codto g Ž. 0 0 s eplaced by deadg that thee exsts a test fucto such that g og fo all dces. Ž2 1., does ot vash at the

15 LINEAR COMBINATIONS OF RIDGE FUNCTIONS N We ae gve C ad the set N as defed the Itoducto. Because we do ot kow the a, ths s a specfc subset of R. We ca ad wll assue, by usg the ethods of the pevous secto, that we ae able to detfy the a up to ultplcato by costats. That s, a d a fo soe deteed a but udeteed d. Ž Note that we ust hee aga assue that the a ae pawse lealy depedet.. Futheoe we also assue that, usg the ethods of Secto 3, we ca fact detfy g Ž 1. Ž a x. c Ž 1. a x b c Ž 1. d Ž a x. b, fo each. Thus, kowg aleady a, we have educed ou poble to the followg. Gve ad ad c Ž 1. Ž dt b., Ž 5.1. fd the costats c, b, ad d. Ufotuately, we kow of o geeal ethods fo solvg ths poble. Nueous ad hoc ethods peset theselves depedg o the patcula. Fo stace, assue s bouded ad Ž. t whe t, whee. Ž Ths s a typcal case eual etwok applcatos whee usually 0, 1.. Because s bouded, we ca fd c Ž dt b. a, Ž 5.2. fo Ž 5.1. by tegato, whee the ucetaty egadg the polyoal of degee at ost 2 that coes fo the tegato s educed to a costat a Žbecause othe ocostat polyoals ae uled out by boudedess.. Now, lettg t we obta the two values f ad f, oe of whch coespods to c a ad the othe whch coespods to c a, depedg o the sg of d. Thus we obta two optos fo Ž c, a.. If, addto, s stctly ootoe, we ca fd b ad d fo Ž 5.2. by evaluatg at sutable t1 ad t 2. Ths wll gve us oe o two possble choces fo the costats a, b, c, ad d. Howeve, t should be oted that t s vey possble that these costats ae ot uquely deteed. ACKNOWLEDGMENT We thak Joach Stockle ad Bukhad Leze fo potg out the ovesght 3.

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