NECESSARY AND SUFFICIENT CONDITIONS FOR ALMOST REGULARITY OF UNIFORM BIRKHOFF INTERPOLATION SCHEMES. by Nicolae Crainic
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1 NECESSARY AND SUFFICIENT CONDITIONS FOR ALMOST REGULARITY OF UNIFORM BIRKHOFF INTERPOLATION SCHEMES by Ncolae Cranc Abstract: In ths artcle usng a combnaton of the necessary and suffcent condtons for the almost regularty of an nterolaton scheme we wll determne all lane unform Brkhoff schemes when the set of nterolated nodes has n elements and the set of dervatves we are nterolatng wth s A = {(( } For the same A we wll determne all rectangular Brkhoff unform nterolaton schemes (the set of nodes has rectangular shae and we wll take note of the fact that these two results dffer sgnfcantly Two crtera normalty condton and Pólya condton are known to be used when establshng the almost regularty (regularty of a multdmensonal olynomal nterolaton general scheme Only one suffcent condton s known for the almost regularty (regularty of a multdmensonal nterolaton scheme and ths ales for a more restrctve doman namely for the schemes of tye Brkhoff Of course for the nterolaton schemes whose comonents are more and more lmted the number of the necessary and suffcent condtons for regularty (almost regularty s ncreasng and these are more and more exlct The two tyes of nterolaton schemes that we wll resent n ths artcle also rove ths asect We wll resent two necessary and suffcent condtons for bdmensonal unform Brkhoff schemes when the set that descrbes the dervatves s A = {(( } For the begnnng we resent the followng notons: The fnte set L IN s nferor f R( u v L for any ( u v L where R ( u v = {( IN : u v} A set of nodes Z s ( q - rectangular or smle rectangular ( and q are natural numbers f t can be wrtten n the form Z = {( x : q} where x x x are ar-wse dstnctve real numbers (smlarly for y yq 3 The bvarate unform Brkhoff nterolaton scheme s the trlet ( Z S consstng of a set (of nodes 6
2 Z ={ t ( x } n t t IR t= z = an nferor set S IN and a subset A of S The assocated (unform bvarate Brkhoff nterolaton roblem conssts n determnng the olynomals P P S = P IR[ x y] : P( z = a x y z = ( x y IR ( S that satsfy the equatons: + β P ( zt = c ( zt ( ( β A zt Z β β x y where c β ( zt are arbtrary real constants Moreover f Z s rectangular then we have the rectangular unform Brkhoff scheme 4 An nterolaton scheme ( Z S s called normal f Z A = S (where Z A and S s the cardnalty of the corresondng sets etc In case of normalty the determnant of the nterolated system exsts and we denote t by D ( Z S 5 The scheme ( Z S s regular (sngular f D ( Z S does not vansh (does vansh for any choce of the set Z of nodes and s almost regular f D ( Z S s not dentcal null 6 The nterolaton scheme ( Z S satsfes the Pólya condton f Z A L L for any nferor set L S 7 The condtons (crtera necessary for the almost regularty of a general nterolaton scheme ( Z S are the normalty condton and the Pólya condton 8 A suffcent condton for the almost regularty of a unform Brkhoff nterolaton scheme s the followng: f S admts a avement (coverage of unque tye { A A A} ( A n tmes then ( Z A S s almost regular (a restatement of the results from [] Next we resent the two romsed necessary and suffcent condtons Prooston We consder the unform Brkhoff scheme ( Z S wth A = {( ( } S Then ( Z S s almost regular wth resect to the sets Z 6
3 of n nodes f and only f the followng condtons are satsfed: ( S = n and ( S contans at most n elements on the axs Oy Proof: The condton that must be satsfed at ( when the scheme ( S s almost regular results from the Pólya condton (6 aled to the nferor set L = S Oy For the reverse mlcaton we use (8 and we wll show that S admts a avement of unque tye wth n coes of A We use the nducton after n We consder the shft Λ whch moves ( n (3 and the orgn n ( Thus Λ s the shft relatve to A whch moves A mnmal to the rght Then we consder the shft Λ relatve to A Λ ( A whch moves A mnmal to the rght (thus t moves ( n (5 and the orgn n (4 We contnue n ths mode and when the frst lne of S s covered we move to the next lne that we fll n the same manner e from left to rght In ths way we cover S wth coes of A by the obvous method from left to rght lne by lne If ths rocess does not block u ths rovdes us a unque tye avement of S wth coes of A and the roof s ended wthout even usng the hyothess of nducton On the other hand the rocess can block u only because the onts on the axs Oy can be covered movng the orgn ( ( A can bot be moved back toward left In other words when the rocess stos the only onts of S whch have not been covered are on the axs Oy (and such onts exst Thus n ths case f β s the bggest number wth the roerty that ( β S then ( β S In ths case we start everythng from the begnnng We consder the shft Λ whch moves A maxmal uwards and then maxmal to the rght (n S Then ( β Λ( and we can see that S \ Λ ( s a new nferor set whch has ( n elements and whch has one element less than S on the axs Oy e at most n = ( n + elements Usng the nducton hyothess we fnd a avement of the set S \ Λ ( wth n coes of A and ths fact together wth Λ gves us the desred avement of the set S Prooston We consder the set of nodes ( q rectangular Z = {( x : q} If S s an nferor set then ( Z S s regular f and only f S = R( + q Moreover n ths case the solutons P PS of the nterolaton equatons 63
4 P( x = c P ( x = c' x (where q and c c ' IR are arbtrary constants wll be gven by P( x y = c ϕ ( x φ ( y + [ c' ϕ ' ( x c ]( x x ϕ ( x φ ( y where ( x x ( x x ( x x+ ( x x ϕ ( x = ( x x ( x x ( x x ( x x 64 + ( y y ( y y ( y y+ ( y y φ ( y = ( y y ( y y ( y y ( y y Proof: We suose frst that ( Z S s regular and we defne a = max{ a IN : ( a S} b = max{ b IN : ( b S} Frst t s clear that S R( a b Usng the normalty condton S = Z A we deduce that ( + ( q + ( a + ( b + ( On the other hand we wll show that a ( + We suose the contrary e a Then a non null olynomal + P R[X ] of degree a exsts such that P( x = P'( x = for any q} Indeed ths s a lnear system (the unknowns are the coeffcents of P In whch the number of the unknowns ( a + s strctly bgger than the number of the equatons ( ( + Thus such P exsts It s clear that P PS and ths contradcts the regularty of the scheme ( Z S It results analogously that b q (3 Combnng ( ( and (3 we see that the above nequaltes as well as the +
5 ncluson S R a b become equaltes Thus S = R a b a = and ( ( + b = q For the recrocty s suffcent to check that the olynomal from the statement satsfes the nterolaton condtons Because of the symmetry t s suffcent to check the equatons n the node x We have: ( y ( x ϕ daca = daca = ϕ' ( x = f φ ( y β = f β In artcular the olynomals ( x x ϕ ( x and ( x x ϕ' ( x vansh for any x x x } We deduce that for any β q} qed { P( x β = c ϕ ( x φ ( yβ = c β P( x β = c ϕ ' ( x φ ( yβ + [ c' ϕ ' ( x c } ϕ ( x φ ( yβ = x c βϕ ' ( x + ( c' β ϕ ' ( x c β = c' β = 3 Remark The dfference between the generc case and the rectangular case s clear We consder for examle = q = so we have n = 4 nodes In the frst case (the generc one s ( Z = and n the second case (the rectangular one s ( Z = where by s ( Z we denoted the number of the nferor sets S for whch ( Z S s almost regular References Cranc N Scheme de nterolare Brkhoff e domen lane Annales Unvestats Aulenss Unverstarea Decembre 98 Alba-Iula Sera matematcanformatca 4-54 Cranc M Cranc N Brkhoff nterolaton wth rectangular sets of nodes Utrecht Unversty Prernt nr 66 January 3 (trms sre ublcare la Journal of Numercal Analyss 3Lorentz R A Multvarate Brkhoff Interolaton Srnger Verlag Berln 99 65
6 4Stancu D Dmtre Coman Gheorghe Agratn Octavan Trâmbţaş Radu Analză numercă ş teora aroxmăr Vol I Presa Unverstară Cluană Unverstatea Babeş Bolya Ncolae Cranc Decembre 98 Unversty of Alba Iula str N Iorga No 3 Alba Iula Romana 66
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