The Degree of Shape Preserving Weighted Polynomial Approximation

Transcription

1 The Degree of Shape Preservig eighted Polyomial Approximatio Day Leviata School of Mathematical Scieces, Tel Aviv Uiversity, Tel Aviv, Israel Doro S Lubisky School of Mathematics, Georgia Istitute of Techology, Atlata, GA , USA Abstract e aalyze the degree of shape preservig weighted polyomial approximatio for expoetial weights o the whole real lie I particular, we establish a Jackso type estimate Key words: Shape Preservig Polyomials, k-mootoe, Expoetial eights, Jackso Theorem, Freud eights 1 Itroductio Shape preservig polyomial approximatio has bee a active research topic for decades There are may iterestig features, ad a great may complex examples, ad exceptioal cases Perhaps the oldest moder result is due to O Shisha [14] For cotiuous f : [ 1, 1] R, let E [f] = if f P L degp [ 1,1] I additio, let } E 1 [f] = if { f P : P mootoe i [ 1,1] L [ 1,1] degp Research supported by NSF grat DMS11182 ad US-Israel BSF grat addresses: leviata@posttauacil Day Leviata, lubisky@mathgatechedu Doro S Lubisky Preprit submitted to Approximatio Theory October 6, 211

2 Shisha [14] essetially proved that whe f is o-egative ad cotiuous, for 1, E 1 [f] 2E 1 [ f ] 11 This simple estimate is disappoitig, i that oe loses a factor of 1, whe compared to Jackso-Favard estimates However, it is best possible i the class of fuctios to which it applies [13] Similar results hold for covex fuctios, ad more geerally, k-mootoe fuctios Recall that a fuctio f is called k-mootoe, if for ay distict x,x 1,,x k i the iterval of defiitio, [x,x 1,,x k,f] = k i= f x i ω x i, where ω x = k x x j j= The case k = 1 correspods to mootoe fuctios, ad k = 2 to covex fuctios The atural geeralisatio of 11 to k-mootoe fuctios is [ E k [f] 2E k f k], for k Agai, this is a disappoitig estimate, as oe loses a factor of k whe compared with ucostraied approximatio However, it turs out that this estimate may ot, i geeral, be improved, see [4] See also [3], [5] A recet iterestig paper of O Maizlish [1] seems to be the first extedig shape preservig approximatio to weighted polyomial approximatio o the whole real lie Recall that for α >, α x = exp x α, x R, is a expoetial weight, ofte called a Freud weight The polyomials are dese i the weighted space of cotiuous fuctios geerated by α iff α 1 Thus, if α 1, ad f : R R is cotiuous, with lim f α x =, x while E [f] α = if f P α, L R degp 2

3 we have lim E [f] α = This is a special case of the classical solutio of Berstei s weighted polyomial approximatio problem, ivolvig more geeral weights, by Achieser, Mergelya, ad Pollard [2], [7], [9] For α, α > 1, the Jackso theorem takes the form E [f] α C 1+1/α f α L R, provided f is cotiuous i R Here C is idepedet of f ad Iterestigly eough, there is o estimate of this type for 1, eve though the polyomials are dese There are Jackso theorems ivolvig weighted moduli of cotiuity, see [1], [8], [9] Let k 1, ad let E k [f] α = if degp { f P α L R : P is k-mootoe i R } 12 Maizlish proved that if f is k times cotiuously differetiable o R ad f k is o-egative, the Somewhat more is true: let ad µ x = lim Ek [f] α = r = 4 f k 2 1/α x, x R, 2 1/α, 1 α Maizlish also proved that the there exists a polyomial P of degree at most 2 + k that is k-mootoe, ad such that ad f P α L [ r,r ] M 1E [µ] α µ α L R f P α L R\[ r,r ] M 1 1+1/α E [µ] α µ α L R Here M 1 is idepedet of f ad Note that µ ca be somewhat less smooth tha f k I this paper, we prove results of this type that are closer i spirit to the uweighted Shisha type theorems Throughout, [x] deotes the greatest iteger x 3

4 Theorem 11 Let α > 1, ad k 1 Let A > 1 There exist B,C > with the followig property: for every f : R R that is k times cotiuously differetiable ad k-mootoe, satisfyig we have for 1, E k [A]+k [f] α C lim x f k α x =, 13 [ [E f k] ] + f k α α L R e B 14 Coversely, give ay B >, there exists sufficietly large A for which this last iequality holds for all 1 e may replace the geometric factors e B by factors that decay more slowly, ad the allow [A] to be replaced by somethig smaller e may also cosider more geeral Freud weights, or eve expoetial weights o a fiite iterval For simplicity, we shall cosider oly eve weights = e Q, defied o a symmetric iterval I = d,d, where < d Accordigly, we defie E [f] = if f P L, 15 degp I ad } E k [f] { f = if P : P is k-mootoe i I L I 16 degp e start with a geeralizatio of Theorem 11 for Freud weights: Theorem 12 Let = e Q, where Q : R R is eve, ad Q is cotiuous i R, while Q exists i, Assume i additio, that i Q > i, ad Q = ; ii Q > i, ; iii For some Γ,Λ > 1, Γ tq t Qt Λ, t, ; 17 4

5 iv Q t Q t C Q t 1, t, 18 Qt Let A > 1 ad 2 l A + 1, 1 Let k 1 There exist B,C > with the followig property: for every f : R R that is k times cotiuously differetiable ad k-mootoe, satisfyig lim x f k x =, we have for 1, [ E k +l [f] +k C E [f k] f + k l e B 1/2 3/2 L R ] 19 Observe that Theorem 11 is the special case i which Qx = x α ad l = [A 1 ] Give a positive iteger j, if we choose [ l = r 1/3 log 2/3], with large eough r, we obtai [ E k [f] +[r 1/3 log 2/3 ]+k C E [f k] ] f + k j 11 L R Fially, we tur to geeral eve expoetial weights For these, we eed the cocept of the th Mhaskar-Rakhmaov-Saff umber a, associated with = e Q This is the positive root of the equatio = 2 π 1 a tq a t dt 1 t It is uiquely defied if tq t is positive ad strictly icreasig i,d with limits ad at ad d respectively Oe of its features is the Mhaskar-Saff idetity [6], [12] P L I = P L [ a,a ], 112 for all polyomials P of degree Moreover, a is essetially the smallest umber for which this holds e shall also eed the fuctio T x = xq x, x,d 113 Qx 5

6 e shall say that T is quasi-icreasig i,d if there exists C > such that T x CT y for all < x < y < d Our most geeral theorem is: Theorem 13 Let I = d,d, where < d Let = e Q, where Q : I R is eve, ad Q is cotiuous i I, while Q exists i,d Assume i additio, that i Q = ad lim t d Qt = ; ii Q > i,d; iii Q > i,d; iv For some Λ > 1, v while T is quasi-icreasig there T t Λ, t,d, 114 Q t Q t C Q t 1, t,d 115 Qt Let A > 1 ad 2 l A + 1, 1 Let k 1 There exist B,C > with the followig property: for every f : I R that is k times cotiuously differetiable ad k-mootoe, ad for which we have for 1, E k lim x d f k x =, +l [f] +k 116 [ C E f k] f + k L I e BTa 1/2 l 3/2 Here a is the th Mhaskar-Rakhmaov-Saff umber for Q Examples of such weights o the iterval 1,1 iclude x = exp 1 1 x 2 α 117 or 1 x = exp exp k 1 exp k x 2 α, 118 6

7 where α > 1, ad exp k = exp exp exp k times }{{} is the kth iterated expoetial O the whole real lie, i additio to the Freud weights, oe may choose x = exp exp k exp k x α, 119 where k 1 ad α > 1 For of 117, [6, p 31, Example 3] T a 1 α+ 1 2 This meas that the ratio of the two sides is bouded above ad below by positive costats idepedet of For of 118, [6, p 33, Example 4] where k 1 T a log k 1+ 1 α log j, j=1 log k = log log log k times }{{} is the kth iterated logarithm For of 119, [6, p 3, Example 2] T a k log j j=1 e ote that all our weights lie i the class F C 2 cosidered i [6, p 7] e may actually cosider the o-eve weights there, as well as the more geeral class F Lip 1 2, but avoid this for otatioal simplicity The mai ew idea i this paper over that of Maizlish is the use of oegative polyomials, obtaied from discretizig potetials, ad that were costructed i [6, Theorem 74, p 171] e shall use may of Maizlish s ideas, as well as devices from the uweighted theory of shape preservig approximatio The proofs are cotaied i the ext sectio 2 Proof of Theorem 13 e begi with some backgroud o potetial theory with exteral fields [12] Let us assume the hypotheses of Theorem 13 The Mhaskar-Rakhmaov- Saff umber a t may be defied by 111, for ay t >, ot just for iteger 7

8 : thus for t >, t = 2 π 1 a t uq a t u du 1 u 2 The fuctio t a t is a cotiuous strictly icreasig fuctio of t, so has a iverse fuctio b, defied by ba t = t, t > For each t >, there is a equilibrium desity σ t, that satisfies at a t σ t = t The equilibrium potetial satisfies V σt z = at a t log 1 z u σ t udu V σt + Q = c t i [ a t,a t ], where c t is a characteristic costat e shall eed mostly the fuctio It satisfies U t x = V σt x + Qx c t, x I U t x =, x [ a t,a t ]; U t x <, x I\[ a t,a t ] e shall eed a alterative represetatio for U t For a iterval [a,b], the Gree s fuctio for C\[a,b] with pole at, is g [a,b] z = log 2 z a + b + z a z b b a 2 It vaishes o [a,b], is o-egative i the plae, ad behaves like log z + O 1, as z There is the represetatio [6, Corollary 29, p 5] bx U t x = g [ aτ,a τ] x dτ, x [,d 21 t It is really this that we shall eed, ot so much the other quatities above 8

9 Lemma 21 a For 1, ad polyomials P of degree, P x e Ux P L R, x > a 22 b Let D > 1 For m D, ad x a m, U U m x C T a 1/2 Here C is idepedet of m,,x 1 m 3/2 23 Proof a This is a classical iequality of Mhaskar ad Saff that ca be foud, for example, i [6, Lemma 44, p 99] or [12, p 153, Thm 21] b From 21, for x > a m, U x U m x = m g [ aτ,a τ] xdτ, x [,d Here for each τ [,m], g [ aτ,a τ] x is a icreasig fuctio of x a m, as the Gree s fuctio g [a,b] icreases as we move to the right of [a,b] It follows that for x a m, U x U m x U a m U m a m = m g [ aτ,a τ] a m dτ 24 Next, by Lemma 45a i [6, p 11], followed by 351 of Lemma 311a i [6, p 81], for τ [,m], 1/2 am C m 1/2 g [ aτ,a τ] a m C 1 a τ T a 1/2 τ 1 Note that i the eve case, i [6], δ = a, ad a 2 Ca The 23 follows easily from 24 e also eed polyomials costructed by discretizig the potetial V σt The method is due to Totik, but the form we eed was proved i [6, Theorem 74, p 171]: Lemma 22 There exists C > 1 with the followig property: for eve 2, there exists a polyomial R of degree such that 1 R C i [ a,a ]; 25 ad moreover, R e U i I 26 9

10 Now we ca use this to geerate o-egative weighted polyomial approximatios to o-egative fuctios: Lemma 23 Let g : I R be a cotiuous o-egative fuctio such that ad g L I = 1, 27 lim gx = x d Assume that D > ad {l } is a sequece of positive itegers with 2 l D + 1The there exist B,C >, ad for 1, a polyomial P # of degree + l such that P # i I 28 ad g P # C E [g] + e BTa 1/2 l 3/2 29 L I Here C C,g Proof Choose a polyomial P such that As g, we have g P L I = E [g] P E [g] i [ a,a ] 21 Let m = m = 2 [ +l ] 2, a eve iteger Note that m + 1 Let Rm be the polyomial of Lemma 22 Let S x = P x + E [g] + e BTa 1/2 l 3/2 R m x, a polyomial of degree m From 25 ad 21, we have i [ a m,a m ], From Lemma 21a, for x a,d, S x P x P L I eux g + E L I [g] e Ux 2e Ux, 1

11 recall our ormalizatio 27 The from Lemma 22, for x a m,d, S x 2e Ux + E [g] + e BTa 1/2 l 3/2 e Umx This will be o-egative if U U m x log E [g] + e BTa 1/2 l 3/2 From Lemma 21b, it suffices i tur that for some large eough C, or C T a 1/2 1 m 3/2 log 2 e BTa 1/2 l 3/2 C T a 1/2l3/2 2B T a 1/2 l 3/2 2, So we ca choose B = C/2, ad esure o-egativity of S i [,d The iterval d, may be hadled similarly Fially, g S L I g P L I + E [g] + e BTa 1/2 l 3/2 E [g] + C E [g] + e BTa 1/2 l 3/2, R m L I where C is as i Lemma 22 Proof of Theorem 13 By the last lemma, we ca choose a polyomial P of degree + l such that P i I, ad f k P L I [ C E f k] f + k =: M, L I e BTa 1/2 l 3/2 say e have take accout of the eed to divide f k by f k L I, i order to satisfy the ormalizatio 27 Now, let P x = x tk 1 t1 k 1 P t dt dt 1 dt k j= f j x j j!

12 The P is k mootoe For x >, we have, followig Maizlish s ideas, f P x x = x M x = M x tk 1 x tk 1 x t k 1 t1 t1 tk 1 f k P t dt dt 1 dt k t dt dt 1 dt k 1 t k 1 t k 2 t1 Fix r,d Here by mootoicity of Q, for t 1 >, t1 while by its covexity, for t 1 r, t1 r It follows that for all t,d, t 1 t dt t 1 t 1 t dt dt 1 dt k t 1 t1 t dt e Q rt 1 t dt 1 r Q r t1 Applyig this repeatedly to 211 gives t 1 t dt r + 1 Q r f P x M r + 1 k Q r The case x < is similar, so we obtai E k +l [f] +k r + 1 Q r k [ C E f k] f + k l e BTa 1/2 3/2 L I Proof of Theorem 12 This is a special case of Theorem 13, where T is bouded above ad below by positive costats 12

13 Proof of Theorem 11 This is the special case = α of Theorem 12 e ca choose l = [A 1 ] whe that is at least 2 For the remaiig fiitely may, we ca set l = 2 ad use the elemetary iequality E k k [f] α C f k L R The fact that we may choose B as large as we please, with correspodigly large A, is easily see from the proof of Lemma 23 Ackowledgemet e gratefully ackowledge commets from the referees that improved the presetatio of the paper, ad corrected some mior errors Refereces [1] Z Ditzia ad V Totik, Moduli of Smoothess, Spriger, New York, 1987 [2] P Koosis, The Logarithmic Itegral I, Cambridge Uiversity Press, Cambridge, 1988 [3] D Leviata, Shape Preservig Approximatio by Polyomials, J Comp Appl Math, 121 2, [4] D Leviata ad I A Shevchuk, Couterexamples i Covex ad Higher Order costraied approximatio, East J Approx, , [5] D Leviata ad I A Shevchuk, Uiform ad Poitwise Shape Preservig Approximatio SPA by Algebraic Polyomials, Surveys i Approximatio Theory forthcomig [6] E Levi ad D S Lubisky, Orthogoal Polyomials for Expoetial eights, Spriger, New York, 21 [7] G G Loretz, M vo Golitschek, Y Makovoz, Costructive Approximatio: Advaced Problems, Spriger, Berli, 1996 [8] D S Lubisky, hich eights o R admit Jackso Theorems?, Israel Joural of Mathematics, ,

14 [9] D S Lubisky, A Survey of eighted Polyomial Approximatio with Expoetial eights, Surveys i Approximatio Theory 27 [1] O Maizlish, Shape Preservig Approximatio o the Real Lie with expoetial eights, J Approx Theory, , [11] H N Mhaskar, Itroductio to the Theory Of eighted Polyomial Approximatio, orld Scietific, Sigapore, 1996 [12] E B Saff ad V Totik, Logarithmic Potetials with Exteral Fields, Spriger, New York, 1997 [13] I Shevchuk, Oe example i mootoe approximatio J Approx Theory , [14] O Shisha, Mootoe Approximatio, Pacific J Math, ,