ON MÜNTZ RATIONAL APPROXIMATION IN MULTIVARIABLES

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1 C O L L O Q U I U M M A T H E M A T I C U M VOL. LXVIII 1995 FASC. 1 O MÜTZ RATIOAL APPROXIMATIO I MULTIVARIABLES BY S. P. Z H O U EDMOTO ALBERTA The present paper shows that or any s sequences o real numbers each with ininitely many distinct elements {λ j n} j = 1... s the rational combinations o x λ1 m 1 1 x λ2 m x λs ms s are always dense in C I s. 1. Introduction. Let C [01] be the class o all real continuous unctions in [0 1]. For C [01] ω t = Given a subspace S o C [01] let max x + h x 0<h<tx [01 h] = max x [01] x. RS = {P x/qx : P x S Qx S Qx > 0 x 0 1]} where we assume that lim x 0+ P x/qx = P 0/Q0 is inite in the case Q0 = 0. For a sequence o real numbers Λ = {λ n } n=0 write RΛ = Rspan{x λ n }. From Müntz s theorem c. [2] it is well-nown that the combinations o x λ n or 1 0 = λ 0 < λ 1 < λ 2 <... are dense in C [01] i only i 1 =. λ n n=1 As to the rational case in 1976 Somorjai [6] showed a beautiul result that under 1 RΛ is always dense in C [01]. In 1978 Ba ewman [1] proved that i λ n is a sequence o distinct positive numbers then RΛ is dense in C [01] as well. Recently our wor [7] showed that the above result 1991 Mathematics Subject Classiication: 41A20 41A30 41A63. [39]

2 40 S. P. Z H O U also holds or any sequence o real numbers with ininitely many distinct elements. On the other h S. Ogawa K. Kitahara [5] gave a generalization o Müntz s theorem to multivariable cases. They proved 1 that or two given positive monotone sequences {α i } {β j } the set {1} {x α i } {y β j } is complete in C I 2 i only i i=1 1/α i j=1 1/β j diverge where I s = {X = x 1... x s : 0 x j 1 1 j s} C I s is the class o all continuous unctions on I s. For many reasons it is quite reasonable to conjecture that the conclusion corresponding to that o [7] will hold or Müntz rational approximation in the multivariable case that is or any s sequences o real numbers {λ j n} j = 1... s each with ininitely many distinct elements the rational combinations o {x λ1 m 1 1 x λ2 m x λs ms s } are always dense in C I s. Since rational combinations are not linear it is not a trivial wor. The present paper will prove that this is true. 2. Result proo Theorem. Let Λ j = {λ j n} j = 1... s be s sequences o real numbers each with ininitely many distinct elements. Then RΛ 1... Λ s is dense in C I s. We need the ollowing lemmas rom the univariable case the irst two o which are due to Somorjai [6] the author [7]. We will however give the setch o proos here or the sae o completeness. Lemma 1 Somorjai [6]. Let {λ n } be a sequence o real numbers such that λ n + as n. Given 1 or any C [01] there are an integer n an operator Z x Zx =: r1 x R{λ j } n j=0 with such that 0 Z x span{x λ j } n j=0 Zx = r 1 = Oω 1. Z x P r o o. We select a sequence {λ nj } n j=1 rom Λ by induction. Let λ n0 be any element rom Λ Z 0 x = x λ n 0. Choose λ nj+1 with the ollowing properties: 1 For convenience we only state their result or two variables.

3 RATIOAL APPROXIMATIO 41 Deine Z j+1 x = λnj+1 j + 1 x 1 Z j x or x < j Z j+1 x > Z j x or x > j + 2. r 1 x = or C [01]. Then by calculation Z x v=0 Z vx x r 1 x = Oω 1. Lemma 2 Zhou [7]. Let {λ n } be a sequence o real numbers such that λ n as n. Given 1 or any C [01] there are an integer n an operator C x Cx =: r2 x R{λ j } n j=0 with such that 0 C x span{x λ j } n j=0 Cx = r 2 = Oω 1. C x P r o o. Similar to Lemma 1 let λ n1 be any element rom Λ C1 x = x λ n 1. Choose λ nj+1 satisying λnj+1 Cj+1x = j x C j x or x < j 1 Cj+1x < 1 Cj x or x > j + 2. For C [01] deine + 1 r 2 C x = x v=1 C v x. =1 Then the required result ollows. Lemma 3. Let {λ n } be a sequence o real numbers with ininitely many distinct elements such that λ n l as n with < l <. Given 1 ε > 0 there are an integer n an operator e 1 / D x Dx =: r3 x R{λ j } n j=0

4 42 S. P. Z H O U with D x Dx span{x λ j } n j=0 such that r 3 2ωg 1/2 + ε where gu = e 1 1/u. Precisely we have 2 3 G x = D x Dx = G x + H x lnx/e 4 H x ε lnx/e P r o o. There are two possibilities: i There is a subsequence {λ n } o {λ n } which strictly increases to λ < + in symbols λ n λ < + as ; ii there is a subsequence {λ n } which strictly decreases to λ > in symbols λ n λ > as. We will prove Lemma 3 in these two cases separately. C a s e i. For convenience we still write λ n as λ n. So under the hypothesis λ n λ < + as n. Let α 0 < α 1 <... let P x denote the th divided dierence o x/e α at α = α 2 1 α α 2 1 or = that is in general P 0 x = P 0 x α 2 1 = x/e α 2 1 P 1 x = P 1 x α 2 1 α 2 2 = x/eα 2 1 x/e α 2 2 α 2 1 α 2 2 P x = P x α α 2 1 = P 1x α α 2 P 1 x α α 2 1 α 2 1 α 2 1 By the mean value theorem P x = P x α... α P x = x/eη ln x/e! α 2 1 η α 2 1 = P x = x/eη ln x/e α 0 η α.!

5 RATIOAL APPROXIMATIO 43 ow let C [01]. Then gu = e 1 1/u C [01]. Write B x = = = x 1 x j=0 1 j j x j+ 1 j x j. j For given 1 the well-nown Bernstein theorem implies that that is j= gu B g u < 3 2 ωg 1/2 7 x B g 1/ lnx/e < 3 2 ωg 1/2. Choose suiciently large m such that or m 0 < λ λ < ε/ Set α = λ m+ = Deine r 3 x = e 1 / j= 12j j! j P j x.!p x Then r 3 x is a rational combination o {xλ j } m+2 1 j=m by 5 6 j 1 r 3 x = e 1 / 1 j x/e η j j lnx/e j= with η 0 = 0 0 < η j λ m+ λ m λ λ m j = ow write j 1 1 j x/e η j j lnx/e j= = = 1 1 j j lnx/e j= + j= j j 1 1 j x/e η j 1 j lnx/e 1 lnx/e Σ 1. lnx/e

6 44 S. P. Z H O U Since or η > 0 1 x/e η lnx/e we have 1 x/e η ln x/e η or 1. Consequently Σ ε j= η ε j ε thus 2 4 are proved. ow rom 4 7 together with H x = Σ1 x r 3 x x B g 1/ lnx/e + H x 3 2 ωg 1/2 + ε that is Lemma 3 holds true in Case i. C a s e ii. We may assume that λ n λ > as n. Tae r 3 x = P x = P x λ m... λ m+ 0 1 P x = P x λ m λ m+2 1 e 1 / j= 12j j! j P j x.!p x Similar to Case i or given ε > 0 1 we can prove 2 4 or suiciently large m x r 3 x 3 2 ωg 1/2 + ε. The proo o Lemma 3 is now complete. P r o o o t h e T h e o r e m. Given a sequence with ininitely many distinct elements {λ n } there are three possibilities: i {λ n } has at least one inite cluster point; ii one cluster point o {λ n } is + ; iii one cluster point o {λ n } is. Without loss o generality we may assume λ j n + as n 1 j r λ j n as n r + 1 j t λ j n l as n t + 1 j s < l < +.

7 RATIOAL APPROXIMATIO 45 For given 1 ε > 0 rom Lemmas 1 3 we may select a common n such that X r 1 x j = Oω xj 1 1 j r X r 2 x j = Oω xj 1 r + 1 j t X B g 1/ lnx j /e = Oω xj g 1/2 t + 1 j s hold at the same time where ω xj δ Deine = max 0 h δ x 1... x j + h x j+1... x s x 1... x j x j+1... x s. r X = Evidently From 2 0 j j s j1... j t... e1 /j t+1... e 1 /j s Z j 1 x 1 Zx 1... Z j r x r C jr+1 x r+1... C j t x t Zx r Cx r+1 Cx t D j t+1 x t+1 Dx t+1... D j s x s Dx s. r X Rspan{x λ1 i } n i=0 span{x λ2 i } n i=0... span{x λs i } n i=0. X r X =... 0 j 1 0 j s j1 X... j t e1 /j t+1... e 1 /j s Z j 1 x 1 Zx 1... Z j r x r C jr+1 x r+1... C j t x t Zx r Cx r+1 Cx t G jt+1 x t+1... G js x s + Σ 3 := Σ 2 + Σ 3 where by Lemma 3 8 Σ 3 4 s ε. Because X C I s there is a δ > 0 such that or X Y = s j=1 x j y j 2 < δ X Y < ε

8 46 S. P. Z H O U while or X Y δ thereore in any case ow Σ 2 ε + 2δ 2 X Y 2δ 2 X Y ε + 2δ 2 0 j 1... t 0 j s i=1 s x j y j 2 j=1 s x j y j 2. j=1 x i j 2 i s x i e 1 /j i 2 Z j 1 x 1 Zx i=t Z j r x r Zx r C j r+1 x r+1... C j t x t Cx r+1 Cx t G j t+1 x t+1... G js x s r = ε + 2δ 2 x i j 2 i Z ji x i Zx i t i=r+1 j i =0 i=1 j i =0 x i j i oting that x 2 Z x Zx = 2x x 2 C ji x i Cx i s i=t+1 j i =0 Z x Zx x i e 1 /j i 2 G ji x i. x 2 rom Lemma 1 we deduce that x 2 Z x Zx = O 1/2. 2 Z x Zx The same results also hold or x /2 C x/cx or x e1 / 2 G x by applying Lemmas 2 3. Altogether we have 9 Σ 2 ε + 2δ 2 O 1/2 thus combining 8 9 we get which is the required result. X r X = Oε + O 1/2

9 RATIOAL APPROXIMATIO 47 REFERECES [1] J. Ba D. J. ewman Rational combinations o x λ λ 0 are always dense in C [01] J. Approx. Theory [2] E. W. Cheney Introduction to Approximation Theory McGraw-Hill [3] G. G. Lorentz Bernstein Polynomials Toronto [4] D. J. ewman Approximation with Rational Functions Amer. Math. Soc. Providence R.I [5] S. Ogawa K. Kitahara An extension o Müntz s theorem in multivariables Bull. Austral. Math. Soc [6] G. Somorjai A Müntz-type problem or rational approximation Acta Math. Acad. Sci. Hungar [7] S. P. Z h o u On Müntz rational approximation Constr. Approx DEPARTMET OF MATHEMATICS UIVERSITY OF ALBERTA EDMOTO ALBERTA CAADA T6G 2G1 Reçu par la Rédaction le ; en version modiiée le