The error curve for rational best approximation of f ∈ C[−1, 1] is characterized by the well-known equioscillation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the

equilibrium distribution. It is known that these points need not to be dense in [−1, 1]. The reason is the influence

of the distribution of the poles of the rational approximants. In this paper, we generalize the results known so

far to situations where the requirements for the degrees of numerators and denominators are less restrictive.Крива похибок для раціонального найкращого наближення f∈C[−1,1] характеризується відомою властивістю еквіосциляцій. На відміну від поліноміального випадку розподіл цих змін знаку не визначається рівноважним розподілом. Відомо, що ці точки не обов'язково мають бути щільними в [−1,1], що зумовлено впливом розподілу полюсів раціональних наближень. У даній роботі узагальнено відомі результати на випадки, де на степені чисельників та знаменників накладаються менш жорсткі умови

Blatt, H.-P.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 517.5H.-P. Blatt (Kathol. Univ. Eichsta¨tt-Ingolstadt, Germany)THE INFLUENCE OF POLES ON EQUIOSCILLATIONIN RATIONAL APPROXIMATIONVPLYV POLGSIV NA EKVIOSCYLQCI}U RACIONAL\NOMU NABLYÛENNIThe error curve for rational best approximation of f ∈ C[−1, 1] is characterized by the well-known equioscil-lation property. Contrary to the polynomial case, the distribution of these alternations is not governed by theequilibrium distribution. It is known that these points need not to be dense in [−1, 1]. The reason is the influenceof the distribution of the poles of the rational approximants. In this paper, we generalize the results known sofar to situations where the requirements for the degrees of numerators and denominators are less restrictive.Kryva poxybok dlq racional\noho najkrawoho nablyΩennq f ∈ C[−1, 1] xarakteryzu[t\sq vidomogvlastyvistg ekvioscylqcij. Na vidminu vid polinomial\noho vypadku rozpodil cyx zmin znaku ne vy-znaça[t\sq rivnovaΩnym rozpodilom. Vidomo, wo ci toçky ne obov’qzkovo magt\ buty wil\nymy v[−1, 1], wo zumovleno vplyvom rozpodilu polgsiv racional\nyx nablyΩen\. U danij roboti uza-hal\neno vidomi rezul\taty na vypadky, de na stepeni çysel\nykiv ta znamennykiv nakladagt\sq menßΩorstki umovy.1. Introduction. Let f ∈ C[−1, 1] be a real-valued function and let Rn,m denote thefamily of real rational functions with numerator in Pn and denominator in Pm, wherePk is the set of algebraic polynomials of degree at most k, k ∈ N0. For each pair ofnonnegative integers (n,m), there exists a unique function r∗n,m ∈ Rn,m that is the bestuniform approximation to f on I = [−1, 1] in the sense that‖f − r∗n,m‖ < ‖f − r‖ for all r ∈ Rn,m, r = r∗n,m,where ‖· ‖ denotes the sup norm on I . Writing r = pn/qm,where pn ∈ Pn and qm ∈ Pmhave no common factor and qm is monic, the defect of r is defined bydn,m(r) := min(n− deg pn,m− deg qm). (1)Let us definel(r) = n+m+ 1− dn,m(r); (2)then l(r) is the dimension of the tangential space with respect to the coefficients of the nu-merator and denominator as parameter space. We write r∗n,m := p∗n/q∗m with no commonfactors and define for abbreviationln,m := l(r∗n,m).Then it is well known that the best approximation of f is characterized by the followingequioscillation property:There exist ln,m + 1 points x(n,m)k ,−1 ≤ x(n,m)0 < . . . < x(n,m)ln,m ≤ 1,c© H.-P. BLATT, 2006ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1 34 H.-P. BLATTsuch thatλn,m(−1)k(f − r∗n,m)(x(n,m)k ) = ‖f − r∗n,m‖, 0 ≤ k ≤ ln,m, (3)where λn,m = +1 or λn,m = −1 is fixed. Such a point set {x(n,m)k } is called alternationset. In general, it is not unique. Therefore, in the following, we denote byAn,m = An,m(f) = {x(n,m)k }ln,mk=0an arbitrary but fixed alternation set for the best approximation r∗n,m of f out of Rn,m.Let νn,m denote the normalized counting measure of An,m, i.e.,νn,m([α, β]) :=#{x(n,m)k : α ≤ x(n,m)k ≤ β}ln,m + 1. (4)Kadec [1] has shown that there exists a subsequence Λ of N such thatνn,0∗−→ µ as n ∈ Λ, n −→∞, (5)where µ is the equilibrium measure of [−1, 1], i.e., the density of µ on I isdµ(x) =dx√1− x2 .For rational approximation, Borwein et al. [2] have proved that denseness on [−1, 1] holdsfor a subsequence of alternation sets An,m whenever m = m(n) andnm(n)−→ κ > 1as n −→ ∞. Moreover, they have shown in the case limn→∞m(n)n= 0 that there existsΛ ⊂ N such thatνn,m(n)∗−→ as n ∈ Λ, n −→∞.More quantitative results were obtained by Kroo´ and Peherstorfer in [3]. Namely, letus denote by Nn,m(α, β) the number of points of An,m in [α, β]. Then the main resultcan be stated as follows: let m(n) < n, thenNn,m(n)(α, β)n−m(n) ≥ µ([α, β])− c√log nn−m(n) , (6)where c is an absolute constant independent of f and n.Braess et al. [4] considered the case m(n) = n + κ, κ ∈ Z, fixed. Their re-sults were based on the number γn(ε) of poles of best approximants lying outside anε-neighbourhood of [−1, 1]. Roughly speaking, if γn(ε) is sufficiently big, then there is aconnection of the distribution of An,m with the equilibrium distribution µ.The intimate relation between An,m(f) and the poles of r∗n,m was investigated in [5].To be precise, let f be not a rational function and let n and m(n) satisfym(n) ≤ n; m(n) ≤ m(n+ 1) ≤ m(n) + 1. (7)Moreover, letQn(x) = q∗m(n)(x)q∗m(n+1)(x) =κn∏i=1(x− yi) (8)ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 5be the product of the denominators of r∗n,m(n) and r∗n+1,m(n+1), thenτn(∆) :=#{yi : yi ∈ A}κn(A ⊂ C)denotes the normalized counting measure of all finite poles of r∗n,m(n) and r∗n+1,m(n+1)counted with their multiplicities. Then it was proved in [5] that there exists a subsequenceΛ ⊂ N such thatνn,m(n) − αnτ̂n − (1− αn)µ ∗−→ 0 as n→∞, n ∈ Λ, (9)in the weak∗-topology, whereαn =κnln,m(n) + 1and τ̂n denotes the balayage measure of τn onto [−1, 1]. The purpose of the present paperis to obtain a convergence result of type (9), where the restriction m(n) ≤ n is weakenedto m(n) ≤ n + 1. We point out that this weaker condition implies that the original proofin [5] has to be substantially modified. Moreover, the weaker condition m(n) ≤ n + 1allows to apply and to understand examples of [2].Borwein et al. [2] have proved in the case m(n) = n + 1 that there exists a functionwith no alternation points in a certain interval.It is a challenge to generalize results of type (8) to m(n) > n+ 1.2. Main results. We assume that m(n) depends on the parameter n ∈ N. LetEn,m(n) := infr∈Rn,m(n)‖f − r‖ = ‖f − r∗n,m(n)‖and define for abbreviationr∗n := r∗n,m(n), p∗n = p∗n,m(n), q∗n = q∗n,m(n),En = En,m(n), ln = ln,m(n), dn = dn,m(n),x(n)k := x(n,m(n))k , k = 0, 1, . . . , ln.Again, we use the normalized counting measure νn of the alternation set {x(n)k }lnk=0and the normalized counting measure τn of the union of the (finite) poles of r∗n and r∗n+1.All poles are counted with their multiplicities. For any finite Borel measure ν, the loga-rithmic potential of ν is defined byUν(z) :=∫log1|z − t| dν(t).A crucial role is played by the balayage measure τ̂n of τn onto [−1, 1]. τ̂n is the uniquemeasure supported on [−1, 1], for which ‖τ̂n‖ = ‖τn‖ andU τ̂n(z) = Uτn(z) + c, z ∈ [−1, 1],wherec =∫G(t,∞)dτn(t)and G(z, a) denotes Green’s function of Ω = C \ [−1, 1] with pole at a ∈ Ω (cf. [6]).Furthermore, τ̂n has the following properties:ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 16 H.-P. BLATTa) U τ̂n(z) ≤ Uτn(z) + c, z ∈ C;b) if h is continuous on C and harmonic in Ω, then∫hdτn =∫hdτ̂n.Our main result can be formulated in the following theorem.Theorem. Let f be not a rational function and let the parameters m(n), n ∈ N,satisfym(n) ≤ n+ 1, m(n) ≤ m(n+ 1) ≤ m(n) + 1. (10)Then there exists a subsequence Λ ⊂ N such thatνn − αnτ̂n − (1− αn)µ ∗−→ 0 as n→∞, n ∈ Λ,whereαn =deg q∗n + deg q∗n+1ln + 1.We note that condition (10) is less restrictive than (7).It is possible to formulate the above result in a more concise manner such that only thealternation counting measure νn and pole counting measures of r∗n and rn+1 are involved.LetRn = r∗n+1 − r∗n =pq,where p and q have no common divisor. Then the degree of pqis defined bydegpq:= max(deg p, deg q).Then the number of zeros, resp. poles, of Rn in the closed complex plane C is degRn,where all zeros and poles are counted with their multiplicity.We define the normalized pole counting measure σpole,n of Rn in C byσpole,n(A) =#{poles of Rn inA}degRn(A ⊂ C)and the normalized zero counting measure σzero,n of Rn in C byσzero,n(A) =#{zeros of Rn inA}degRn(A ⊂ C).Corollary. Under the conditions of Theorem 1, there exists Λ ⊂ N such thatσ̂zero,n − σ̂pole,n ∗−→ 0 as n→∞, n ∈ Λ.Especially,νn − σ̂pole,n ∗−→ 0 as n→∞, n ∈ Λ,if limn→∞m(n)n≤ 1.Let us discuss the second part of the corollary in Kadec’s case, i.e., (n,m(n)) =(n, 0). Then Rn = p∗n+1 − p∗n and pn, p∗n+1 are the best approximating polynomials to fwith respect to Pn, resp. Pn+1 and Rn has a pole of multiplicity n+1 at∞ if p∗n = p∗n+1.ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 7Now, for the Dirac measure δ∞ at the point at∞ we know that the balayage mesure δ̂∞ isjust the equilibrium measure µ (cf. [6]). Moreover, all zeros of p∗n+1 − p∗n are separatingthe alternation points. Hence, σ̂zero,n = σzero,n andlimn→∞n∈Λσzero,n = limn→∞n∈Λνn,0 = µfrom the corollary. That is Kadec’s result (5).3. Proofs. Since limn→∞En = 0, by a well-known argument, there exists a subsequenceΛ ⊂ N such thatEn + En+1En − En+1 ≤ n2 for n ∈ Λ (11)(cf. [7, p. 243], Lemma 7.3.3). In particular, for n ∈ Λ we have r∗n = r∗n+1 and, by (3),(−1)k(r∗n+1 − r∗n)(x(n)k ) ≥ En − En+1 (12)for 0 ≤ k ≤ ln, where we have assumed without loss of generality that the numberλn,m(n) = 1 in (3). WritingRn = r∗n+1 − r∗n =p∗n+1q∗n − p∗nq∗n+1q∗nq∗n+1=Pnq∗nq∗n+1=PnQn,we obtain(−1)kRn(x(n)k ) ≥ En − En+1, 0 ≤ k ≤ ln. (13)In the following, we assume that an is the highest coefficient of Pn(x), i.e.,Pn(x) = anxln + . . . .By (13), Pn orRn = PnQnhas at least ln zeros in (−1, 1). Since r∗n = r∗n+1, condition (10)implies that all zeros of Pn are in (−1, 1). As in [5], our next intention is to reconstructthe polynomial Qn by interpolation at the points x(n)k , 0 ≤ k ≤ ln. Sinceκn = degQn = deg q∗n + deg q∗n+1 ≤ m(n)− dn + n+ 2 = ln + 1,the degree of Qn is, in general, too big to be reconstructed by interpolation at x(n)k , 0 ≤≤ k ≤ ln.In the case κn ≤ ln, we can use the method of proof in [5]. Therefore, we can restrictourselves in the following to the case κn = ln + 1.First, we have to modify the polynomial Pn(x): Let ξn be such thatξn ≥ n max(1, |y1|, |y2|, . . . , |yκn |), (14)where y1, . . . , yκn are all zeros of Qn(x) in C. Then we defineP˜n(x) := (x− ξn)Pn(x) and R˜n := P˜nQn. (15)Thendeg P˜n = degQn = ln + 1ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 18 H.-P. BLATTand we can reconstruct Qn by interpolation at the points x(n)k , 0 ≤ k ≤ ln, and at thepoint ξn. We obtainQn(z) =ln∑k=0Qn(x(n)k )w(z)(z − x(n)k )w′(x(n)k )+Qn(ξn)w(z)(z − ξn)w′(ξn) , (16)wherew(z) = (z − ξn)ln∏k=0(z − x(n)k ). (17)For z = ξn, x(n)k , 0 ≤ k ≤ ln, relation (16) can be written asQn(z)w(z)=ln∑k=0Qn(x(n)k )(z − x(n)k )w′(x(n)k )+Qn(ξn)(z − ξn)w′(ξn) . (18)By definition, we have‖Rn‖ ≤ ‖f − r∗n‖+ ‖f − r∗n+1‖ ≤ En + En+1and, therefore, ∥∥∥R˜n∥∥∥ ≤ (ξn + 1)(En + En+1). (19)Moreover, by (13) we get(−1)k+1R˜n(x(n)k ) ≥ (ξn − 1)(En − En+1). (20)Next, we consider the functionh(z) := log∣∣∣R˜n(z)∣∣∣− κn∑i=1G(z, yi) +G(z, ξn).The function h(z) is subharmonic in C; hence, the maximum principle applies andh(∞) ≤ maxz∈Ih(z) = maxz∈Ilog∣∣∣R˜n(z)∣∣∣ = log ‖R˜n‖,and we obtainh(∞) = log |an| −κn∑i=1G(∞, yi) +G(∞, ξn).Therefore, with (19)log |an| ≤ log((ξn + 1)(En + En+1))+κn∑i=1G(∞, yi)−G(∞, ξn). (21)Next, let us consider the approximation of the function P˜n(x) at the pointsx(n)k , 0 ≤ k ≤ ln,with interpolation at the zero ξn with respect to Pln and the weight function1Qn(x). Itturns out that de la Valle´e Poussin’s theorem implies together with (20) that the minimalerror ρ satisfiesρ ≥ (ξn − 1)(En − En+1). (22)ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 9On the other hand, for any P ∈ Pln with P (ξn) = 0, we haveρ =∣∣∣∣∑lnk=0 βk(P˜n − P )(x(n)k )∣∣∣∣∑lnk=0∣∣βkQn(x(n)k )∣∣ , (23)where1β k= w′(x(n)k)=(x(n)k − ξn) ∏i=k(x(n)k − x(n)i). (24)Now, fix the polynomial P ∈ Pln by P (ξn) = 0 and P(x(n)k)= P˜n(x(n)k), 1 ≤ k ≤ ln.Hence,(P˜n − P )(x) = an(x− ξn)ln∏k=1(x− x(n)k )and, therefore,(P˜n − P )(x(n)0 ) = an(x(n)0 − ξn)ln∏k=1(x(n)0 − x(n)k ).By (22) – (24) we obtainρ =|an|∑lnk=0∣∣βkQn(x(n)k )∣∣ ≥ (ξn − 1)(En − En+1).Using representation (18), for z /∈ I we get∣∣∣∣Qn(z)w(z)∣∣∣∣ ≤ D(z) ln∑k=0|βkQn(x(n)k )|+1|z − ξn|∣∣∣∣Qn(ξn)w′(ξn)∣∣∣∣ ≤≤ D(z) |an|(ξn − 1)(En − En+1) +1|z − ξn|∣∣∣∣Qn(ξn)w′(ξn)∣∣∣∣ , (25)whereD(z) = max0≤k≤ln∣∣∣z − x(n)k ∣∣∣−1.Since κn = ln + 1 ≤ 2n+ 3, for n ≥ 2 we obtain∣∣∣∣Qn(ξn)w′(ξn)∣∣∣∣ =∣∣∣∣∣∣∣∏κni=1(ξn − yi)∏lnk=0(ξn − x(n)k )∣∣∣∣∣∣∣ ≤[ξn(1 + 1/n)ξn − 1]κn≤≤(1 + 1/n1− 1/ξn)κn≤(1 + 1/n1− 1/n)κn≤ c1, (26)where c1 is independent of n.In the following, we consider the level lineΓ1/n :={z ∈ C : G(z,∞) = log(1 +1n)}ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 110 H.-P. BLATTof Green’s function G(z,∞). Then for z ∈ Γ1/n, n ≥ 2, we havelog1|z − ξn| ≤ log1|ξn| + c2,where c2 is independent of n. Sincelimz→∞(G(z,∞) + log 12− log |z|)= 0and limn→∞ξn = ∞, there exists c3 > 0 such thatlog1|z − ξn| ≤ −G(ξn,∞) + c3for all n ≥ 2. Then from (26) we obtain for z ∈ Γ1/n and n ≥ 2 thatlog(1|z − ξn|∣∣∣∣Qn(ξn)w′(ξn)∣∣∣∣) ≤ c4 −G(ξn,∞), (27)where c4 > 0 is independent of n.Define for abbreviationAn :=|an|(ξn − 1)(En − En+1) . (28)Then inequality (21) together with (11) implieslogAn ≤ log ξn + 1ξn − 1 + logEn + En+1En − En+1 +κn∑i=1G(∞, yi)−G(∞, ξn) ≤≤ log n+ 1n− 1 + log(n2) +κn∑i=1G(∞, yi)−G(∞, ξn)and for z ∈ Γ1/nlog(D(z)An) ≤ c5 logn+κn∑i=1G(∞, yi)−G(∞, ξn) (29)for n ∈ Λ, n ≥ 2, with some constant c5 independent of n.Comparing the right-hand sides of (27) and (29), we conclude from (25) that, forn ∈ Λ, n ≥ 2, and z ∈ Γ1/n,log∣∣∣∣Qn(z)w(z)∣∣∣∣ ≤ c6 logn+ κn∑i=1G(∞, yi)−G(∞, ξn) (30)with an absolute constant c6 independent of n.The last inequality can be written with the logarithmic potentials Uνn(z), Uτn(z) andthe Dirac measure δξn at the point ξn asUνn(z)− αnUτn(z) + 1ln + 1Uδξn (z) ≤≤ 1ln + 1(c6 logn+κn∑i=1G(∞, yi)−G(∞, ξn)).ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 11Next, we use the balayage measure δ̂ξn of δξn onto the interval [−1, 1]. SinceU δ̂ξn (z) ≤ Uδξn (z) +G(∞, ξn), z ∈ C,we obtain for z = ξnUνn(z)− Uτn(z) + 1ln + 1U δ̂ξn (z) ≤ 1ln + 1(c6 log n+κn∑i=1G(∞, yi)). (31)Taking into account that we can choose the point ξn arbitrarily large on the positive realaxis andlimξn→∞δ̂ξn = µin the weak∗-sense (cf. [6], Chapter II, formula 4.46), we can choose ξn such that|U δ̂ξn − Uµ(z)| < 1n, z ∈ Γ1/n.Then we obtain for z ∈ Γ1/n thatUνn(z)− Uτn(z) ≤ c log nn+1κnκn∑i=1G(∞, yi).The last inequality is of the same structure as inequality (30) in [5]. Hence, the remainingproof follows the same lines as in this paper and is therefore omitted.The proof of the corollary is left to the reader.1. Kadec M. I. On the distribution of points of maximal deviation in the approximation of continuous func-tions by polynomials // Uspekhi Mat. Nauk. – 1960. – 15. – P. 199 – 202.2. Borwein P. B., Kroo´ A., Grothmann R., Saff E. B. The density of alternation points in rational approxima-tion // Proc. Amer. Math. Soc. – 1989. – 105. – P. 881 – 888.3. Kroo´ A., Peherstorfer F. On the asymptotic distribution of oscillation points in rational approximation //Anal. Math. – 1993. – 19. – P. 225 – 232.4. Braess D., Lubinsky D. S., Saff E. B. Behavior of alternation points in best rational aproximation // ActaAppl. Math. – 1993. – 33. – P. 195 – 210.5. Blatt H.-P., Grothmann R., Kovacheva R. K. On the distribution of alternation points in uniform rationalapproximation // Compt. rend. Acad. bulgare Sci. – 2002. – 55, # 8. – P. 5 – 8.6. Saff E. B., Totik V. Logarithmic potentials with external fields. – Heidelberg: Springer, 1997.7. Andrievskii V. V., Blatt H.-P. Discrepancy of signed measures and polynomial approximation. – New York:Springer, 2002.Received 16.05.2005ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1

Influence of poles on equioscillation in rational approximation

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Influence of poles on equioscillation in rational approximation

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)