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APPEARED IN BULLETIN OF THE

arXiv:math/9301217v1 [math.CA] 1 Jan 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 116-122BEST UNIFORM RATIONAL APPROXIMATION OF xα ON [0, 1]Herbert StahlAbstract. A strong error estimate for the uniform rational approximation of xαon [0, 1] is given, and its proof is sketched.

Let Enn(xα, [0, 1]) denote the minimalapproximation error in the uniform norm. Then it is shown thatlimn→∞e2π√αnEnn(xα, [0, 1]) = 41+α| sin πα|holds true for each α > 0.1.

IntroductionLet Πn denote the set of all polynomials of degree at most n ∈N with realcoefficients; Rmn denote the set {p/q|p ∈Πm, q ∈Πn, q ̸≡0}, m, n ∈N, ofrational functions; and the best rational approximant r∗mn ∈Rmn, m, n ∈N, andthe minimal approximation error Emn = Emn(xα, [0, 1]) be defined by(1.1)Emn := ∥xα −r∗mn∥[0,1] =infr∈Rmn ∥xα −r∥[0,1],where ∥· ∥K denotes the sup norm on K ⊆R.It is well known that the bestapproximant r∗mn exists and is unique within Rmn (cf. [Me, §§9.1, 9.2] or [Ri,§5.1]).

The unique existence also holds in the special case (n = 0) of best polynomialapproximants.Since fα(x) := |x|α is an even function on [−1, 1], the same is true for its uniqueapproximant r∗mn = r∗mn(fα, [−1, 1]; ·), and consequently a substitution of z2 byz shows that approximating |x|2α on [−1, 1] and xα on [0, 1] poses an identicalproblem. We have(1.2)E2m,2n(|x|2α, [−1, 1]) = Emn(xα, [0, 1])for all m, n ∈N.From Jackson’s and Bernstein’s theorems about the interdependence of approx-imation speed and smoothness of the function to be approximated (cf.

[Me, §§5.5,5.6]) we know that in case of α ∈R+\N the minimal error Em,0(|x|α, [−1, 1]) be-haves like O(m−α) as m →∞. In [Be1, Be2] Bernstein proved a result that isdeeper and much more difficult to obtain; he showed that the limit(1.3)limm→∞mαEm,0(|x|α, [−1, 1]) := β(α)1991 Mathematics Subject Classification.

Primary 41A20, 41A25, 41A50.Received by the editors April 22, 1992Research supported by the Deutsche Forschungsgemeinschaft (AZ: Sta 299/4-2)c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2HERBERT STAHLexists for each α > 0; however, an explicit expression for the constant β(α), α > 0,still is not known.In case α = 1 the number β := β(1) = 0.28016 . .

. is known as Bernstein’sconstant.

In [Bel, p. 56] Bernstein raised the question whether β can be expressedby known transcendentals or whether it defines a new one, and based on numericalupper and lower bounds for β, which he calculated up to a precision of ±0.005, hemade the tentative conjecture β?= 1/(2√π), which, however, has been disproved in[VC1] by extensive and nontrivial high precision calculations.For large values of α, Bernstein was able to establish in [Be2] an asymptoticexpression. He showed that(1.4)limα→∞β(α)Γ(α)| sin(πα/2)| = 1π .While Bernstein’s investigations on best polynomial approximation of |x| and|x|α were published in the period between 1909 and 1938, the study of best rationalapproximation of |x|α was only started in 1964 by Newman’s surprising (at thetime) result in [Ne] that(1.5)12e−9√n ≤Enn(|x|, [−1, 1]) ≤3e−√nfor all n = 4, 5, .

. .

.A comparison of this result with (1.3) shows that the convergence behavior ofrational approximants is essentially better than that of polynomials. Newmann’sinvestigation has triggered a whole series of contributions, from which we select ashort list with papers that contain substantial improvements of the error estimatein the uniform norm.Enn(xα, [0, 1]) ≤e−c(α) 3√n,α ∈R+in [FrSz];Enn(x1/3, [0, 1]) ≤e−c√n,in [Bu1];Enn(xα, [0, 1]) ≤e−c(α)√n,α ∈R+,in [Go1];13e−π√2n ≤Enn(x1/2, [0, 1]) ≤e−π√2n(1−O(n−1/4)),in [Bu2];e−c(α)√n ≤Enn(xα, [0, 1]),α ∈Q+\N,in [Go2];e−4π√αn(1+ε) ≤Enn(xα, [0, 1]) ≤e−π√αn(1−ε),α ∈R+\N, ε > 0, n ≥n0(ε, α)in [Go3];Enn(x1/2, [0, 1]) ≤cne−π√2n,in [Vj1];13e−π√2n ≤Enn(x1/2, [0, 1]) ≤ce−π√2nin [Vj2];e−c1(s)√n ≤Enn( s√x, [0, 1]) ≤e−c2(s)√n,s ∈Nin [Tz].Here c, c(α), .

. .

denote constants. Relation (1.2) allows us to transfer these resultsto the problem of approximating |x|α on [−1, 1].It may be appropriate to repeat a remark from [Go2], where it was pointed outthat Newman’s result can be obtained rather immediately from an old result (from1877) by Zolotarev.

In this sense, the investigation of rational approximants datesback even further than Bernstein’s work on polynomial approximants.

UNIFORM RATIONAL APPROXIMATION OF xα ON [0, 1]3The best result known so far for the rational approximation of xα on [0, 1] wasobtained independently by Ganelius [Ga] in 1979 and by Vjacheslavov [Vj3] in 1980.They proved that for α ∈R+\N there exists a constant c1 = c1(α) > 0 such that(1.6)lim infn→∞e2π√αnEnn(xα, [0, 1]) ≥c1(α)and, conversely, that for each positive rational number α ∈Q+ there exists aconstant c2 = c2(α) < ∞such that(1.7)lim supn→∞e2π√αnEnn(xα, [0, 1]) ≤c2(α).Both authors were not able to show that c2 = c2(α) depends continuously on α.Thus, inequality (1.7) remained open for α ∈R+\Q; however, in [Ga] Ganelius wasable to prove the somewhat weaker estimate(1.8)Enn(xα, [0, 1]) ≤c2(α)e2π√αn+c3(α) 4√nfor n ≥n0(c2(α), c3(α)),which holds for each α > 0. In (1.8) c2(α) and c3(α) are constants depending on α.The results (1.6)–(1.8) give the correct exponent −2π√αn in the error formula;however, nearly nothing is said about the coefficient in front of the exponential term.The determination of this coefficient is the subject of the present note.

Practically asa byproduct, we prove the upper estimate (1.7) for irrational exponents α ∈R+\Q.2. The resultTheorem 1.

The limit(2.1)limn→∞e2π√αnEnn(xα, [0, 1]) = 41+α| sin πα|holds for each α > 0.Remarks. (1) From (2.1) we deduce that the approximation error Enn(xα, [0,1]) hasthe asymptotic behavior(2.2)Enn(xα, [0, 1]) = 41+α| sin πα|e−2π√αn(1 + o(1))as n →∞,and equivalently it follows with (1.2) that(2.3)Enn(|x|α, [−1, 1]) = 41+α/2| sin πα/2|e−π√αn(1 + o(1))as n →∞for each α > 0.

(2) Not only the explicit expression on the right-hand side of (2.1) but already theexistence of the limit represents a result difficult to obtain. The value 41+α| sin πα|,α > 0, is the analogue of Bernstein’s constant β(α) in (1.3) for the case of ratio-nal approximation.

It has already been noted in the introduction that an explicitexpression for β(α) is still not known. The best we know is Bernstein’s asymp-totic formula (1.4).

Since in (1.4) we have considered approximation on [−1, 1],the counterpart of the asymptotic value1πΓ(α)| sin πα/2| for β(α) is the value41+α/2| sin(πα/2)| in case of rational approximation.

4HERBERT STAHL(3) If we turn our attention to the special case of rational approximation of |x|on [−1, 1], then it follows from (1.2) that(2.4)E2n,2n(|x|, [−1, 1]) = Enn(√x, [0, 1])for n ∈N,and hence we deduce from (2.1) that(2.5)limn→∞e−π√nEnn(|x|, [−1, 1]) = 41+1/2| sin π2 | = 8.Limit (2.5) has been conjectured in [VRC] on the basis of high precision calculationsand was proved in [St].It may be surprising that in case of rational approximation, which is in manyrespects more complex than the polynomial case, limit (2.5) has a rational value,while in the polynomial case Bernstein’s question in [Be1] about the character ofthe number β = β(1) is still open and the numerical results in [VC1] show that βcannot be a rational number with a moderately small denominator.3. Outline of the proof of Theorem 1If limit (2.1) is proved for one of the paradiagonal sequences{En+k,n(xα, [0, 1])}∞n=|k|,k ∈Z fixed, then it holds also for the diagonal sequence{Enn(xα, [0, 1])}∞n=1.It turns out that(3.1)n + k = m := n + 1 + [α],α ∈R+\N, n ∈N,is a good choice for the numerator degree m. From the theory of best rationalapproximants we learn that the error function(3.2)en(z) := zα −r∗mn(z),z ∈C\R−,has exactly 2n + 2 + [α] zeros in the interval (0, 1).

Hence the theory of multipointPad´e approximants is applicable, and it gives us rather precise information aboutthe structure of the numerators and denominators of r∗mn (cf. [GoLa; StTo, §§6.1,6.2]).In the next step the error function en and the approximant r∗mn will be trans-formed in such a way that the resulting function Ψn is analytic in C\R with possibleexceptions in a disc ∆(R) with radius R > 0 around the origin.

The function Ψnhas boundary values from both sides of R\∆(R) that allow a comparison with aspecial logarithmic potential. The potential will be introduced after Theorem 2below.

UNIFORM RATIONAL APPROXIMATION OF xα ON [0, 1]5The transformation of en and r∗mn into the function Ψn is carried out in severalsteps. The intermediate functions are defined asrn(z) := zα −r∗mn(z)zα + r∗mn(z);(3.3)Rn(w) := 4w2α −1wαrn(ε1/αnw) −1wα ,εn := Emn(xα, [0, 1]);(3.4)Φn(w) :=18wαRn(w) +pRn(w)2 −4;(3.5)Ψn(w) := ψ(Φn(w))for Im(w) ≥0,ψ(Φn(w))for Im(w) < 0,(3.6)with(3.7)ψ(z) :=zsin πα −i(cos πα)z .In (3.4) a new variable w is introduced implicitly by(3.8)w := ε−1/αnz,z ∈C.The properties of each new function rn, Rn, Φn have to be studied carefully.

Theproperties of the last function Ψn is summerized inLemma 1. The function Ψn is analytic and different from zero in C\(R ∪∆(R)),R > 0 appropriately chosen, and there exist constants c1, .

. .

, c4 such that| log |Ψn(w)∥≤c1|w|−2αfor w ∈R−\∆(R), n ≥n0(c1, R);(3.9)| log |Ψn(w)wα4 sin πα∥≤c2|w|−αfor w ∈R+\∆(R), n ≥n0(c2, R);(3.10)| log |Ψn(w)∥≤c3for w ∈∂∆(R), n ≥n0(c3, R). (3.11)If we consider the representation(3.12)log |Ψn(w)| = ψn(w) −ZgD(w, t) dµn(t),where D := C\(R−∪∆(R)), gD(w, t) the Green function in D, ψn a harmonicfunction in D with(3.13)ψn(w) = log |Ψ(w)|for w ∈∂D,and µn a measure with(3.14)supp(µn) ⊆[R, ∞]andµn ≥0on [R, ε−1/αn],then in addition to (3.9)–(3.11) we have the inequalities∥µn|[ε−1/α,∞]∥≤1,(3.15)|µn([R, ε−1/αn]) −2n| ≤c4for n ≥n0(c4, R).(3.16)Remark.

Estimates (3.9), (3.10), and (3.16) contain the information that is mostrelevant for the proof of Theorem 1.The function log |Ψn| will be compared with a special Green potential pn. Thedefinition of this potential is based on the following.

6HERBERT STAHLTheorem 2. For each a ∈(1, ∞) there uniquely exists a Green potential(3.17)pa(w) :=ZgD(w, t) dνa(t),D := C\R−,with(3.18)νa ≥0,supp(νa) = [ba, a],0 < ba < a,that satisfies(3.19)pa(w) = log wfor w ∈[ba, a],> log wfor w ∈(0, ba).For the constant ba and the measure νa appearing in (3.17)–(3.19) we have(3.20)ba →√2and(3.21)1aeπ√2∥νa∥→4as a →∞.A proof of Theorem 2 can be derived from [St, Theorem 2].

It is necessary tochange the domain of definition D by the transformation w 7→1/√w. Basic toolsin the proof of Theorem 2 are estimates for certain elliptical integrals.The potential pn for a comparison with the function log |Ψn| is now defined as(3.22)pn(w) := −αpa(cw)with(3.23)a :=4εnsin πα1/αandc := |4 sin πα|1/α.From (3.21), (3.23), and (3.16) together with other information provided by Lemma1 it then follows that(3.24)εn4 sin πα1/αeπ√2(2n)/αα=εn4| sin πα|e2π√αn →4αas n →∞.Theorem 1 then immediately follows from (3.24).What we have given here is only a sketch of the overall structure of the proof ofTheorem 1.

Some of the steps demand rather subtle analysis.Note Added in Proof. In [VC2] Varga and Carpenter have calculated numericalapproximations for the right-hand side of (2.1) for the six values α = j/8, j =1, 2, 3, 5, 6, 7, and conjectured formula (2.1) on the basis of these numerical val-ues.

The numerical results for the two cases α = 1/4 and α = 3/4 are especiallyinteresting since here the right-hand side of (2.1) is rational.

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