Best uniform rational approximation of $x^α$ on $[0,1]$

7 Pages·0.12 MB·English

by  Herbert Stahl

APPEAREDINBULLETINOFTHE
AMERICANMATHEMATICALSOCIETY
Volume28,Number1,January1993,Pages116-122
BEST UNIFORM RATIONAL APPROXIMATION OF xα ON [0,1]
Herbert Stahl
3
9
9 Abstract. A strong error estimate for the uniform rational approximation of xα
1 on [0,1] is given, and its proof is sketched. Let Enn(xα,[0,1]) denote the minimal
approximationerrorintheuniformnorm. Thenitisshownthat
n
a lim e2π√αnEnn(xα,[0,1])=41+α|sinπα|
J n
→∞

1
holdstrueforeachα>0.

]
A
C 1. Introduction
.
h Let Π denote the set of all polynomials of degree at most n N with real
n
at coefficients; mn denote the set p/q p Πm, q Πn, q 0 ,∈m,n N, of
R { | ∈ ∈ 6≡ } ∈
m rational functions; and the best rational approximant r , m,n N, and
m∗n ∈ Rmn ∈
the minimal approximation error E =E (xα,[0,1]) be defined by
[ mn mn

1 (1.1) E := xα r = inf xα r ,
v mn k − m∗nk[0,1] r∈Rmnk − k[0,1]
7
1 where denotes the sup norm on K R. It is well known that the best
K
2 approxkim·aknt r exists and is unique with⊆in (cf. [Me, 9.1, 9.2] or [Ri,
1 m∗n Rmn §§
5.1]). Theuniqueexistencealsoholdsinthespecialcase(n=0)ofbestpolynomial
0 §
approximants.
3
9 Since fα(x):= xα is anevenfunction on[ 1,1],the same is true forits unique
| | −
/ approximant r = r (f ,[ 1,1]; ), and consequently a substitution of z2 by
h m∗n m∗n α − ·
z shows that approximating x2α on [ 1,1] and xα on [0,1] poses an identical
t
a problem. We have | | −
m
: (1.2) E (x2α,[ 1,1])=E (xα,[0,1]) for all m,n N.
v 2m,2n | | − mn ∈
i
X From Jackson’s and Bernstein’s theorems about the interdependence of approx-
r imation speed and smoothness of the function to be approximated (cf. [Me, 5.5,
a §§
5.6]) we know that in case of α R N the minimal error E (xα,[ 1,1]) be-
+ m,0
∈ \ | | −
haves like O(m α) as m . In [Be1, Be2] Bernstein proved a result that is
−
→ ∞
deeper and much more difficult to obtain; he showed that the limit
(1.3) lim mαE (xα,[ 1,1]):=β(α)
m,0
m | | −
→∞
1991Mathematics Subject Classification. Primary41A20,41A25, 41A50.
ReceivedbytheeditorsApril22,1992
ResearchsupportedbytheDeutsche Forschungsgemeinschaft(AZ:Sta299/4-2)
c1993AmericanMathematicalSociety
(cid:13) 0273-0979/93 $1.00+$.25perpage
1
2 HERBERT STAHL
exists for eachα>0; however,an explicit expressionfor 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’s
constant. In [Bel, p. 56] Bernstein raised the question whether β can be expressed
by knowntranscendentalsor whether itdefines anew one,andbasedonnumerical
upper and lower bounds for β, which he calculated up to a precision of 0.005,he
±
?
madethe tentativeconjectureβ =1/(2√π),which,however,hasbeendisprovedin
[VC1] by extensive and nontrivial high precision calculations.
For large values of α, Bernstein was able to establish in [Be2] an asymptotic
expression. He showed that
β(α) 1
(1.4) lim = .
α Γ(α) sin(πα/2) π
→∞ | |
While Bernstein’s investigations on best polynomial approximation of x and
| |
xα werepublishedintheperiodbetween1909and1938,thestudyofbestrational
| |
approximation of xα was only started in 1964 by Newman’s surprising (at the
| |
time) result in [Ne] that
(1.5) 1e 9√n E (x,[ 1,1]) 3e √n for all n=4,5,....
2 − ≤ nn | | − ≤ −
A comparison of this result with (1.3) shows that the convergence behavior of
rational approximants is essentially better than that of polynomials. Newmann’s
investigation has triggered a whole series of contributions, from which we select a
short list with papers that contain substantial improvements of the error estimate
in the uniform norm.
E (xα,[0,1]) e c(α)√3n, α R in [FrSz];
nn − +
≤ ∈
E (x1/3,[0,1]) e c√n, in [Bu1];
nn −
≤
Enn(xα,[0,1]) e−c(α)√n, α R+, in [Go1];
≤ ∈
1e π√2n E (x1/2,[0,1]) e π√2n(1 (n−1/4)), in [Bu2];
3 − ≤ nn ≤ − −O
e−c(α)√n Enn(xα,[0,1]), α Q+ N, in [Go2];
≤ ∈ \
e 4π√αn(1+ε) E (xα,[0,1]) e π√αn(1 ε),
− nn − −
≤ ≤
α R N, ε>0,n n (ε,α) in [Go3];
+ 0
∈ \ ≥
E (x1/2,[0,1]) cne π√2n, in [Vj1];
nn −
≤
1e π√2n E (x1/2,[0,1]) ce π√2n in [Vj2];
3 − ≤ nn ≤ −
e c1(s)√n E (√sx,[0,1]) e c2(s)√n, s N in [Tz].
− nn −
≤ ≤ ∈
Herec,c(α),... denoteconstants. Relation(1.2)allowsus totransfertheseresults
to the problem of approximating xα on [ 1,1].
| | −
It may be appropriate to repeat a remark from [Go2], where it was pointed out
thatNewman’s resultcanbe obtainedratherimmediately fromanoldresult(from
1877) by Zolotarev. In this sense, the investigation of rational approximants dates
back even further than Bernstein’s work on polynomial approximants.
UNIFORM RATIONAL APPROXIMATION OF xα ON [0,1] 3
The best result known so far for the rational approximation of xα on [0,1] was
obtainedindependentlybyGanelius[Ga]in1979andbyVjacheslavov[Vj3]in1980.
They proved that for α R N there exists a constant c =c (α)>0 such that
+ 1 1
∈ \
(1.6) liminfe2π√αnE (xα,[0,1]) c (α)
nn 1
n ≥
→∞
and, conversely, that for each positive rational number α Q there exists a
+
∈
constant c =c (α)< such that
2 2
∞
(1.7) limsupe2π√αnE (xα,[0,1]) c (α).
nn 2
≤
n
→∞
Both authors were not able to show that c = c (α) depends continuously on α.
2 2
Thus,inequality(1.7)remainedopenforα R Q;however,in[Ga]Ganeliuswas
+
∈ \
able to prove the somewhat weaker estimate
(1.8) E (xα,[0,1]) c (α)e2π√αn+c3(α)√4n for n n (c (α),c (α)),
nn 2 0 2 3
≤ ≥
whichholdsforeachα>0. In(1.8)c (α)andc (α)areconstantsdependingonα.
2 3
The results (1.6)–(1.8) give the correctexponent 2π√αn in the error formula;
−
however,nearlynothingissaidaboutthecoefficientinfrontoftheexponentialterm.
Thedeterminationofthiscoefficientisthesubjectofthepresentnote. Practicallyas
abyproduct,weprovetheupperestimate(1.7)forirrationalexponentsα R Q.
+
∈ \
2. The result
Theorem 1. The limit
(2.1) lim e2π√αnE (xα,[0,1])=41+α sinπα
nn
n | |
→∞
holds for each α>0.
Remarks. (1)From(2.1)wededucethattheapproximationerrorE (xα,[0,1])has
nn
the asymptotic behavior
(2.2) Enn(xα,[0,1])=41+α sinπαe−2π√αn(1+o(1)) as n ,
| | →∞
and equivalently it follows with (1.2) that
(2.3) E (xα,[ 1,1])=41+α/2 sinπα/2e π√αn(1+o(1)) as n
nn −
| | − | | →∞
for each α>0.
(2)Notonlytheexplicitexpressionontheright-handsideof(2.1)butalreadythe
existenceofthelimitrepresentsaresultdifficulttoobtain. Thevalue41+α sinπα,
| |
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 explicit
expression 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 value 1Γ(α) sinπα/2 for β(α) is the value
π | |
41+α/2 sin(πα/2) in case of rational approximation.
| |
4 HERBERT 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) E (x,[ 1,1])=E (√x,[0,1]) for n N,
2n,2n nn
| | − ∈
and hence we deduce from (2.1) that
(2.5) lim e π√nE (x,[ 1,1])=41+1/2 sinπ =8.
n − nn | | − | 2|
→∞
Limit(2.5)hasbeenconjecturedin[VRC]onthebasisofhighprecisioncalculations
and was proved in [St].
It may be surprising that in case of rational approximation, which is in many
respects 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 of
the number β = β(1) is still open and the numerical results in [VC1] show that β
cannot be a rational number with a moderately small denominator.
3. Outline of the proof of Theorem 1
If limit (2.1) is proved for one of the paradiagonalsequences
E (xα,[0,1]) ,
{ n+k,n }∞n=|k|
k Z fixed, then it holds also for the diagonal sequence
∈
E (xα,[0,1]) .
{ nn }∞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 rational
approximants we learn that the error function
(3.2) e (z):=zα r (z), z C R ,
n − m∗n ∈ \ −
has exactly 2n+2+[α] zeros in the interval (0,1). Hence the theory of multipoint
Pad´e approximants is applicable, and it gives us rather precise information about
the structure of the numerators and denominators of r (cf. [GoLa; StTo, 6.1,
m∗n §§
6.2]).
In the next step the error function e and the approximant r will be trans-
n m∗n
formedinsuchawaythattheresultingfunctionΨ isanalyticinC Rwithpossible
n
\
exceptions in a disc ∆(R) with radius R > 0 around the origin. The function Ψ
n
has boundary values from both sides of R ∆(R) that allow a comparison with a
\
special logarithmic potential. The potential will be introduced after Theorem 2
below.
UNIFORM RATIONAL APPROXIMATION OF xα ON [0,1] 5
The transformationof e and r into the function Ψ is carriedout in several
n m∗n n
steps. The intermediate functions are defined as
zα r (z)
(3.3) r (z):= − m∗n ;
n zα+r (z)
m∗n
4w2α 1 1
(3.4) R (w):= − r (ε1/αw) , ε :=E (xα,[0,1]);
n wα n n − wα n mn
1
(3.5) Φ (w):= R (w)+ R (w)2 4 ;
n 8wα (cid:16) n p n − (cid:17)
ψ(Φ (w)) for Im(w) 0,
n
(3.6) Ψ (w):= ≥
n
(cid:26) ψ(Φ (w)) for Im(w)<0,
n
with
z
(3.7) ψ(z):= .
sinπα i(cosπα)z
−
In (3.4) a new variable w is introduced implicitly by
(3.8) w :=ε 1/αz, z C.
−n ∈
Thepropertiesofeachnewfunctionr ,R ,Φ havetobestudiedcarefully. The
n n n
properties of the last function Ψ is summerized in
n
Lemma 1. The function Ψ is analytic and different from zero in C (R ∆(R)),
n
\ ∪
R>0 appropriately chosen, and there exist constants c ,...,c such that
1 4
(3.9) log Ψ (w) c w 2α for w R ∆(R),n n (c ,R);
n 1 − 0 1
| | k≤ | | ∈ −\ ≥
(3.10) log Ψn(w)wα4sinπα c2 w−α for w R+ ∆(R),n n0(c2,R);
| | k≤ | | ∈ \ ≥
(3.11) log Ψ (w) c for w ∂∆(R),n n (c ,R).
n 3 0 3
| | k≤ ∈ ≥
If we consider the representation
(3.12) log Ψ (w) =ψ (w) g (w,t)dµ (t),
n n D n
| | −Z
where D := C (R ∆(R)), g (w,t) the Green function in D, ψ a harmonic
D n
function in D w\ith− ∪
(3.13) ψ (w)=log Ψ(w) for w ∂D,
n
| | ∈
and µ a measure with
n
(3.14) supp(µn)⊆[R,∞] and µn ≥0 on [R,ε−n1/α],
then in addition to (3.9)–(3.11) we have the inequalities
(3.15) kµn|[ε−1/α,∞]k≤1,
(3.16) µ ([R,ε 1/α]) 2n c for n n (c ,R).
| n −n − |≤ 4 ≥ 0 4
Remark. Estimates (3.9), (3.10), and (3.16) contain the information that is most
relevant for the proof of Theorem 1.
The function log Ψ will be compared with a special Green potential p . The
n n
| |
definition of this potential is based on the following.
6 HERBERT STAHL
Theorem 2. For each a (1, ) there uniquely exists a Green potential
∈ ∞
(3.17) p (w):= g (w,t)dν (t), D :=C R ,
a D a
Z \ −
with
(3.18) ν 0, supp(ν )=[b ,a], 0<b <a,
a a a a
≥
that satisfies
=logw for w [b ,a],
a
(3.19) p (w) ∈
a
(cid:26) >logw for w (0,b ).
a
∈
For the constant b and the measure ν appearing in (3.17)–(3.19) we have
a a
(3.20) b √2
a
→
and
(3.21) a1eπ√2kνak →4 as a→∞.
A proof of Theorem 2 can be derived from [St, Theorem 2]. It is necessary to
change the domain of definition D by the transformation w 1/√w. Basic tools
7→
in the proof of Theorem 2 are estimates for certain elliptical integrals.
The potential p for a comparison with the function log Ψ is now defined as
n n
| |
(3.22) p (w):= αp (cw)
n a
−
with
1/α
4
(3.23) a:= sinπα and c:= 4sinπα1/α.
(cid:12)ε (cid:12) | |
(cid:12) n (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
From(3.21),(3.23),and(3.16)togetherwithotherinformationprovidedbyLemma
1 it then follows that
(3.24) εn 1/αeπ√2(2n)/α α = εn e2π√αn 4α as n .
(cid:20)(cid:12)4sinπα(cid:12) (cid:21) 4 sinπα → →∞
(cid:12) (cid:12) | |
(cid:12) (cid:12)
Theorem 1 then immediately follows from (3.24).
What we have given here is only a sketch of the overallstructure of the proof of
Theorem 1. Some of the steps demand rather subtle analysis.
Note Added in Proof. In [VC2] Varga and Carpenter have calculated numerical
approximations 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 especially
interesting since here the right-hand side of (2.1) is rational.
UNIFORM RATIONAL APPROXIMATION OF xα ON [0,1] 7
References
[Be1] S. Bernstein, Sur meilleure approximation de |x| par des polynˆomes de degr´es donn´es,
ActaMath.37(1913), 1–57.
[Be2] ,Aboutthebestapproximationof|x|pbymeansofpolynomialsofveryhighdegree,
Bull. Acad. Sci. USSR Cl. Sci. Math. Natur. 2 (1938), 169–190; also Collected Works,
vol.II,262–272. (Russian)
[Bu1] A.P.Bulanow,Asymptoticsforthe least derivation of|x|from rational functions,Mat.
Sb.76(1968), 288–303; Englishtransl.inMath.USSRSb.5(1968), 275–290.
[Bu2] ,Theapproximationofx1/3 byrationalfunctions,Vests¯ıAkad.NavukBSSRSer.
F¯ız.-Mat.Navuk2(1968), 47–56. (Russian)
[FrSz] G.FreudandJ.Szabados,Rationalapproximationtoxα,ActaMath.Acad.Sci.Hungar.
18(1967), 393–399.
[Ga] T.Ganelius,Rational approximation toxα on [0,1],Anal.Math.5(1979), 19–33.
[GoLa] A.H.GoncharandG.Lopez,OnMarkov’stheoremformultipoint Pade´e approximants,
Mat.Sb.105(1978); Englishtransl.inMath.USSRSb.34(1978), 449–459.
[Go1] A. A. Gonchar, On the speed of rational approximation of continuous functions with
characteristicsingularities,Mat.Sb.73(1967),630–638;Englishtransl.inMath.USSR
Sb.2(1967).
[Go2] , Rational approximation of the function xα, Constructive Theory of Functions
(Proc. Internat. Conf., Varna1970), Izdat. Bolgar. Akad. Nauk, Sofia, 1972, pp. 51–53.
(Russian)
[Go3] , The rate of rational approximation and the property of single-valuedness of an
analytic function in a neighborhood of an isolated singular point, Mat. Sb. 94 (1974),
265–282; Englishtransl.inMathUSSRSb.23(1974), 254–270.
[Me] G. Meinardus, Approximation of functions: Theory and numerical methods, Springer-
Verlag,NewYork,1967.
[Ne] D.J.Newman,Rational approximation to |x|,MichiganMath.J.11(1964), 11–14.
[Ri] T. J. Rivlin, An introduction to the approximation of functions, Blaisdell Publ. Co.,
Waltham,MA,1969.
[St] H.Stahl,Bestuniformrationalapproximationof|x|on[−1,1],Mat.Sb.183(1992),no.
8,85–118; Englishtransl.inRussianAcad.Sci.Sb.Math.76(1993), no.2(toappear).
[StTo] H. Stahl and V. Totik, General orthogonal polynomials, Cambridge Univ. Press, New
York,1992.
[Tz] J.Tzimbalario,Rational approximation to xα,J.Approx.Theory16(1976), 187–193.
[VC1] R.S.VargaandA.J.Carpenter,OntheBernsteinConjectureinapproximation theory,
Constr. Approx. 1 (1985), 333–348; Russian transl. in Mat. Sb. 129 (1986), 535–548;
EnglishalsoinMath.USSRSb.57(1987), 547–560.
[VC2] ,Some numerical results on best uniform rational approximation of xα on [0,1],
Numer.Algorithms(toappear).
[VCR] R.S.Varga,A.Ruttan,andA.J.Carpenter,Numericalresultsonbestuniformrational
approximationof|x|on[−1,1],Mat.Sb.182(1991),1523–1541;Englishtransl.inMath.
USSRSb.74(1993), no.2(toappear).
[Vj1] N. S. Vjacheslavov, The approximation of |x| by rational functions, Mat. Zametki 16
(1974), 163–171; Englishtransl.inMath.Notes 16(1974), 680–685.
[Vj2] , On the uniform approximation of |x| by rational functions, Dokl. Akad. Nauk
SSSR220(1975), 512–515; Englishtransl.inSovietMath.Dokl.16(1975), 100–104.
[Vj3] , On the approximation of xα by rational functions, Izv. Akad. Nauk USSR 44
(1980); Englishtransl.inMath.USSRIzv.16(1981), 83–101.
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