Best rational approximationand "1/9" constant

A.I.Aptekarev

Keldysh Institute of Applied Mathematics RASand Moscow State University

Approximation days in Leuven, BelgiumJuly 2, 2012

Program of the talk

I Introduction (Best Rational Approximants)

I Historical reminiscences (1/9.)

I Weak asymptotics BRA (Gonchar, Rakhmanov and Stahl)

I Strong asymptotics BRA (asymptotically sharpalternances)

Statement of problem

I Class of functions

ρn(z) :=

∫F

ρn(t)t − z

dt2πi

, F ⊂ C, n ∈ N,

I Best Rational Approximants (BRA)

dn := dn(ρn,E) := infr∈Rn

max

E|ρn − r |

, E ∩ F = ∅,

I F – singularities (poles); E – interpolations.

Rate of convergence, sharp constants

I Analytic functions on E

d := limn→∞

d1/(2n)n .

I Refinement of the rate

limn→∞

dn(ρn,E)

αnd2n = ? , where α1/nn = 1 + o(1).

The rate dn(e−x , [0,∞])

I

d := d(e−x , [0,∞]) = limn→∞

d1/(2n)n (e−x , [0,∞]).

I M.J. Cody, G. Meinardus, and R.S. Varga, Chebyshevrational approximation to e−x on [0,+∞) and applicationsto heat-conduction problems, J. Approx. Theory, (1969), 2,50–65

R.S. Varga, Functional analysis and approximation theoryin numerical analysis , Soc. Industrial Appl. Math.,Philadelphia 1971.

Intermediate results

I Colunm approximants (Rm,n)

limn→∞

d1/(2n)0,n (e−x , [0,∞]) =

13

1973 Schönhage.

I Pade approximants, ....

E.B. Saff and R.S. Varga, Some open questionsconcerning polynomial and rational functions, Padé andrational approximation, ed. E. B. Saff, R. A. Varga, (1977),Academic Press, New York, 483–488

Numerically exact valueI CF (AAK) approximants. N. Trefethen (May 23, 1981)

d =√

H , H =1

9.2890254919208189 . . .

L. N. Trefethen and M. H. Gutknecht,The Carathéodory–Fejér method for real rational approx.,SIAM J. Numer. Anal., 20 (1983), pp. 420–436.

I Remez computations BRA.A.J. Carpenter, A. Ruttan, and R.S. Varga,Extended computations on the ‘1/9’ conjecture in rationalapproximation theory,Lecture Notes in Math., (1984), 1105, 383–411

R.S. Varga, Scientific computation on some mathematicalconjectures, Approximation theory V. Proc. 5th Int. Symp.,College Station, Tex. 1986, (1986), 191–209

Explicit expression – I (A. Magnus 1986):I H = exp

[−πK ′

K

], K (k) = 2E(k),

where K and K ′ are the complete elliptic integrals of the firstkind for moduli k and k ′ =

√1− k2 and E(k) is the complete

elliptic integral of the second kind.I

A.P. Magnus, CFGT determin. of Varga’s constant ‘1/9’,Inst. Preprint B-1348, (1986), Inst. Math. UCL, Belgium;http://www.math.ucl.ac.be/ magnus/onine/index.html

A.P. Magnus, On the use of Carathéodory–Fejér methodfor investigating ‘1/9’ and similar constants, Nonlinearnumerical methods and rational approximation, ed.A. Cuyt, (1988), Dordrecht, D. Reidel, 105–132

A.P. Magnus and L. Meinguet, The elliptic functions andintegrals of the ‘1/9’ problem, Numer. Algorithms, (2000),24, 117–139

Explicit expression – II (Gonchar-Rakhmanov)

I

H :∞∑

l=1

alH l =18, al :=

∣∣∣∣∑b|l

(−1)bb∣∣∣∣.

I

A.A. Gonchar, Rational approximation of analytic functions,Proc. Int. Congr. Math., Berkeley – Calif. 1986,(1987), 1, 739–748

A.A. Gonchar and E. A. Rakhmanov, Equilibriumdistributions and degree of rational approximation ofanalytic functions, Mat. Sb., (1987), 134, 3, 306–352;English transl. in Math. USSR-Sb., (1989), 62

Explicit expression – 0 (Halphen 1886)

I Alphonse discovery (in ?)

H :∑ nHn

1− (−H)n =18.

I G.H. Halphen, Traité des fonctions elliptiques et de leursapplications. I. Théorie des fonctions elliptiques et de leursdéveloppements en séries, Gauthier-Villars, Paris, 1886.

Magnus Conjecture

I

dn(e−x , [0,∞])

d2n → 2d as n→∞, d =√

H .

A.P. Magnus, Asymptotics and super asymptotics of bestrational approximation error norms for the exponentialfunction (the ‘1/9’ problem) by the Carathéodory–Fejérmethod, Nonlinear methods and rational approximation. II,ed. A. Cuyt, (1994), Kluwer Ac. Publ., Dordrecht, 173–185

I Magnus conjecture is true!

A. I. Aptekarev, Sharp constants for rationalapproximations of analytic functions, Mat. Sb. 193 (2002),no. 1, 3–72 (in Russian); Sbornik Math. 193 (2002),no. 1–2, 1–72.

Weak asymptoticsChebyshev’s RA

Gonchar’s method

I Multipoint-Pade approximants, weak asymptoticspolynomials orthogonal with respect to a varying weight viapotential theory approach.

I A.A. Gonchar, On the speed of rational approximation ofsome analytic functions, Mat. Sb., (1978), 105(147), 2,147–163; English transl. in Math. USSR-Sb., (1978), 34

A.A. Gonchar and G. Lopes, On Markov’s theorem formultipoint Padé approximants , Mat. Sb., (1978), 105(147),4, 512–524; English transl. in Math. USSR-Sb., (1978), 34

I The real case understood.

The Complex Case

I Weak asymptotics of OP with complex weight.

H. Stahl, The structure of extremal domains associatedwith an analytic function, Complex Variables Theory Appl.4 (1985), no. 4, 339–354.H. Stahl, Orthogonal polynomials with complex-valuedweight function. I, II, Constr. Approx. 2 (1986), no. 3,225–240; 241–251.

I Complex varying weight and BRA

A.A. Gonchar and E. A. Rakhmanov, Equilibriumdistributions and degree of rational approximation ofanalytic functions, Mat. Sb., (1987), 134, 3, 306–352;English transl. in Math. USSR-Sb., (1989), 62

Equilibrium of the S-condenser with external fieldI Equilibrium measure

M(E ,F ) : µ = µF − µE , µE ∈ ME , µF ∈ MF ,

λ(E ; F , ϕ) ∈ M(E ,F ) :Vλ + ϕ =

γ∆ on ∆ := S(λ) ∩ F ,≥ γ∆ on F ,

Vλ = γE on E .

I S property

(E ; F , ϕ) ∈ S, :∂(Vλ + ϕ)

∂n+=∂(Vλ + ϕ)

∂n−a.e. on ∆

I Rate of convergence

γ := γ(E ; F , ϕ) = γ∆ − γE .

Theorem (Gonchar–Rakhmanov)

I Givenρn(z) :=

∫F

ρn(t)t − z

dt2πi

,

consider dn for its BRA on E .

I Suppose that

I ρn ∈ H(Ω), F ⊂ Ω , and1

2nlog

1|ρn|→Ωϕ as n→∞;

I (E ; F , ϕ) ∈ S.I Then

limn→∞

dn(ρn,E)1/(2n) = d = e−γ .

Gonchar-Rakhmanov plate F for exp(z)

−60 −40 −20 0 20

−20

0

20

bestapproximation

S-property and BVP for a complex potential

I the S-property⇒

ddz

(Vλ++Qα) = − ddz

(Vλ−+Qα) on ∆α, α = E ,F , Qα ∈ H(Ωα), α = E ,F ,

I BVP for complex potentials

Vλ+ + Vλ− =

−2QF + 2ΓF on ∆F ,

−2QE + 2ΓE on ∆E ,

I BVP analytic functions f = exp[−Vλ]:

f ∈ H∞(C \ (∆F ∪∆E )), f 6= 0 in C \ (∆F ∪∆E ),

f (∞) = 1, ∆F

arg f = 2π, ∆E

arg f = −2π,

f+f− = exp2Qα − 2Γα on ∆α, α = E ,F .

Asymptotically sharp alternancesChebyshev’s RA(outline of the proof)

Interpolation points

1) We first construct a sequence of polynomials

ω2n+1(z) =2n+1∏j=1

(z−xj,2n+1), xj,2n+1 ∈ E , j = 1, . . . ,2n+1,

such that

ω2n+1

Φ2n+1E fE

→n→∞⇒ 1 uniformly on K ⊂ C \ E

and

ω2n+1

|ΦE |2n+1= |fE |2 cos

((2n + 1) arg ΦE + arg fE

)+ o(1)

uniformly on the whole interval E .

Multi points PA – rn

I Definition ρn − rn =ω2n+1

PnRn,∫

FPn(z) zν

exp−2(nQ + Q1)ω2n+1

dz = 0, ν = 0, . . . ,n− 1,

I Asymptotics

Pn = (Φ∆e−γ)n(

f∆f∆(∞)

+ o(1)

),

Rn =1

w∆· e−nγ

f∆(∞)· 1 + o(1)

Φn∆f∆

on K ⊂ C \ F .

I Where BVP on ∆

Φ∆+Φ∆−e−2(Q+log ΦE ) = 1 on ∆, f∆+f∆−e−Q1

ΦE fE=

iw∆+

in ∆.

Plugging Asymptotics on EI

ρn − rn =ω2n+1

PnRn

=ω2n+1

|ΦE |2n+1· |ΦE |2n+1

(Φ∆e−γ)n f∆· 1

w∆· e−γn(1 + o(1))

f∆(∞)Φn∆f∆

= 2

∣∣∣∣∣ ΦE

Φ∆

∣∣∣∣∣2n|ΦE fE |f 2∆w∆

(cos((2n + 1) arg ΦE + arg fE ) + o(1)

).

I Alternance BVP on E∣∣∣∣∣ ΦE

Φ∆

∣∣∣∣∣ = exp(−2 γ n),

|ΦE fE |f 2∆w∆

= exp(−2γ∗),

Final BVPs

I Gonchar–Rakhmanov BVP(ΦE

Φ∆

)+

(ΦE

Φ∆

)−

=

e−2Q on ∆,

e−2γ on E ,

where γ := γ∆ − γE .I Szego functions BVP(

ΦE fEf 2∆w∆

)+

(ΦE fEf 2∆w∆

)−

=

e−4Q1 on ∆,

e−2γ∗ on E ,

I 2Q1 = Q ⇒ Magnus conjecture is correct!

Statement of the Theorem

I Input

ρn(z) :=

∫F

ρn(t)t − z

dt2πi

,

ρn is a real-symmetric analytic (in a neighbourhood Ω ofthe arc F ) function such that

ρn := exp−2(nQ + Q1), Q,Q1 ∈ H(Ω);

I Outputdn := dn(ρn,E).

Statement of the TheoremConditions and Conclusion

I "1)" E is a finite or semi-infinite interval on R;I "2)" F is an R-symmetric rectifiable Jordan arc (F is

connected, F = F );I "3)" (E ; F ,<Q) ∈ S, the equilibrium set on F

∆ := z : Vλ(E ;F ,<Q)(z) + <Q(z) = γ∆

is connected and λ′ > 0 at the interior points of the arc ∆.

I THENdn = 2d∗d2n(1 + o(1)),

where

d = exp[−γ(E ; F ,<Q)] = exp[−γ∗(E ; ∆,Q)],

d∗ = exp[−γ∗(E ; ∆,2Q1)],

Halphen versus Remez

Best regards to AlphonseFrom me and