RATIONAL APPROXIMATION &SOBOLEV ORTHOGONAL POLYNOMIALS

(Markov’s Theorem and its extension to Sobolev-type orthogonality)

Hector Pijeira-Cabrera† Abel Dıaz-Gonzalez‡ Ignacio Perez-Yzquierdo∗

Universidad Carlos III de Madrid Universidad Carlos III de Madrid Univ. Autonoma de Santo Domingo

† Research partially supported by Ministry of Science, Innovation and Universities of Spain, under grant PGC2018-096504-B-C33.

‡ Supported by the Research Fellowship Program, Ministry of Economy and Competitiveness of Spain, under grant MTM2015-65888-C4-2-P.

∗ Research partially supported by Fondo Nacional de Innovacion y Desarrollo Cientıfico y Tecnologico (FONDOCYT), Dominican Republic, under grant2015-1D2-164.

Rational Approximation & Sobolev Orthogonality hpijeira@math.uc3m.es 1 / 15

Historically, the orthogonal polynomials pn(x) originated in the the-ory of continued fractions. This relationship is of great importanceand is one of the possible starting points of the treatment of orthogonalpolynomials.

Gabor SzegoOrthogonal Polynomials (§3.5)

Nowadays, the theory of orthogonal polynomials is usually presentedwithout any reference to continued fractions. In fact historically, thetheory of orthogonal polynomials originated from certain types of con-tinued fractions. Special cases of such continued fractions were studiedby C.F. Gauss, C.G. Jacobi, E.B. Christoffel and G. F. Mehler, whilemore general aspects were given by P.L. Chebyshev, H.E. Heine andA.A. Markov.

Claude BrezinskiHistory of Continued Fractions and Pade Approximants (§5.2.3)

Rational Approximation & Sobolev Orthogonality §1.- Classical Markov’s theorem hpijeira@math.uc3m.es 2 / 15

Markov’s Theorem

I µ finite positive Borel measure with support [−1, 1],

I Pn nth monic orthogonal polynomials w.r.t. µ.

I P[1]n (z) =

∫ 1

−1

Pn(z)− Pn(t)z− t

dµ(t) the associated polynomial.

Then

P[1]n (z)

Pn(z)⇒n→∞

µ(z) =

∫ 1

−1

dµ(x)

z− x, K ⊂ Ω∞ = C \ [−1, 1].

A.A. Markov(1856-1922)

Complement of Markov’s theorem

lim supn→∞

∥∥∥∥∥µ− P[1]n

Pn

∥∥∥∥∥1

2n

K

≤ κ < 1,

Rational Approximation & Sobolev Orthogonality §1.- Classical Markov’s theorem hpijeira@math.uc3m.es 3 / 15

Sobolev orthogonality vs. standard orthogonality

〈p, q〉S =

∫p(x)q(x)dµ0(x) +

N∑k=1

∫p(k)(x)q(k)(x)dµk(x), N > 0. (1)

(Sobolev-type or discrete if µkNk=1 are discrete measures)

Let Sn⊥〈·, ·〉S monic, deg(Sn) = n.

1.- 〈xp, q〉S 6= 〈p, xq〉S (non-standard), except trivial examples. Hence, Sn does not satisfy a 3TRR∗†

2.- The matrix of moments is not a Hankel matrix‡.

3.- The zeros of Sn are not necessarily on Co

(N⋃

k=1

suppµk

). For example:

〈p, q〉S =

∫ 1

−1p(x)q(x)dx + p′′(−2)q′′(−2) + p′′′(−2)q′′′(−2) + p′(2)q′(2) + p′′(2)q′′(2).

∗A. Duran, A generalization of Favard’s Theorem for polynomials satisfying a recurrence relation, J. Approx. Theory, 74 (1993), 83-109.†W. D. Evans, L. Littlejohn, F. Marcellan, C. Markett, A. Ronveaux, On recurrence relations for Sobolev orthogonal polynomials, SIAM J. Math. Anal. ,

26(1995), 446–467.‡D. Barrios, G. Lopez, H. Pijeira, The moment problem for a Sobolev inner product, J. Approx. Theory, 100 (1999) 364-380.

Rational Approximation & Sobolev Orthogonality §2.- Sobolev-type orthogonality hpijeira@math.uc3m.es 4 / 15

Sobolev-type inner product

〈f , g〉S =

∫ 1

−1f (x)g(x) dµ(x) +

N∑j=1

dj∑i=0

ηj,i f (i)(cj)g(i)(cj); (2)

where µ is a finite positive Borel measure supported on [−1, 1], N ≥ 0, ηj,i ≥ 0, ηj,dj > 0,cj ∈ R \ [−1, 1] and dj ∈ Z+. Sn the nth monic orthogonal polynomial w.r.t. (2).I Qn the nth monic orthogonal polynomial w.r.t. 〈·, ·〉µρ , where

〈f , g〉µρ =

∫ 1

−1f (x) g(x) dµρ(x), (3)

ρ(z) =∏

cj<−1

(z− cj)dj+1

∏cj>1

(cj − z)dj+1 and dµρ(x) = ρ(x)dµ(x).

I If n > d =

N∑j=1

(dj + 1), Sn satisfies the next quasi-orthogonality relations w.r.t. dµρ

〈Sn, f 〉µρ = 〈Sn, ρf 〉µ =

∫ 1

−1Sn(x) f (x)ρ dµ(x) = 〈Sn, ρ f 〉 = 0, for all f ∈ Pn−d−1.

I If n > d, then Sn has at least (n− d) changes of sign in (−1, 1) (note that d ≤ N).

Rational Approximation & Sobolev Orthogonality §3.- Associated polynomials hpijeira@math.uc3m.es 5 / 15

Definition 3.1 (kth associated polynomial to Sn, k ∈ N and n ≥ 1.)

S[k]n (z) =

∫ 1

−1

Sn+k(z)− Sn+k(x)

z− xQk−1(x) dµρ(x)

S[0]n (z) =Sn(z).

Properties1.I Let S[k]

n be the kth associated polynomial. Then S[k]n is a polynomial of degree n and

leading coefficient equal to ‖Qk−1‖2µρ

.

2.I For n ≥ 2d − 1, the sequences S[k]n ∞n=0 satisfy the following 2d + 1 term recurrence

relation

ρ(z)S[k]n (z) =

n+d∑j=n−d

an+k,j+k S[k]j (z), where an+k,j+k =

〈Sn+k, ρSj+k〉〈Sj+k, Sj+k〉

. (4)

3.I For n ≥ d − 1, the sequences S[k]n ∞n=0, for k ≥ 2, hold the following relation

S[k]n (z) = (z− βk−2)S

[k−1]n+1 (z)− α2

k−2S[k−2]n+2 (z). (5)

Rational Approximation & Sobolev Orthogonality §3.- Associated polynomials hpijeira@math.uc3m.es 6 / 15

Sobolev-type inner product sequentially-ordered

Definition 4.1Let (rj, νj)M

j=1⊂R×Z+ be a finite sequence of M ordered pairs and A ⊂ R. We say that (rj, νj)Mj=1 is

sequentially-ordered with respect to A, if

1.- 0 ≤ ν1 ≤ ν2 ≤ · · · ≤ νM .

2.-(*) rk /∈ Ch (A ∪ r1, r2, . . . , rk−1) for k = 1, 2, . . . ,M.

If A = ∅, we say that (rj, νj)Mj=1 is sequentially-ordered for brevity.

Definition 4.2The inner product (2) is sequentially-ordered, if the set of ordered pairs

(cj, i) : 1 ≤ j ≤ N, 0 ≤ i ≤ dj and ηj,i > 0

may be arranged as a finite sequence of ordered pairs which is sequentially ordered with respect to (−1, 1).

From 2.-(*), the coefficient ηj,dj is the only coefficient ηj,i (i = 0, 1, . . . , dj) different from zero, for eachj = 1, 2, . . . ,N. Hence, the inner product (2) takes the form

〈f , g〉 =

∫ 1

−1f (x)g(x) dµ(x) +

N∑j=1

ηj,dj f (dj)(cj)g(dj)(cj). (6)

Rational Approximation & Sobolev Orthogonality §4.- Sequentially-ordered Sobolev-type inner product hpijeira@math.uc3m.es 7 / 15

I Examples

〈f , g〉 =

∫ 1

−1f (x)g(x) dµ(x) + f ′(2)g′(2) + f ′′(5)g′′(5) + f ′′′(9)g′′′(9).

〈f , g〉 =

∫ 1

−1f (x)g(x) dµ(x) + f ′(2)g′(2) + f ′′(−2)g′′(−2) + f ′′(−3)g′′(−3).

〈f , g〉 =

∫ 1

−1f (x)g(x) dµ(x) + f ′(−2)g′(−2) + f ′′(−2)g′′(−2) + f ′′′(−2)g′′′(−2).

〈f , g〉 =

∫ 1

−1f (x)g(x) dµ(x) + f ′(9)g′(9) + f ′′(5)g′′(5) + f ′′′(2)g′′′(2).

sequentially-ordered non sequentially-ordered

Rational Approximation & Sobolev Orthogonality §4.- Sequentially-ordered Sobolev-type inner product hpijeira@math.uc3m.es 8 / 15

Theorem 4.3

If (6) is a sequentially-ordered, then Sn has at least n− N changes of sign on (−1, 1).

Theorem 4.4If (6) is sequentially-ordered and µ ∈ M(0, 1) (the Nevai class). Then

Sn(z)Pn(z)⇒n→∞

N∏j=1

(ϕ(z)− ϕ(cj))2

2ϕ(z) (z− cj), K ⊂ Ω, (7)

where ϕ(z) = z +√

z2 − 1, with√

z2 − 1 > 0 for z > 1.G. Lopez Lagomasino, F. Marcellan, and W. Van Assche, Relative asymptotics for orthogonal polynomials with respect to a discrete Sobolev inner product, Constr.Approx., 11 (1995), 107-137.

Corollary 4.3-4.41.- For all sufficiently large n, each sufficiently small neighborhood of cj; j = 1, · · · ,N; contains exactly

one zero of Sn and the remaining n− N zeros lie on (−1, 1).

2.- For all sufficiently large n, the zeros of Sn are real and simples.

3.- The set of zeros of Sn∞n=1 is uniformly bounded.

Rational Approximation & Sobolev Orthogonality §4.- Sequentially-ordered Sobolev-type inner product hpijeira@math.uc3m.es 9 / 15

Non-sequentially-ordered Sobolev-type inner product

Example 1.

〈f , g〉 =∫ 1

−1f (x)g(x) dµ(x) + f ′′(2)g′′(2) + f ′(3)g′(3).

S5(x) =x5+

112826251995289

x4+

2022364101795760

x3+

285069001995289

x2

−43841375541901069

x−117588251995289

, whose zeros are

Example 2.

〈f , g〉 =∫ 1

−1f (x)g(x)(1− x) dx + f ′(3)g′(3) + f ′′(2)g′′(2).

S5(x) =x5+

5794314527312164

x4 −24223704513656082

x3 −52227758520484123

x2 −5321481540968246

x +22091264552141404

,

whose zeros are

Rational Approximation & Sobolev Orthogonality §4.- Sequentially-ordered Sobolev-type inner product hpijeira@math.uc3m.es 10 / 15

Markov’s theorem for Sobolev orthogonality

Extended Markov’s TheoremLet (6) be a discrete Sobolev inner product sequentially-ordered with µ ∈M(0, 1).Then, for k ∈ N,

R[k]n =

S[k]n (z)

Sn+k(z)⇒n→∞

µk(z) =

∫ 1

−1

Qk−1(x)

z− xdµρ(x),

K ⊂ C \ ([−1, 1] ∪ c1, c2, · · · , cN) .

We call µk the kth Markov-type function associated to µρ.

Corollary (estimate for the degree of convergence )

lim supn

∥∥∥µk − R[k]n

∥∥∥1/2n

K≤ ‖ϕ−1‖K < 1, where ‖f‖k = sup

z∈K|f |.

Rational Approximation & Sobolev Orthogonality §5.-Markov’s theorem for Sobolev orthogonality hpijeira@math.uc3m.es 11 / 15

Prologue of the proof and auxiliary lemmas

I Let ξn,in−Ni=1 be the n− N simple zeros of Sn on (−1, 1) for (n ≥ d + dN) and let

ξn,n−N+iNi=1 be the remaining N zeros of Sn.

I Sn(x) = Sn,1(x) Sn,2(x), where Sn,1(x) =n−N∏i=1

(x− ξn,i) and Sn,2(x) =N∏

i=1

(x− ξn,n−N+i).

I S+n,2(x) = (−1)νSn,2(x) where ν the number of cj greater than 1.

I If n is sufficiently large, S+n+k,2(x) ρ(x) > 0 for all x ∈ [−1, 1].

I For each k ∈ N,

R[k]n,1(z) =

S[k]n,1(z)

Sn+k,1(z). S[k]

n,1(z) =∫ 1

−1

Sn+k,1(z)− Sn+k,1(x)z− x

Qk−1(x) dµρ,n(x).

dµρ,n(x) = S+n+k,2(x) ρ(x) dµ(x). µk,n(z) =

∫ 1

−1

Qk−1(x)z− x

dµρ,n(x).

Rational Approximation & Sobolev Orthogonality §5.-Markov’s theorem for Sobolev orthogonality hpijeira@math.uc3m.es 12 / 15

Lemma 1 (Gauss-Jacobi type quadrature formula)

Let Sn and ξn,in−Ni=1 as above. If n > d then for every polynomial T with deg(T) ≤ 2n− d − N − 1∫ 1

−1T(x)S+n,2(x)dµρ(x) =

n−N∑i=1

λn,i S+n,2(ξn,i) T(ξn,i),

where λn,i =

∫ 1

−1

Sn(x)

S′n(ξn,i)(x− ξn,i)dµρ(x) (Christoffel-type number).

Moreover, the number of positive coefficients λn,i is greater than or equal to(

n− d+N2

).

Lemma 2 (estimating the remainder)

µk,n(z)− R[k]n,1(z) = S+n+k,2(z)

(µk(z)− R[k]

n (z))

= O(

1z2(n+1)+k−d−N

).

Lemma 3 (Fundamental Lemma)

For n sufficiently large R[k]n,1(z) =

n+k−N∑j=1

S+n+k,2(ξn+k,j) λn+k,j

(z− ξn+k,j).

Furthermore, R[k]n,1 is uniformly bounded on each compact subset K ⊂ Ω.

Rational Approximation & Sobolev Orthogonality §5.-Markov’s theorem for Sobolev orthogonality hpijeira@math.uc3m.es 13 / 15

Outline of the proof of Extended Markov’s Theorem

K ⊂ Ω∞ compact and `τ the level curve `τ = z ∈ C : |ϕ(z)| = τ, where τ > 1.

Since ϕ(K) is a compact set we can take τ sufficiently close to 1 such that 1 < τ < min |ϕ(K)|.

From Lemma 3 the sequences µk,n and R[k]n,1 are uniformly bounded on K. Then, there exist

a constant Cτ , independent of n, such that for all z ∈ `τ∣∣∣(µk,n(z)− R[k]n,1(z)

)ϕ2(n+1)+k−d−N(z)

∣∣∣ ≤ Cτ τ 2(n+1)+k−d−N . (8)

From Lemma 2 we have((µk,n(z)− R[k]

n,1(z))ϕ2(n+1)+k−d−N(z)

)∈ H(Ω∞).

Now, form the maximum modulus principle the bound (8) also holds on K.

Therefore supz∈K

∣∣∣µk,n(z)− R[k]n,1(z)

∣∣∣ ≤ Cτ

(τ

min |ϕ(K)|

)2(n+1)+k−d−N

−→n→∞

0,

which is equivalent to R[k]n,1(z) ⇒

n→∞µk,n(z) K ⊂ Ω∞.

From Corollary 4.3-4.4, there exist a constant C2 > 0, independent of n, such that C2 ≤ |Sn+k,2(z)|for all z ∈ K. Therefore, taking into account Lemma 2 we get

supz∈K

∣∣∣µk(z)− R[k]n (z)

∣∣∣ ≤ CτC2

(τ

min |ϕ(K)|

)2(n+1)+k−d−N

−→n→∞

0.

Rational Approximation & Sobolev Orthogonality §5.-Markov’s theorem for Sobolev orthogonality hpijeira@math.uc3m.es 14 / 15

Rational Approximation & Sobolev Orthogonality hpijeira@math.uc3m.es 15 / 15