Sobolev-Type Extremal Polynomials

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1 Soolev-Type Extremal Polyomials (case of the sigular measures) Ael Díaz-Gozález Joi with Héctor Pijeira-Carera ULL 2016 Soolev-Type Extremal Polyomials 1 / 18

2 Moic extremal polyomials µ 0, µ 1,, µ N fiite positive Borel measures o the real lie. k supp pµ kq, for k 0,, N. 0 cotais ifiitely may poits. All polyomials are itegrale with respect to µ k, for k 0,, N. If Q P P ad 1 ă p ă 8, we defie 1 d Nÿ }Q} S,p Q pkq p p ż, where }Q} k,p }Q} Lppµkq p Q p dµ k. k,p L p, is the th moic extremal polyomials relative to } } S,p if }L p,} S,p if Q pxq x ` }Q}S,p dpx, P 1q. ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 2 / 18

3 Uiqueess Theorem Let } } S,p e a Soolev type orm, the there exists a uique moic extremal polyomial of degree, L p, ˆ}L p,} S,p mi }Q} S,p. Q pxq x ` (1) L p, ad rl p, moic extremal polyomials of degree, the L 1 2,p ` rl,p is a moic extremal polyomials too. (2) }L p, ` rl p,} S,p 1 2 }Lp, ` rl p,} S,p ` 1 2 }Lp, ` rl p,} S,p }L p,} S,p ` }rl p,} S,p Nÿ (3) Lp, pkq ` rl p, pkq p ÿ N ˆ L pkq p, ` r L pkq p, k,p k,p k,p p 1 «1 Nÿ S,p L p, ` rl p, L pkq p, ` rl pkq p, k,p p p ÿ N ˆ L pkq ď p, ` L rpkq p k,p p, k,p pff 1 1 Nÿ ď L pkq p, k,p p p Nÿ ` L rpkq p, p p }L p,} S,p ` }rl p,} S,p k,p (4) }L p, ` rl p,} 0,p }L p,} 0,p ` }rl p,} 0,p (5) L p, αrl p, µ 0 a.e ñ L p, αrl p, ñ α 1 ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 3 / 18

4 Characterizatio p, q p : P ˆ P Ñ R Nÿ ż pf, gq p }f } p S,p pf, f q p pkq ˇˇˇg " y f pkq pkqˇˇˇp 1, if y 0; sg g dµk, where sg pyq y 0, if y 0. Theorem (uless p 2, p, q p does ot defie a ier product) L,p is the th moic extremal polyomial relative to } } S,p if ad oly if pq, L,pq p P P 1. Corollary 1 For ě 1, L,p has at least oe zero of odd multiplicity (or poit of chages of sig) i C h p 0q. Corollary 2 For ě 2, L 1,p has at least oe zero of odd multiplicity i C h p 0 Y 1q ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 4 / 18

5 Proof of the Characterizatio Theorem rñs Let L,p e the th moic extremal polyomial relative to } } S,p. (1) Let Q P P 1, the }L,p } S,p ď }L,p ` α Q} P R., (2) Fpαq }L,p ` αq} p S,p Nÿ ż ˇˇˇL pkq,p ` αq pkqˇˇˇp dµk. (3) From the Uiqueess Theorem ad (2), α 0 is the uique miimum poit of F ad 0 F 1 p0q p Nÿ ż Q pkq sg L pkq,p pkqˇ ˇˇˇL ˇp 1 dµ k ppq, L,p q p.,p ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 5 / 18

6 Proof of the Characterizatio Theorem rðs Let L,p e a moic polyomial such that dgr pl,pq ad pq, L,pq p P P 1. (1) Let rq e a moic polyomial of degree ad Q rq L,p P P 1. (2) Cosider G k sg L,p ˇˇˇL pkq pkqˇ ˇp 1,p (3) k 0, 1,, N ş Gk q dµ k }L,p} p k,p q where q is the cojugate expoet of p p. p 1 Nÿ Nÿ ż Nÿ (4) }L,p} p S,p }L,p} p k,p G k q dµ k }G k } q k,q. (5) α β ď αp p ` βq, α, β ě 0, p ą 1 q (6) }L,p} S,p ď }rq} S,p }L,p} p S,p pl,p, L,pq p pl,p ` Q, L,pq p prq, L,pq p ď Nÿ Nÿ }rq pkq }rq pkq } p k,p } k,p }G k } k,q ď p Nÿ ż rq pkq G k dµ k, ` }G k} q k,q q } rq} p S,p p ` }L,p}p S,p. q ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 6 / 18

7 Selected topics o Logarithmic Potetial Theory Defiitio (µ P Reg) We say that a fiite orel measure o C with compact support K, elog to class Reg, the class of regular measures if }Q,p } µ,p cap pkq, where lim } } µ,p the usual L p pµq orm ( 1 ă p ă 8). Q,p is th moic extremal polyomial with respect to } } µ,p. cap pkq the logarithmic capacity of K. ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 7 / 18

8 Asymptotic distriutio of critical poits ˆ ż ż lim λ λ lim fdλ fdλ for all f cotiuous ad ouded. If Q is a polyomial with degree ad zeros z 1,, z the υpq q 1 ÿ δ zj, is the ormalized zeros coutig measure of Q. j 1 µ K e the equilirium measure o K (compact suset of C) Theorem Let }Q} S,p `}Q} p µ 1,p ` }Q 1 } p µ 1,p piq µ 0, µ 1 P Reg 1 p ad deote 0 Y 1. If piiq For i 0, 1, i Ă R is a regular compact w.r.t. the Dirichlet prolem i Cz i. the lim υ `L,p 1 µ, ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 8 / 18

9 Selected topics o Logarithmic Potetial theory K is a compact set of R regular with respect to the Dirichlet prolem i CzK. µ a fiite positive Borel measures supported o K } } K the supremum orm o K. tq u a sequece of moic polyomials such that dgr pq q. d }Q } K Lemma LPT 1. µ P Reg ô lim 1. }Q } µ,p Lemma LPT 2. lim sup Lemma LPT 3. }T } K Lemma LPT 4. lim sup d }Q 1 } K }Q } K ď 1. a if }Q}K ùñ lim }T} K cap pkq. Q pxq x`... a }Q} K cap pkq ùñ lim υpqq µk. ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 9 / 18

10 Proof of asymptotic distriutio of critical poits Let T e the th moic Cheyshev polyomial for the compact suset, d d }L,p} S,p p (1) lim sup ď lim sup µ 0 p 0 q ` µ 1 p 1 q }T1 } p }T } }T } p ď 1. (2) lim sup (3) lim sup }L,p} p S,p ď }T}p S,p }T}p 0,p ` }T1 }p 1,p, ď µ 0 p 0 q }T } p 0 ` µ 1 p 1 q }T 1 }p 1 }L,p} S,p ď cap p q ď µ 0 p 0 q }T } p ` µ 1 p 1 q }T 1 }p. }L,p} 1 i ď cap p q, i 0, 1. lim sup lim sup (4) cap p q lim sup (5) lim sup }L,p 1 }p ď lim sup }L,p} p 0 ď lim sup }L,p} p 0 0,p ď lim sup }L,p} p S,p ď cap p qp. }L,p 1 }p ď lim sup }L 1 1,p }p 1,p ď lim sup }L,p} p S,p ď cap p qp. }T } ď lim sup d 1 ` 1 }L1 `1,p } ď lim sup }L,p} 1 ď cap p q. }L,p} 1 cap p q, ad the result it follows from Lemma LPT 4. ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 10 / 18

11 Case of two sigular measures µ 0, µ 1 such that pa, q C h p 0 q, pc, dq C h p 1 q ad pa, q X pc, dq H. a,, c, d P R N s pq; Iq umer of zeros of Q o I with odd multiplicity (poits of chages of sig). N z pq; Iq umer of zeros(coutig multiplicities) of Q o I. From Rolle s Theorem Rolle s Lemma for polyomials If Q 1 ı 0 the N z pq; Iq ` N z pq 1 ; CzIq ď dgr pqq. ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 11 / 18

12 Locatio of critical poits Theorem If pa, q X pc, dq H, the we have 1.- If ě 1. the 1 ď N s pl,p ; pa, qq ` N s pl,p; 1 pc, dqq ď. 2.- For all ě 2 the critical poits of L,p are simple, cotaied i pa, dq. Proof. Deote l N s pl,p ; pa, qq ` N s pl 1,p; pc, dqq p ď cq (1) For 1 (1.) is trivial. ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 12 / 18

13 Scheme of the proof (critical poits localizatio) (2) Let ě 2 ad let x 0 e the poit i pa, q closest to rc, ds where L,p chages sig. (3) There are two possiilities, either L 1,p px` 0 q L1,p pc`q ą 0 (I) or L 1,p px` 0 q L1,p pc`q ă 0 (II) where x` 0 x 0 ` ɛ for all sufficietly small ɛ ą 0. (4) If l ď 1 i case (I) ad l ď 2 i case (II) we ca foud a polyomial Q P P 1 such that QpxqL,ppxq ě P ra, s ad Q 1 pxql 1,ppxq ě P rc, ds. (S) the ż ż d 0 pq, L,pq p Q sg pl,pq L,p p 1 dµ 0 ` Q 1 sg `L,p 1 L 1,p p 1 dµ 1 ą 0, a c which is a cotradictio ad (1.) would e proved with l i case (I). (5) l 1 N spl.p; pa, x 0 qq ` N spl.p 1 ; pc, dqq ď NspL1.p ; pa, x 0qq ` N spl.p 1 ; pc, dqq ď 1 ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 13 / 18

14 Case I (L 1,ppc`q L 1,ppx`0 q 1) Suppose that l ď 1 ad take Q e a polyomial such that: Thus dgr pqq ď l (Q ı 0) Q has a zero at each poit of pa, q where L,p chages sig. Q 1 has a zero at each poit of pc, dq where L 1,p chages sig. (1) l ď N zpq; pa, qq ` N zpq 1 ; pc, dqq ď dgr pqq ď l. (2) l N spq; pa, qq ` N spq 1 ; pc, dqq dgr pqq. (3) QL,p ě 0 o ra, s ad `Q 1 L 1,p ď 0 or Q 1 L 1,p ě 0 o rc, ds (4) l 1 N spq; pa, x 0qq ` N spq 1 ; pc, dqq ď N spq 1 ; pa, x 0qq ` N spq 1 ; pc, dqq ď l 1 (5) sg `Q 1 pc`q sg `Q 1 px` 0 q 1 (6) sg `Q 1 pc`q sg `L 1,ppc`q 1 sg `Q 1 pc`q sg `L 1,ppc`q sg `Q 1 px` 0 q sg `L 1,ppx` 0 q sg `Qpx` 0 q sg `L,ppx` 0 q 1 ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 14 / 18

15 Case II (L 1,ppc`q L 1,ppx`0 q 1) Suppose that l ď 2 ad take Q e a polyomial such that: Thus dgr pqq ď l ` 1 (Q ı 0) Q has a zero at each poit of pa, q where L,p chages sig. Q 1 has a zero at each poit of pc, dq where L,p 1 chages sig. Q 1 pcq 0 (1) l ` 1 ď N zpq; pa, qq ` N zpq 1 ; rc, dqq ď dgr pqq ď l ` 1. (2) l ` 1 N spq; pa, qq ` N spq 1 ; rc, dqq dgr pqq. (3) QL,p ě 0 o ra, s ad `Q 1 L,p 1 ď 0 or Q1 L,p 1 ě 0 o rc, ds (4) l N spq; pa, x 0 qq ` N spq 1 ; rc, dqq ď N spq 1 ; pa, x 0 qq ` N spq 1 ; rc, dqq ď l (5) sg `Q 1 pc`q q sg Q 1 px` 0 1 (6) sg `Q 1 pc`q sg `L 1,p pc`q 1 sg `Q 1 pc`q sg `L 1,ppc`q sg sg sg q Q 1 px` 0 q Qpx` 0 sg L 1,ppx` 0 q L,ppx` 0 q 1 ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 15 / 18

16 Theorem of Bieracki Theorem Let K e a compact covex set cotaiig all the critical poits of a polyomial f, ad let f pz q 0. The the zeros of f lie i the uio of all the closed discs with ceters α at the extreme poits of K ad radius ρ α z α. ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 16 / 18

17 Theorem of Bieracki Corollary If pa, q X pc, dq H, the the zeros of L,p lie i # Bpd, d aq Y Bpa, aq, if ă c Bpc, cq Y Bp, aq, if d ă a. where Bpa, rq tz P C : z a ă ru ULL 2016 Soolev-Type Extremal Polyomials adiazgo@math.uc3m.es 17 / 18

18 Thak you very much for your kid attetio ULL 2016 Soolev-Type Extremal Polyomials 18 / 18