Polynomial approximation of piecewise analytic functions on quasi-smooth arcs
PPolynomial approximation of piecewise analyticfunctions on quasi-smooth arcs
Liudmyla Kryvonos
Abstract
For a function f that is piecewise analytic on a quasi-smooth arc L and any < σ < we construct a sequence of polynomials that convergeat a rate e − n σ at each point of analyticity of f and are close to the bestpolynomial approximants on the whole L . Moreover, we give exampleswhen such polynomials can be constructed for σ = 1 . MSC:
Keywords:
Polynomial approximation, Quasi-smooth arcs, Near-best approxima-tion.
Let L be a quasi-smooth arc on the complex plane C , that is, for any z , ζ ∈ L the length |L ( z, ζ ) | of the subarc L ( z, ζ ) of L between points z , ζ satisfies |L ( z, ζ ) | (cid:54) c | z − ζ | for some c = c ( L ) (cid:62) .Consider a piecewise analytic function f on L belonging to C k ( L ) , k ≥ ,that means f is k times continuously differentiable on L and there exist points z , z , ..., z m − such that f is analytic on L\{ z , z , ..., z m } , ( z , z m – endpoints of L ), but is not analytic at points z , z , ..., z m . We call the z i points of singularityof f .The rate of the best uniform approximation of a function f by polynomialsof degree at most n ∈ N := { , , ... } is denoted by E n ( f ) = E n ( f, L ) := inf P n : degP n ≤ n (cid:107) f − P n (cid:107) L . (1 . Here (cid:107)· (cid:107) L means the supremum norm over L . Also, let p ∗ n ( f, z ) be the (unique)polynomial minimizing the uniform norm in (1.1).It is natural to expect the difference f ( z ) − p ∗ n ( z ) to converge faster at pointsof analyticity of f . But, it turns out, singularities of f adversely affect thebehavior over the whole L of a subsequence of the best polynomial approximants p ∗ n ( f, z ) . This so-called "principle of contamination" manifests itself in densityof extreme points of f − p ∗ n , discussed by A. Kroo (cid:48) and E.B. Saff in [8] and a r X i v : . [ m a t h . C V ] F e b ccumulation of zeros of p ∗ n ( f, z ) , showed by H.-P. Blatt and E.B. Saff in [6].For more details, we refer the reader to [10].Surprisingly, such behavior of zeros and extreme points need not hold forpolynomials of "near-best" approximation, that is for polynomials P n that sat-isfy (cid:107) f − P n (cid:107) L ≤ CE n ( f ) , n = 1 , , ..., with a fixed C > . Hence, it is natural to seek "near-best" polynomials whichwould converge faster at points z ∈ L\{ z , z , ..., z m } .For the case of L = [ − , and a piecewise analytic function f belongingto C k [ − , , E.B. Saff and V. Totik in [12] have proved that if non-negativenumbers α, β satisfy α < and β (cid:62) α or α = 1 and β > , then thereexist constants c , C > and polynomials P n , n = 1 , , ..., such that for every x ∈ [ − , | f ( x ) − P n ( x ) | (cid:54) CE n ( f ) e − cn α d ( x ) β , (1 . where d ( x ) denotes the distance from x to the nearest singularity of f in ( − , .Accordingly, the question of constructing "near-best" polynomials ariseswhen [ − , is replaced by an arbitrary quasi-smooth arc L in C . Polyno-mial approximation of functions on arcs is an important case of a more generalproblem of approximation of functions on an arbitrary continuum of the com-plex plane studied in the works of N.A. Shirokov [13], V.K. Dzjadyk and G.A.Alibekov [1], V.V. Andrievskii [3] and others (see, for example, [7]).The behavior of "near-best" polynomials is well studied in the case of ap-proximation on compact sets K with non-empty interior Int ( K ) . The followingresults demonstrate how the possible rate of convergence inside K depends onthe geometry of K . V.V. Maimeskul have proved in [9] that if Ω := C \ K satisfies the α -wedge condition with < α (cid:54) , then for any σ < α/ thereexist "near-best" polynomials converging at a rate e − n σ in the interior of K .E.B. Saff and V.Totik in [11] show the possibility of geometric convergence of"near-best" polynomials inside K if the boundary of K is an analytic curve.Meantime, N.A. Shirokov and V. Totik in [14] discuss the rate of approximationby "near best" polynomials of a function f given on a compact set K with ageneralized external angle smaller than π at some point z ∈ ∂K . They showedthat if f has a singularity at z , then geometric convergence inside K , where f is analytic, is impossible. Taking into account these results, the most interestingcase for us is when singularities of the function f occur at points where the anglebetween subarcs of L is different from π . It turns out that for some such arcsthere are no restrictions on the rate of convergence of "near-best" polynomialsand it can be geometric at points where f is analytic, as opposed to the resultfor compact sets with non-empty interior. We formulate and prove this assertionin Theorem 2. Furthermore, the general case is given by the following Theorem 1.
Let f be a piecewise analytic function on a quasi-smooth arc L , i.e.there exist points z , ..., z m − ∈ L , such that they divide L into L , L , ..., L m − and f ( z ) = f i ( z ) , z ∈ L i , i = 1 , m − , (1 . here f i ( z ) are analytic in some neighborhood of L i , respectively, and satisfy f ( r ) i − ( z i ) = f ( r ) i ( z i ) , f ( k i ) i − ( z i ) (cid:54) = f ( k i ) i ( z i ) (1 . for r = 0 , k i , i = 2 , m − . Then, for any < σ < , there exists a sequence { P n } ∞ of "near-best" polynomial approximants of f on L , such that lim n −→∞ (cid:107) f − P n (cid:107) E e n σ = 0 (1 . holds for any compact set E ⊂ L\{ z , ..., z m − } . On the complex plane consider lemniscates that are level lines of some com-plex polynomials. Namely, take P ( z ) = P N ( z ) := ( z − a )( z − a ) ... ( z − a N ) ,where a k = Re i π ( k − N , k = 1 , N and R > is a fixed number. Then | P ( z ) | = R N is an equation of a lemniscate. Note that the origin is a point of thislemniscate (since | P (0) | = R N ).The lemniscate divides the plane into three parts, namely the curve itself,points { z : | P ( z ) | < R N } and { z : | P ( z ) | > R N } . Consider an arc L = L (cid:48) ∪ L (cid:48)(cid:48) ,where L (cid:48) , L (cid:48)(cid:48) may belong to different petals of the lemniscate, meet at the originand satisfy | P ( z ) | < R N , z ∈ L \ { } . An example for N = 4 , R = 1 you cansee below.In particular, two line segments meeting at the origin at angle < ϕ (cid:54) π satisfy this property: if πm +1 < ϕ (cid:54) πm for some integer m , it is enough to take R to be sufficiently large and N = m . 3et f be a piecewise analytic function on L given by f ( z ) = (cid:40) f ( z ) , if z ∈ L (cid:48) f ( z ) , if z ∈ L (cid:48)(cid:48) where f , f are functions, analytic on L (cid:48) and L (cid:48)(cid:48) correspondingly, satisfying f ( r )1 (0) = f ( r )2 (0) , r = 0 , k, f ( k +1)1 (0) (cid:54) = f ( k +1)2 (0) . With these assumptions we prove the following result
Theorem 2.
Let L and f be as above. Then there exist a constant c > anda sequence of "near-best" polynomials { P n } ∞ , such that lim n −→∞ (cid:107) f − P n (cid:107) E e cnd ( E ) = 0 , where d ( E ) > for any compact set E ⊂ L \ { } . In this section we give some results which allow us to get estimates for the E n ( f ) and are needed for constructing "near-best" polynomials.For a > and b > we will use the notation a (cid:52) b if a (cid:54) cb , with someconstant c > . The expression a (cid:16) b means a (cid:52) b and b (cid:52) a .Let L be a quasi-smooth arc and Ω := C \L . Consider a conformal mapping Φ : Ω −→ ∆ := { ω : | ω | > } , normalized in such a way that Φ( ∞ ) = ∞ , Φ (cid:48) ( ∞ ) > , and denote Ψ := Φ − .By (cid:101) Ω we denote compactification of the domain Ω by prime ends in theCaratheodory sense, and (cid:101) L := (cid:101) Ω \ Ω . For the endpoints z , z of L and u > , j = 1 , , let Φ( z j ) := τ j ;∆ := { τ : τ ∈ ∆ , arg τ < arg τ < arg τ } ;∆ := ∆ \ ∆ , (cid:101) Ω j := Ψ(∆ j ) , Ω j := Ψ(∆ j ); (cid:101) L j := (cid:101) Ω j ∩ (cid:101) L ; L ju := { ζ : ζ ∈ (cid:101) Ω j , | Φ( ζ ) | = 1 + u } ; ρ ju ( z ) := dist ( z, L ju ); ρ ∗ u ( z ) := max j =1 , ρ ju ( z ) . Let z be a point of L , distinct from endpoints of the arc. Then point z divides L into two parts, L (cid:48) and L (cid:48)(cid:48) . Consider the function f ( z ) = (cid:40) f ( z ) , if z ∈ L (cid:48) f ( z ) , if z ∈ L (cid:48)(cid:48) (2 . f , f are functions, analytic on L (cid:48) and L (cid:48)(cid:48) , i.e. analytic in some neigh-borhoods of L (cid:48) and L (cid:48)(cid:48) correspondingly, and satisfying f ( r )1 ( z ) = f ( r )2 ( z ) , r = 0 , k, f ( k +1)1 ( z ) (cid:54) = f ( k +1)2 ( z ) . (2 . By U we will denote an open circular neighborhood of the point z , whereboth f , f are analytic.Let Z , Z ∈ (cid:101) L be the prime ends, s.t. | Z j | = z , j = 1 , . Set τ j := Φ( Z j ) , j = 1 , . Points τ j , j = 1 , we define by τ j = λτ j , with λ > such that Г , Г ⊂ U, where Г j = Г j := { ζ : 1 < | Φ( ζ ) | < λ, arg Φ( ζ ) = arg τ j } , j = 1 , . (2 . The arcs Г , Г are rectifiable (see [4, Chap. 5]), thus, can be oriented insuch a way that for all z ∈ L\{ z } function f can be represented, by the Cauchyformula, as f ( z ) = h ( z ) + h ( z ) , where h ( z ) = 12 πi (cid:90) Г ∪ Г f ( ζ ) − f ( ζ ) ζ − z dζ, (2 . and h ( z ) is analytic for all z ∈ L , therefore it can be approximated with ageometric rate on L .We will make use of the following lemma. Lemma 1.
Let L be a quasi-smooth arc. Then for any fixed non-negative integer k , a positive integer n and ζ ∈ Г ∪ Г there exists a polynomial kernel K n ( ζ, z ) ofthe form K n ( ζ, z ) = n (cid:88) j =0 a j ( ζ ) z j with continuous in ζ coefficients a j ( ζ ), j = 0 , n ,satisfying for z ∈ L and ζ with | ζ − z | ≥ ρ ∗ /n ( z ) | ζ − z − K n ( ζ, z ) | ≤ c [ ρ ∗ /n ( z )] k +2 | ζ − z | − ( k +3) , (2 . where c = c ( L ) > . Proof.
To show (2.5), we repeat word by word the proof for k = 0 , ([4, Lemma5.4]). 5et n be sufficiently large. For fixed m and r we consider the Dzyadykpolynomial kernel K ,m,r,n ( ζ, z ) (see, e.g., [4, Chap. 3]). Then, for r (cid:62) and z ∈ L , ζ ∈ Г j , j = 1 , , (cid:12)(cid:12)(cid:12)(cid:12) ζ − z − K ,m,r,n ( ζ, z ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:52) | ζ − z | (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) ζ − ζ (cid:101) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) rm where (cid:101) ζ := (cid:101) ζ j /n := Ψ[(1 + 1 /n )Φ( ζ )] .Since (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) ζ − ζ (cid:101) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) (cid:52) (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) ζ − ζζ − z (cid:12)(cid:12)(cid:12)(cid:12) (cid:52) (cid:12)(cid:12)(cid:12)(cid:12) ρ j /n ( z ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) c (cid:52) (cid:12)(cid:12)(cid:12)(cid:12) ρ ∗ /n ( z ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) c , it is enough to take r and m such that rmc (cid:62) k + 2 , and set K n ( ζ, z ) := K ,m,r, [ εn ] ( ζ, z ) , where ε = ε ( r, m ) > is sufficiently small.The next theorem is also a generalization of the case k = 0 in (2.2) and theproof essentially repeats the proof of [4, Theorem 5.2]. Theorem 3.
Let L be a quasi-smooth arc, and let function f be given by (2.1),(2.2). Then c (cid:48) [ ρ ∗ /n ( z )] k +1 ≤ E n ( f , L ) ≤ c (cid:48)(cid:48) [ ρ ∗ /n ( z )] k +1 , (2 . where c (cid:48) , c (cid:48)(cid:48) don’t depend on n. Proof.
First, we estimate E n ( f , L ) from above.Without loss of generality, we can assume z = 0 and n is sufficiently large.Let d n := ρ ∗ /n (0) , γ = γ n := { ζ : ζ ∈ Г ∪ Г , | ζ | ≥ d n } , P n = 12 πi (cid:90) γ ( f ( ζ ) − f ( ζ )) K n ( ζ, z ) dζ. From (2.2), for all ζ in some neighborhood U of the point z = 0 f ( ζ ) = c + c ζ + ... + c k ζ k + c k +1 ζ k +1 + ϕ ( ζ ) ζ k +2 (2 . f ( ζ ) = c + c ζ + ... + c k ζ k + (cid:101) c k +1 ζ k +1 + ϕ ( ζ ) ζ k +2 , (2 . where c k +1 (cid:54) = (cid:101) c k +1 and ϕ ( ζ ) , ϕ ( ζ ) are functions, analytic in U .Hence, there exists a constant C such that | f ( ζ ) − f ( ζ ) | ≤ C | ζ k +1 | , ζ ∈ U. (2 . By (2.5), (2.9), for all z ∈ L (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) Г ∪ Г f ( ζ ) − f ( ζ ) ζ − z dζ − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ π (cid:90) γ | f ( ζ ) − f ( ζ ) | (cid:12)(cid:12)(cid:12)(cid:12) ζ − z − K n ( z, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | dζ | + 12 π (cid:90) ( Г ∪ Г ) \ γ (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ ) − f ( ζ ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) | dζ | Cd k +2 n π (cid:90) γ | dζ || ζ | + C π (cid:90) ( Г ∪ Г ) \ γ | ζ k +1 || ζ − z | | dζ | . (2 . Integration by parts of (cid:82) γ | dζ || ζ | yields (cid:82) γ | dζ || ζ | (cid:52) d n . Since dist ( ζ, L ) (cid:16) | ζ | , (see[4, Chap. 5]), and | ( Г ∪ Г ) \ γ | (cid:52) d n , it implies (cid:82) ( Г ∪ Г ) \ γ | ζ k +1 || ζ − z | | dζ | (cid:52) d k +1 n .Thus, combining with (2.10), we obtain the estimate from above in (2.6).Now, we estimate E n ( f , L ) from below.Let p ∗ n be the polynomial of the best approximation, that is | f ( z ) − p ∗ n ( z ) | ≤ E n ( f ) , z ∈ L (2 . Without loss of generality we can assume that E n ( f ) ≤ d n = ρ /n (0) . Denote by l ⊂ Ω any arc of a circle { ζ : | ζ | = d n } , separating the primeend Z from ∞ .Let z (cid:48) ∈ L (cid:48) and z (cid:48)(cid:48) ∈ L (cid:48)(cid:48) be the endpoints of the arc l . Denote l := L (0 , z (cid:48) ) , l := L (0 , z (cid:48)(cid:48) ) . Next, take a point z so that z ∈ Г , | z | = εd n (we’ll choose the constant ε later). With a corresponding choice of orientation of arcs l j , j = 1 , , I := (cid:90) l ∪ l (cid:101) f ( ζ )( ζ − z ) k +2 dζ = (cid:90) l ∪ l (cid:101) f ( ζ ) − (cid:101) p ∗ n ( ζ )( ζ − z ) k +2 dζ + (cid:90) l (cid:101) p ∗ n ( ζ )( ζ − z ) k +2 dζ, (2 . where (cid:101) f ( ζ ) = f ( ζ ) − ( c + c ζ + ... + c k ζ k ) and (cid:101) p ∗ n ( ζ ) = p ∗ n − ( c + c ζ + ... + c k ζ k ) .Notice that f ( ζ ) − p ∗ n ( ζ ) = (cid:101) f ( ζ ) − (cid:101) p ∗ n ( ζ ) .In the following estimates we use notations a i , (cid:101) a i , (cid:101) C, (cid:98) C, C i for constants.For the left hand side we have (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) l ∪ l (cid:101) f ( ζ )( ζ − z ) k +2 dζ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) c k +1 (cid:90) l ζ k +1 ( ζ − z ) k +2 dζ + (cid:101) c k +1 (cid:90) l ζ k +1 ( ζ − z ) k +2 dζ + (cid:90) l ϕ ( ζ ) ζ k +2 ( ζ − z ) k +2 dζ + (cid:90) l ϕ ( ζ ) ζ k +2 ( ζ − z ) k +2 dζ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) c k +1 log zz − z (cid:48) + (cid:101) c k +1 log z − z (cid:48)(cid:48) z + a z (cid:48) k +1 + a z (cid:48) k z + ... + a k +1 z (cid:48) z k ( z (cid:48) − z ) k +1 + (cid:101) a z (cid:48)(cid:48) k +1 + (cid:101) a z (cid:48)(cid:48) k z + ... + (cid:101) a k +1 z (cid:48)(cid:48) z k ( z (cid:48)(cid:48) − z ) k +1 + (cid:101) C + (cid:90) l ϕ ( ζ ) ζ k +2 ( ζ − z ) k +2 dζ + (cid:90) l ϕ ( ζ ) ζ k +2 ( ζ − z ) k +2 dζ (cid:12)(cid:12)(cid:12)(cid:12) (cid:62) (cid:12)(cid:12)(cid:12)(cid:12) ( (cid:101) c k +1 − c k +1 ) log z − z (cid:48)(cid:48) z + c k +1 log z − z (cid:48)(cid:48) z − z (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) − C ε (1 − ε ) k +1 − (cid:98) C | (cid:101) c k +1 − c k +1 | log 1 − εε − C ε (1 − ε ) k +1 − C Next, we estimate the right hand side of (2.12). By (2.11) and by the choice of z (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) l ∪ l (cid:101) f ( ζ ) − (cid:101) p ∗ n ( ζ )( ζ − z ) k +2 dζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C E n ε k +1 d k +1 n . To estimate the integral over l notice that by (2.7) and (2.8) | (cid:101) f ( ζ ) | ≤ c | ζ k +1 | , ζ ∈ L for some constant c . Without loss of generality, we assume c = 1 (otherwise thearc l must be considered with a radius d n c instead). Since the estimate | (cid:101) p ∗ n ( ζ ) | ≤ | (cid:101) p ∗ n ( ζ ) − (cid:101) f ( ζ ) | + | (cid:101) f ( ζ ) | ≤ d k +1 n (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) ζd n (cid:12)(cid:12)(cid:12)(cid:12) k +1 (cid:19) , ζ ∈ L holds, [4, Theorem 6.1] implies | p ∗ n ( ζ ) | ≤ C d k +1 n , ζ ∈ l . The last inequality yields (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) l (cid:101) p ∗ n ( ζ )( ζ − z ) k +2 dζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ πC (1 − ε ) k +2 . Combining the estimates above, for some small but fixed ε we get C E n ε k +1 d k +1 n (cid:62) | (cid:101) c k +1 − c k +1 | log 1 − εε − C ε (1 − ε ) k +1 − C − πC (1 − ε ) k +2 (cid:62) | (cid:101) c k +1 − c k +1 | − εε . Consequently, the estimate from below in (2.6) holds.With reasoning completely similar, we obtain the following.
Theorem 4.
Let L be a quasi-smooth arc, and let function f be given by (1.3),(1.4). Then c (cid:48) [ ρ ∗ /n ( z )] k +1 ≤ E n ( f , L ) ≤ c (cid:48)(cid:48) [ ρ ∗ /n ( z )] k +1 , where k := min i =2 ,m − { k i } and c (cid:48) , c (cid:48)(cid:48) don’t depend on n. Proof of Theorem 1
As it was mentioned above, f can be represented as f ( z ) = m − (cid:88) j =2 ( h j ( z ) + h j ( z )) , where h j ( z ) are analytic functions on L and h j ( z ) = 12 πi (cid:90) Г j ∪ Г j f j − ( ζ ) − f j ( ζ ) ζ − z dζ, with Г j , Г j being the arcs given by (2.3), that correspond to the point z j .Therefore, it’s enough to construct polynomial approximants for h j ( z ) only.To approximate the integral over Г ij , i = 1 , , consider a function F ij : L ∪ Г ij −→ L iϕ,j , such that F ij is one-to-one and satisfies | F ij ( z ) − F ij ( ζ ) | (cid:54) c | z − ζ | , z, ζ ∈ L ∪ Г ij ,F ij ( z j ) = 0 ,F ij ( L ( z , z j )) = L (cid:48) ,F ij ( L ( z j , z m )) = L (cid:48)(cid:48) ,F ij ( Г ij ) = (cid:101) Г , where z , z m are endpoints of L , L (cid:48) is a line segment in [0 , ∞ ) , L (cid:48)(cid:48) is a linesegment in the upper half plane that form an anle ϕ > with L (cid:48) , (this anglewill be determined below), and (cid:101) Г – a line segment at an angle ϕ to the L (cid:48) .Such a mapping F ij ∈ Lip [ L ∪ Г ij ] always exists, and to see this it is enoughto note that L and Г ij are quasi-smooth and dist ( ζ, L ) (cid:16) | ζ − z j | holds for all ζ ∈ Г ij (see [4, Chap. 5]).By [2, Theorem 4] the function F ij can be approximated by polynomials Q n ( z ) := Q in,j ( z ) with the rate n α , for some α > , that is | F ij ( z ) − Q n ( z ) | (cid:54) Cn α , z ∈ L ∪ Г ij , (3 . where constant C does not depend on z and n .For fixed < σ < take an integer k (cid:62) , such that − σ > kα . Now,for ϕ = πk consider corresponding mapping F ij and approximating polynomials Q n . 9et (cid:98) P in,j ( z, ζ ) = 1 − (cid:18) Q k [ nβ ] ( z ) − ζ Q k [ nβ ] ( ζ ) − ζ (cid:19)(cid:2) n − β k (cid:3) ζ − z + (cid:32) Q k [ n β ] ( z ) − ζ Q k [ n β ] ( ζ ) − ζ (cid:33)(cid:2) n − β k (cid:3) K [ n ] ( z, ζ ) . It is not hard to see that (cid:98) P in,j ( z, ζ ) is a polynomial in z of degree at most n .The idea of constructing such a polynomial is motivated by [5, p. 380].We will show that for some choice of β and ζ the term (cid:12)(cid:12)(cid:12)(cid:12) Q k [ nβ ] ( z ) − ζ Q k [ nβ ] ( ζ ) − ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) n − β k (cid:3) is bounded uniformly on L by a constant that does not depend on n , and atpoints of analyticity of f it can be bounded by q (cid:2) n − β k (cid:3) , for some q < .For n sufficiently large the arc L can be written as a disjoint union L = A ∪ A ∪ A , where A := { z ∈ L : | Q n ( z ) | < Cn α sin π k } , (3 . A := { z ∈ L : | Q n ( z ) | (cid:62) Cn α sin π k , dist ( Q n ( z ) , L (cid:48) ) (cid:54) Cn α } , (3 . A := { z ∈ L : | Q n ( z ) | (cid:62) Cn α sin π k , dist ( Q n ( z ) , L (cid:48)(cid:48) ) (cid:54) Cn α } , (3 . C is the constant from (3.1).Points of A satisfy | Q kn ( z ) | < C k n αk ( sin π k ) k . (3 . For A we have | sin ( arg Q n ( z )) | (cid:54) Cn α | Q n ( z ) | (cid:54) sin π k , that implies − π (cid:54) arg Q kn ( z ) (cid:54) π . (3 . Similarly, for A | sin ( 2 πk − arg Q n ( z )) | (cid:54) Cn α | Q n ( z ) | (cid:54) sin π k , that yields − π (cid:54) arg Q kn ( z ) (cid:54) π . (3 . Г ij can also be written as a disjoint unionГ ij = B ∪ B , where B := { ζ ∈ Г ij : | Q n ( ζ ) | < Cn α sin π k } , (3 . B := { ζ ∈ Г ij : | Q n ( ζ ) | (cid:62) Cn α sin π k } . (3 . Points of B satisfy | Q kn ( ζ ) | < C k n αk ( sin π k ) k . (3 . For B we have | sin ( πk − arg Q n ( ζ )) | (cid:54) Cn α | Q n ( ζ ) | (cid:54) sin π k ,π − π (cid:54) arg Q kn ( ζ ) (cid:54) π + π . (3 . Now, if we choose ζ to be a point in (0 , ∞ ) with ζ > max {|L (cid:48) | k , |L (cid:48)(cid:48) | k } ,then (3.5), (3.6) and (3.7) imply | Q kn ( z ) − ζ | (cid:54) ζ + C k n αk ( sin π k ) k , z ∈ L . Also, by(3.10) and (3.11) the estimate | Q kn ( ζ ) − ζ | (cid:62) ζ − C k n αk ( sin π k ) k holds for ζ ∈ Г ij .According to these observations, we have (cid:12)(cid:12)(cid:12)(cid:12) Q k [ n β ] ( z ) − ζ Q k [ n β ] ( ζ ) − ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) n − β k (cid:3) (cid:54) (cid:18) (cid:101) Cn αβk (cid:19)(cid:2) n − β k (cid:3) (3 . (cid:101) C = C k ζ n αβk ( sin π k ) k − C k (cid:54) C k for n large enough.Let β be such that − σ > β > kα , so that − β < αβk and σ < − β .From (3.12) it follows (cid:12)(cid:12)(cid:12)(cid:12) Q k [ n β ] ( z ) − ζ Q k [ n β ] ( ζ ) − ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) n − β k (cid:3) (cid:54) e (cid:101) cn − αβk n − β (cid:54) (cid:98) C, (3 . where (cid:98) C does not depend on n .Also, for all points z of a compact set E ⊂ L\{ z , z , ..., z m } and n sufficientlylarge the estimate (cid:12)(cid:12)(cid:12)(cid:12) Q k [ n β ] ( z ) − ζ Q k [ n β ] ( ζ ) − ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) n − β k (cid:3) (cid:54) q (cid:2) n − β k (cid:3) , (3 . holds with some q = q ( E ) < .Therefore, if we denote d n := ρ ∗ /n ( z j ) , γ = γ n := { ζ : ζ ∈ Г ij , | ζ − z j | ≥ d n } and consider polynomial P in,j ( z ) = 12 πi (cid:90) γ ( f j − ( ζ ) − f j ( ζ )) (cid:98) P in,j ( z, ζ ) dζ + 12 πi (cid:90) Г ij \ γ ( f j − ( ζ ) − f j ( ζ )) − (cid:18) Q k [ nβ ] ( z ) − ζ Q k [ nβ ] ( ζ ) − ζ (cid:19)(cid:2) n − β k (cid:3) ζ − z dζ, by (3.13) and Theorem 4, for all z ∈ L we get (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) Г ij f j − ( ζ ) − f j ( ζ ) ζ − z dζ − P in,j ( z ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) π (cid:90) γ | f j − ( ζ ) − f j ( ζ ) | (cid:12)(cid:12)(cid:12)(cid:12) Q k [ n β ] ( z ) − ζ Q k [ n β ] ( ζ ) − ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) n − β k (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) ζ − z − K [ n ] ( z, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | dζ | + 12 π (cid:90) Г ij \ γ (cid:12)(cid:12)(cid:12)(cid:12) f j − ( ζ ) − f j ( ζ ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q k [ n β ] ( z ) − ζ Q k [ n β ] ( ζ ) − ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) n − β k (cid:3) | dζ | (cid:52) d k j +2 n (cid:90) γ | dζ || ζ | + (cid:90) Г ij \ γ (cid:12)(cid:12)(cid:12)(cid:12) ζ k j +1 ζ − z (cid:12)(cid:12)(cid:12)(cid:12) | dζ | (cid:52) E n ( f, L ) , (3 . where the last inequality follows by the reasoning, similar to the one we use in(2.10).If z ∈ E , by (3.14), (2.10) and Theorem 4 we have (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) Г ij f j − ( ζ ) − f j ( ζ ) ζ − z dζ − P in,j ( z ) (cid:12)(cid:12)(cid:12)(cid:12) q (cid:2) n − β k (cid:3) π (cid:90) γ | f j − ( ζ ) − f j ( ζ ) | (cid:12)(cid:12)(cid:12)(cid:12) ζ − z − K [ n ] ( z, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | dζ | + q (cid:2) n − β k (cid:3) π (cid:90) Г ij \ γ (cid:12)(cid:12)(cid:12)(cid:12) f j − ( ζ ) − f j ( ζ ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) | dζ | (cid:52) E n ( f, L ) q (cid:2) n − β k (cid:3) (cid:52) E n ( f, L ) e − (cid:101) cn − β . (3 . Let P n ( z ) = Σ m − j =2 ( P n,j ( z ) + P n,j ( z )) .By (3.15), (3.16), polynomials { P n } are "near best" polynomials, approxi-mating (cid:80) m − j =2 h j ( z ) and satisfying (1.5). Since changing the R corresponds to scaling the lemniscate, we can always scalethe picture and without loss of generality assume for simplicity R = 1 .As it was shown above, it’s enough to approximate the function h ( z ) = 12 πi (cid:90) Г ∪ Г f ( ζ ) − f ( ζ ) ζ − z dζ. Here Г and Г we choose in such a way that | P ( ζ ) | > for all ζ ∈ ( Г ∪ Г ) \{ } .While the image of Г ∪ Г under the mapping P belongs to the complement ofthe unit disc, the image of L is inside the disc, that yields (cid:12)(cid:12)(cid:12)(cid:12) P ( z ) P ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) , z ∈ L , ζ ∈ Г ∪ Г (4 . Moreover, due to geometry of L the equality in (4.1) occurs only if ζ = z = 0 .Let (cid:98) P n ( z, ζ ) = 1 − (cid:16) P ( z ) P ( ζ ) (cid:17) [ n N ] ζ − z + (cid:18) P ( z ) P ( ζ ) (cid:19) [ n N ] K [ n ] ( z, ζ ) . (4 . One may check that (cid:98) P n ( z, ζ ) is a polynomial in z of degree at most n .Let d n := ρ ∗ /n (0) , γ = γ n := { ζ : ζ ∈ Г ∪ Г , | ζ | ≥ d n } , and consider P n ( z ) = 12 πi (cid:90) γ ( f ( ζ ) − f ( ζ )) (cid:98) P n ( z, ζ ) dζ + 12 πi (cid:90) ( Г ∪ Г ) \ γ ( f ( ζ ) − f ( ζ )) − (cid:16) P ( z ) P ( ζ ) (cid:17) [ n N ] ζ − z dζ. By virtue of Theorem 3, estimates (2.10) and (4.1), for all z ∈ L (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) Г ∪ Г f ( ζ ) − f ( ζ ) ζ − z dζ − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) π (cid:90) γ | f ( ζ ) − f ( ζ ) | (cid:12)(cid:12)(cid:12)(cid:12) P ( z ) P ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) [ n N ] (cid:12)(cid:12)(cid:12)(cid:12) ζ − z − K [ n ] ( z, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | dζ | + 12 π (cid:90) ( Г ∪ Г ) \ γ (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ ) − f ( ζ ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ( z ) P ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) [ n N ] | dζ |≤ π (cid:90) γ | f ( ζ ) − f ( ζ ) | (cid:12)(cid:12)(cid:12)(cid:12) ζ − z − K [ n ] ( z, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | dζ | + 12 π (cid:90) ( Г ∪ Г ) \ γ (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ ) − f ( ζ ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) | dζ | (cid:52) d k +2 n (cid:90) γ | dζ || ζ | + (cid:90) ( Г ∪ Г ) \ γ (cid:12)(cid:12)(cid:12)(cid:12) ζ k +1 ζ − z (cid:12)(cid:12)(cid:12)(cid:12) | dζ |(cid:22) [ ρ ∗ /n (0)] k +1 (cid:52) E n ( f, L ) . If E is a compact set in L \ { z , , z } , then for all z ∈ E | P ( z ) | < q, (4 . for some q = q ( E ) < .Let d ( E ) := min z ∈ E { − | P ( z ) |} . By (4.3), d ( E ) > for any compact set E ⊂ L \ { z , , z } .Therefore, for all z ∈ E (cid:12)(cid:12)(cid:12)(cid:12) P ( z ) P ( ζ ) (cid:12)(cid:12)(cid:12)(cid:12) [ n N ] ≤ | P ( z ) | [ n N ] ≤ | − d ( E ) | [ n N ] ≤ e − cnd ( E ) , where the constant c > does not depend on n and E .Hence, for z ∈ E (cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) Г ∪ Г f ( ζ ) − f ( ζ ) ζ − z dζ − P n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − cnd ( E ) π (cid:90) γ | f ( ζ ) − f ( ζ ) | (cid:12)(cid:12)(cid:12)(cid:12) ζ − z − K [ n ] ( z, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) | dζ | + e − cnd ( E ) π (cid:90) ( Г ∪ Г ) \ γ (cid:12)(cid:12)(cid:12)(cid:12) f ( ζ ) − f ( ζ ) ζ − z (cid:12)(cid:12)(cid:12)(cid:12) | dζ | (cid:52) [ ρ ∗ /n (0)] k +1 e − cnd ( E ) (cid:52) E n ( f, L ) e − cnd ( E ) . Acknowledgment
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