Not all limit points of poles of the Pad´e approximants are obstructions for poinwise convergence * Victor M. Adukov 5 0 Department of Differential Equations and Dynamical Systems, 0 Southern Ural State University, Lenin avenue 76, 454080 Chelyabinsk, 2 Russia n a E-mail address: [email protected] J 9 Abstract 1 In the work it is shown that not all limit points of poles of the Pad´e approx- ] imants for the last intermediate row are obstructions for pointwise convergence A of the whole row to an approximable function. The corresponding examples are N constructed. . h t The notion of intermediate rows was introduced in the work [2], where a m sufficient conditions for the convergence of the whole intermediate row were [ obtained. We will use the explicit description of the set of limit points of 2 poles of the Pad´e approximants for the last intermediate row obtained in [1]. v The paper is the continuation of the works [1]. 9 We will construct the examples in the class of rational functions. Let 1 1 r(z) = N(z) be a strictly proper rational fraction. Here N(z),D(z) are co- 1 D(z) prime polynomials, degD(z) = λ, and all poles of r(z) lie in the disk |z| < R. 0 5 Let (V (z),U (z)) be the unique solution of the Bezout equation k k 0 h/ N(z)Vk(z)+D(z)Uk(z) = zk, k ≥ 0, t a where the polynomial V (z) satisfy the inequality degV (z) < λ. This solu- m k k tion is called the minimal solution of the Bezout equation. It is easily seen : v that for n ≥ 0 the polynomials V (z),−U (z) are the denominator and i n+λ n+λ X numerator of the Pad´e approximants πr (z) of type (n,λ − 1) for r(z), n,λ−1 r respectively: a U (z) πr (z) = − n+λ n,λ−1 V (z) n+λ (see [1], Theorem 4.1). For V (z) there is the following explicit formula: k λ i−1 V (z) = d v zj k i k+i−j−1 i=1j=0 XX *This work was supported by Russian Foundation for Basic Research under Grant # 04-01-96006. (see [1], formula (4.8)). Here D(z) = zλ +d zλ−1 +···+d and v is the λ−1 0 k coefficient of zλ−1 in the polynomial V (z). k Wewill assume that all roots z ,...,z of the polynomial D(z) are simple. 1 λ Then v is found as follows k v = C zk +···+C zk k 1 1 λ λ (see [1], formula (4.10)), the numbers C are defined in this case by the j formula 1 C = , j D2(z )A j j j where D (z) = D(z) and A is a residue of r(z) at z (see [1], formula (4.14)). j z−zj j j Substituing v into V (z), we obtain k k λ V (z) = C zkD (z). (0.1) k j j j j=1 X We will also suppose that the dominant poles z ,...,z , 1 < ν < λ−1, 1 ν are vertices of a regular σ-gon, σ ≥ ν, and ρ ≡ |z | = ... = |z | > |z | > 1 ν ν+1 |z | ≥ ... ≥ |z |. ν+2 λ Let us introduce the following family of polynomials: ν ω (z) = C zm∆ (z), m = 0,1,...,σ −1, m j j k j=1 X where ∆(z) = (z −z )···(z −z ), ∆ (z) = ∆(z). 1 ν k z−z k Let n+λ ≡ m(modσ). If we divide the sum in (0.1) by two sums over the dominant and nondominant poles, we get λ C zn+λ V (z) = (z −z )···(z −z )zn+λ−mω (z)+D(z) j j . (0.2) n+λ ν+1 λ 1 m z −z j=ν+1 j X Here we take into account the equality zn+λ−m = ··· = zn+λ−m for n+λ ≡ 1 ν m(modσ). From the representation (0.2) we see that for n → ∞, n ∈ Λ ≡ m n ∈ N n+λ ≡ m(modσ) there exists n (cid:12) o (cid:12) (cid:12) limz−(n+λ−m)V (z) = (z −z )···(z −z )ω (z). 1 n+λ ν+1 λ m 2 This means that the set of limit points of poles of the Pad´e approximants πr (z) consists of the poles z ,...,z and the additional limit n,λ−1 n∈Λm ν+1 λ npoints thaot are the roots of the polynomials ω (z), m = 0,1,...,σ − 1 m (see also Theorem 2.7 in [1]). The additional limit points are different from the dominant poles z ,...,z (see Proposition 2.1 in [1]), but they can be 1 ν coincided with the poles z ,...,z . The corresponding example will be ν+1 λ given below. It should be noted that every additional limit point ζ generates 0 the sequence of defects of the Pad´e approximants πr (z), i.e. the sequence n,λ−1 of their zeros and poles that converge to ζ . 0 Theorem 1. Let ζ be a zero of one of polynomials ω (z), m = 0,1,..., 0 m σ −1, and ζ does not coincide with the poles z ,...,z . 0 ν+1 λ If |ζ | < |z |, then at the point ζ the whole sequence πr (z) converges 0 ν+1 0 n,λ−1 to r(z). If |ζ | ≥ |z |, then limπr (ζ ) does not exist and also for |ζ | > |z | 0 ν+1 n,λ−1 0 0 ν+1 lim πr (ζ ) = ∞, n ∈ Λ . n→∞ n,λ−1 0 m Proof. If ω (ζ ) 6= 0, then by Theorem 2.3 from [1] the subsequence m 0 πr (z) uniformly converges to r(z) in a neighborhood of ζ . It n,λ−1 n∈Λm 0 nremains tooconsider those sequences Λ for which ω (ζ ) = 0. In this case m m 0 for n ∈ Λ we have by formula (0.2) m λ C zn+λ j j V (ζ ) = D(ζ ) . n+λ 0 0 ζ −z j=ν+1 0 j X Taking into account our assumption |z | > |z | ≥ ... ≥ |z |, we ν+1 ν+2 λ obtain C V (ζ ) = zn+λD(ζ ) ν+1 +o(1) . (0.3) n+λ 0 ν+1 0 "ζ0 −zν+1 # −(n+λ) Hence, for all sufficiently large n ∈ Λ the polynomial z V (z) is m ν+1 n+λ bounded away from zero in a neighborhood of ζ . From the Bezout equation 0 it follows zn+λ r(z)−πr (z) = . (0.4) n,λ−1 D(z)V (z) n+λ Thus at ζ we have 0 ζn+λ r(ζ )−πr (ζ ) = 0 0 n,λ−1 0 D2(ζ )zn+λ Cν+1 +o(1) 0 ν+1 ζ0−zν+1 h i 3 Hence, if |ζ | < |z |, then at ζ the sequence πr (z) converges to 0 ν+1 0 n,λ−1 n∈Λm the function r(z). Therefore the whole sequencne πr (zo) converges to r(z) n,λ−1 at ζ . 0 If |ζ | ≥ |z |, then from (0.4) we obtain for n → ∞, n ∈ Λ , 0 ν+1 m n+λ ζ 1 |r(ζ )−πr (ζ )| = 0 , i.e. |r(ζ )−πr0 (ζn),λ|−→1 ∞0 for(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)|zζν+|1>(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)D|2a(nζ)dh|ζr0C−(νζz+ν1)+1−+πro(1)i((cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ζ )| 6→ 0 for 0 n,λ−1 0 0 ν+1 0 n,λ−1 0 |ζ | = |z |. The theorem is proved. 0 ν+1 Now we construct the examples that illustrate Theorem 1.. Example 1. Let λ = 3, ν = 2, σ = 2, z = 1,z = −1,z = 1/2, A = 1, 1 2 3 1 A = 1/18, A = 1, i.e. 2 3 1 1 1 r(z) = + + . z −1 18(z +1) z −1/2 Then C = 1,C = 2,C = 16/9 and zeros of the polynomials ω (z) 1 2 3 0,1 are ζ = 1/3,ζ = 3. The point ζ = 1/3 lies in the unit disk D and |ζ | < 0 1 0 0 |z |. Hence, ζ does not an obstruction for the poinwise convergence and 3 0 the sequence πr (z) uniformly converges to r(z) on compact subsets of the n,2 domain UF = D\{1/3,1/2} and pointwise converges on D\{1/2}. Example 2. Let A = A = A = 1, 1 2 3 1 1 1 r(z) = + + . z −1 (z +1) z −1/2 Then C = 1,C = 1/9,C = 16/9 and ζ = −4/5, ζ = −5/4. Now the 1 2 3 0 1 point ζ is an obstruction for the poinwise convergence and 0 lim πr (−4/5) = ∞. m→∞ 2m+1,2 In this case the sequence πr (z) uniformly converges to r(z) on compact n,2 subsets of the domain U˜F = D\{1/2,−4/5}. The following example show that an additional limit point can be coin- cided with a nondominant pole. 4 Example 3. Let A = 2,A = 2/27,A = 1, 1 2 3 2 2 1 r(z) = + + . z −1 27(z +1) z −1/2 Then C = 1/2,C = 3/2,C = 16/9 and ζ = 1/2,ζ = 2. The addi- 1 2 3 0 1 tional limit point ζ coincides with the pole z . 0 3 The following example show that there are a point ζ and a sequence Λ ⊂ N such that there is lim π (ζ) 6= a(ζ),n ∈ Λ. n→∞ n,λ−1 Example 4. Let A = 2/3,A = 2/9,A = 1, 1 2 3 2 2 1 r(z) = + + . 3(z −1) 9(z +1) z −1/2 Then C = 3/2,C = 1/2,C = 16/9 and ζ = −1/2,ζ = −2. By for- 1 2 3 0 1 mula (0.4) we obtain πr (−1/2) = 0. Thus πr (−1/2) is a stationary 2m+1,2 2m+1,2 sequence and πr (−1/2) 6= r(−1/2) = −1. 2m+1,2 References [1] V.M.Adukov, Theuniformconvergenceofsubsequences ofthelastinter- mediate row of the Pad´e table, J. Approx. Theory 122 (2003), 160–207. [2] A. Sidi, Quantitative and constructive aspects of the generalized Koenig’s and de Montessus’s theorems for Pad´e approximants, J. Com- put. Appl. Math. 29 (1990), 257–291. 5