HADAMARD GAP THEOREM AND OVERCONVERGENCE FOR FABER-EROKHIN EXPANSIONS. PATRICE LASSÈRE & NGUYEN THANH VAN

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1 HADAMARD GAP THEOREM AND OVERCONVERGENCE FOR FABER-EROKHIN EXPANSIONS. PATRICE LASSÈRE & NGUYEN THANH VAN Résumé. Principally, we extend the Hadamard-Fabry gap theorem for power series to Faber-Erokhin ones.. A short survey on Faber-Erokhin basis Let Ω C a simply connected domain, K Ω a compact such that Ω \ K is doubly connected. Under these hypothesis, we know that (up to a rotation) there exists a biholomorphic mapping Φ : Ω \ K C( ;, R) = { z C : < z < R}, where R > is the modulus of the condensor C = (Ω, K). Let h Ω,K (z) := sup{u(z) : u SH(Ω) : u, u /K } the relative extremal function and Ω α = {z Ω levels sets ( < α < ) ; we have : h α,k (z) < α} its Ω α = Φ (D(, R α ) = {z C : z < R α }), α ], [. Let f O(Ω), then f Φ is holomorphic on the annulus C(;, R), we have by the Laurent expansion () f Φ (ξ) = where (2) c n = ζ =ρ c n ξ n, < ξ < R f Φ (ζ) ζ n+ dζ, < ρ < R, n Z, 99 Mathematics Subject Classification. 3B5, 3B4. Key words and phrases. Expansions, Faber-Erokhin basis, Hadamard Theorem, Overconvergence, Schauder basis, Change of sign, Gap.

2 2 PATRICE LASSÈRE & NGUYEN THANH VAN and the series converges normay on compact sets of the annulus. Changing ξ C(;, R) by Φ(z) Ω \ K the formula () is f(z) = c n Φ(z) n, z Ω \ K with normal convergence on compact sets of Ω \ K. But now, unlike f Φ, the function f is holomorphic on the whole Ω and by Cauchy formula we have forall α ], [ and z Ω α So f(z) = = (3) f(z) = and c n (4) E n (z) = where α ], [ and z Ω α. Ω α f(t) t z dt c n E n (z), Ωα Φ(t) n t z dt. z Ω Ωα Φ(t) n t z dt, In the exeptional case where Φ extends to a conformal mapping of C \ K with Φ( ) =, then E n =, n <. With (4) it is easy to see that E n, (n ) is a polynomial of degree n, they are the classical Faber polynomials [5]. The Faber polynomial sequence (E n ) is a basis of O(U) for all open level set U of the Green function G K = G(, C \ K, ) associated to K. The pioneer work of Erokhin [2], [5] consists to extend the notion of Faber polynomial to a regular condenser (Ω, K) where Ω\K is a doubly connected domain. His work is built on a «fundamental lemma» about the decomposition of a conformal map onto an annulus : Erhokin s Fundamental Lemma. Every conformal map Φ from a doubly connected domain Ω \ K onto an annulus C(,, R) = { w C : < w < R} can be decomposed into Φ = F 2 F where F and F 2 are conformal maps between simply connected domains, precisly : ) F maps conformly the simply connected domain C \ K onto a simply connected domain C \ L where L is compact in C. The image

3 HADAMARD GAP THEOREM AND OVERCONVERGENCE FOR FABER-EROKHIN EXPANSIONS.3 by F of the boundary of Ω : F ( Ω) define a simply connected domain Ω which contains L. 2) F 2 is the biholomorphic map F 2 : Ω D(, R) such that F 2 ( L) = C(, ). So we are in the following situation : F C \ L = F (C \ K) Ω K Ω L Ω = F ( Ω) Φ = F 2 F F 2 R The Faber-Erokhin basis : With this decomposition, the Faber- Erokhin basis is define by analogy with the Faber one by formula (4) with n N only E n (z) = Φ(t) Ωα n t z dt, α ], [ et z Ω α. Erokhin shows that the sequence (E n ) n is a common basis for the spaces O(Ω), O(Ω α ), ( < α < ) but generally E n when n <. The trivial expansion (3) being always tranformed in f(z) = a n E n (z), z Ω, where the a n are in general new coefficients given by an integral formula usually more complicated than (2). Precisely, we have for all f O(Ω α ), < ρ < α < : a n = ϕ f (ζ) dζ ζn+ ζ =ρ

4 4 PATRICE LASSÈRE & NGUYEN THANH VAN with for all ζ < R ρ (5) ϕ f (ζ) = a n ζ n = τ =Rρ f(φ (τ))(f2 ) (τ) F2 (τ) F2 (ζ) dτ.. 2. Hadamard type results for Faber-Erokhin expansions Let f an holomorphic function on the level set Ω α such that f O(Ω γ ), α < γ <. Let f = n a ne n its expansion in the Faber- Erokhin basis, then power series ϕ f (ζ) := a n ζ n has R α as radius of convergence. Moreover, (5) implies that for all < β < α and ζ < R β : (6) ϕ f (ζ) = a n ζ n = τ =Rβ f(φ (τ))(f2 ) (τ) F2 (τ) F2 (ζ) dτ. Theorem : f extends holomorphically across a point z Ω α iff ϕ f extends holomorphically across the point ζ := Φ(z ) C(, R α ). Proof : Necessary condition : Suppose that there exists a neighbourhood V z Ω \ K of z such that f extends holomorphically on Ω α V z. Let r > such that D(ζ, r) Φ(V z ) C(;, R), and choose < β < α enough close to α so that D(ζ, r) D(, R β ), now, consider the oriented path γ z bellow K V z z Φ ζ γz R β R α R Ω α Ω

5 HADAMARD GAP THEOREM AND OVERCONVERGENCE FOR FABER-EROKHIN EXPANSIONS.5 Then the function defined by the formula (7) ψ(ζ) = f(φ γz (τ))(f2 ) (τ) F2 (τ) F2 (ζ) dτ, ζ D(ζ, r) D(, R β ). is clearly holomorphic on D(ζ, r) D(, R β ). On the other hand with Cauchy (8) C(ζ,r)+ f(φ (τ))(f 2 ) (τ) F 2 (τ) F 2 (ζ) dτ =, ζ D(ζ, r). Formula (8) combined with (6) and (7) assure that ψ = ϕ f on D(, R β ) D(ζ, r), so we succed to extends holomorphically ϕ f across ζ. Sufficient condition : The proof is the same, it is built on the dual formula of (5) (5 ) f(z) = a n E n (z) = ϕ f (Φ(t)) dt, z Ω β. t z 2.. Applications : By contradiction, we have the following property : f O(Ω α ) has Ω α as domain of holomorphy, if and only if, ϕ f has the disc D(, R α ) as domain of holomorphy. So we are able to extend for expansions following the Faber-Erokhin basis some theorems on the boundary behaviour of an powers series. For example, we have (Hadamard) : Let f(z) = + a nk E nk (z) O(D α ) such that f O(D β ), β > α. If there exists a constant c > such that n k+ n k > c n k, k N, then D α is the domain of holomorphy of f. Or stronger (Fabry-Pólya) : Let f(z) = + n k E nk (z) O(D α ) such that n f O(D β ), β > α. If lim k k k = then Ω α is the domain of holomorphy of f. Conversely (Pólya), every increasing sequence of integers n < n <... such that every series + a nk E nk has Ω α as domain n of holomorphy, satisfies lim k k k =. For example, the functions f(z) = + R 2nα E 2 n(z) (Hadamard) or g(z) = + R n2α E n 2(z) (Fabry) admits Ω α as domain of holomorphy Ω β

6 6 PATRICE LASSÈRE & NGUYEN THANH VAN but this is not the cases for h(z) = + R nα E n (z) who presents an unique singular (which of course is Φ ()...) on the boundarys Ω α. 3. The case of an arbitrary common basis. With the same hypothesis on the pair (K, Ω) let us consider now an arbitrary common basis (ϕ n ) n for the spaces O(K), O(Ω). It extends as a common basis of the intermediate spaces O(Ω α ), ( < α < ). This is not difficult to see that the preceeding results are no longer true for any common basis (ϕ n ) n : consider the simple example where K = D(, /2) Ω = D(, 2). This condensor admits as level sets the discs Ω α = D(, 2 3α ). Consider the common basis ϕ n (z) = z π(n), n N where π : N N is a bijection such that π(2 n ) = 2n. Then the function f(z) = + ϕ 2 n(z) satisfies the Hadamard lacunary condition but f(z) = ϕ 2 n(z) = z 2n = z 2 holomorphic on D(, ) = Ω /3 admits C \ {±} as domain of holomorphy. Remarks : In [], J.A.Adepoju proved the Fabry-type gap theorem for Faber polynomials, his proof followed the classical one for entire series and is rather complicated. In [4] we extend Fatou-type theorems to all common bases of the pair (O(K), O(Ω)) in a more general situation. 4. Overconvergence. In the spirit of the proof of theorem, the formulas (5) and (5 ) lead us to transport overconvergence phenomena to Faber-Erokhin series. Let f = + a n E n O(Ω α ), if f is not holomorphic on larger level sets Ω β, α < β then we will say that the series + a n E n is overconvergent if there exists a subsequence (m k ) k such that the corresponding partial sums m k s mk (f, z) := a ν E ν (z), converges compactly in a domain that contains properly Ω α. ν=

7 HADAMARD GAP THEOREM AND OVERCONVERGENCE FOR FABER-EROKHIN EXPANSIONS.7 Unicity of coefficients in the Faber-Erokhin expension and formula (5) gives m k (9) s mk (ϕ f, ζ) := ν= a ν z ν = τ =Rβ s mk (f, Φ (τ))(f2 ) (τ) F2 (τ) F2 (ζ) Suppose now that the sequence (s mk (f, )) k converges uniformly on a neighboorought V z of a boundary point z Ω α, then as in theorem, we have sup s mk (ϕ f, ζ) s mk (ϕ f, ζ) sup s mk (f, z) s mk (f, z) ζ D(ζ,r) ζ V z dτ. F2 γz ) (τ) dτ F2 (τ) F2 (ζ) C sup s mk (f, z) s mk (f, z) ζ V z where as before ζ = Φ(z ), D(z, r) Φ(V z ). This implies that (s mk (ϕ f, )) k is an unifomly convergent Cauchy sequence on the disc D(ζ, r) : the series + a k z k is overconvergent. By duality the overconvergence of + a k z k implies the one for + a k E k. Références [] Adepoju J.A. «Fabry-type gap theorem for Faber series», Demonstration Math. 2-3 (988), [2] Erokhin V.D. «Best linear approximation of functions analytically continuable from a given continuum into a given region», Uspehi Math. Nauk 23- (968), [3] Kahane J.P., Melas A. & Nestoridis V. «Sur les séries de Taylor universelles», C.R. Acad. Sci. Paris, Série I, 33 (2), 3-6. [4] Lassère P. & Nguyen T.V. «Gaps and Fatou Theorem for series in Schauder basis of holomorphic functions», Complex Variables and Elliptic Equations. 5-2 (26), [5] Suetin P.K.»Series of Faber Polynomials» Gordon and Breach Science Publishers (998). [6] Titchmarsh E.C. «Theory of functions», Oxford University Press. Lassère Patrice & Nguyen Thanh Van : Laboratoire de Mathématiques E.Picard, UMR CNRS 558, Université Paul Sabatier, 8 route de Narbonne, 362 TOU- LOUSE. lassere@picard.ups tlse.fr & nguyen@picard.ups tlse.fr