Universal matrix transforms of holomorphic functions

Houston Journal of Mathematics c 2006 University of Houston Volume 32, No. 1, 2006 UNIVERSAL MATRIX TRANSFORMS OF HOLOMORPHIC FUNCTIONS L. BERNAL-GONZ´ ALEZ, M.C. CALDER´ ON-MORENO AND W. LUH Communicated by Herbert Amann Abstract. The phenomenon of overconvergence is related with the convergence of subsequences of the sequence of partial sums of Taylor series at points outside their disk of convergence. During the seventies Chui and Parnes and the third author provided a holomorphic function in the unit disk which is universal with respect to overconvergence. The generic nature of this kind of universality has been recently shown by Nestoridis. In this paper, we connect the overconvergence with the summability theory. We show that there are “many” holomorphic functions in the unit disk such that their sequences of A-transforms have the overconvergence property, A being an infinite matrix. This strengthens Nestoridis’ result. 1. Introduction A century ago Porter discovered that certain Taylor series with radius of convergence 1 enjoy the property that some subsequences {snk(z)}∞ k=0 of their sequences of partial sums {sn(z)}∞ n=0 converge at some points outside the closed unit disk {z:|z| ≤ 1}of the complex plane C. This phenomenon is called overconvergence. This idea was developed by the third author in 1970 [7] and by Chui and Parnes in 1971 [3]. They proved the existence of holomorphic functions f(z) = P∞ ν=0 aνzν in the open unit disk Dwith the property that, given a compact set Khaving connected complement and satisfying K∩ {z:|z| ≤ 1}=∅, and given g∈A(K) –that is, gis continuous in Kand holomorphic in its interior K0– there exists a 2000 Mathematics Subject Classification. Primary 30E10. Secondary 40C05, 42A10. Key words and phrases. Holomorphic function, unit disk, overconvergence, infinite matrix, A-transforms. The first two authors have been partially supported by Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 and by DGES Grant BFM2003-03893-C02-01. 315 316 L. BERNAL-GONZ ´ ALEZ, M.C. CALDER ´ ON-MORENO AND W. LUH subsequence {snk(f, z)}∞ k=0 of the sequence {sn(f, z) = Pn ν=0 aνzν}∞ n=0 of partial sums such that snk(f, z)→g(z) uniformly on K(k→ ∞).(1) Let us denote, as usual, by H(D) the space of holomorphic functions in the unit disk, endowed with the topology of uniform convergence on compact subsets. It is well known that H(D) is a Fr´echet space (= completely metrizable locally convex space), so it is a Baire space. In a Baire space X, a subset Ais residual whenever its complement is of first category (= a countable union of sets whose closures have empty interior) or, equivalently, whenever Acontains some dense Gδsubset. Hence, topologically speaking, a residual set is “very large” in X. In 1996 Nestoridis [10] gave a new impulse to the idea of overconvergence. He was able to prove that this is in fact a generic phenomenon, in the sense that “most” holomorphic functions in Dare universal with respect to overconvergence, even in a stronger way than before: There exists a residual set of functions f∈ H(D) satisfying that for each compact set Kwith K∩D=∅and connected complement, and for given g∈A(K), the approximation property (1) holds for some {nk}[10, Theorem 2.6]. Observe that this time Kis allowed to intersect the boundary ∂D. The results of Luh-Chui-Parnes-Nestoridis have been recently continued in many ways, for instance with properties of non-continuation, covering the plane, holomorphic monsters, and others (see [5, Section 4d] or [6, Section 4] for references). In this paper we want to provide a new way, connecting with summability methods given by infinite matrices. Specifically, we produce generic universality with respect to overconvergence, but this time the sequence of partial sums sn(f, z) = n X ν=0 aνzνof the Taylor series f(z) = ∞ X ν=0 aνzνis replaced to the sequence σn(f, z) := ∞ X ν=0 αnνsν(f, z) of their A-transforms,A= [αnν]∞ n,ν=0 being an adequate infinite matrix with complex entries, see Theorem 2.2 below. 2. Preliminaries and statement of the main result We will make use later of the following purely topological auxiliary assertion. Its special case R0= +∞can be found in [10, Lemma 2.1], which in turn is also a special instance of [9, Lemma 2.1]; see also [2, Lemma 2.9] for an earlier, similar property on general domains of the plane. The proof of Lemma 2.1 can be achieved by modifying suitably the proof of [10, Lemma 2.1]. As usual, C UNIVERSAL MATRIX TRANSFORMS OF HOLOMORPHIC FUNCTIONS 317 is the same as {|z|<+∞},Nwill stand for the set of positive integers, and N0:= N∪ {0}. Lemma 2.1. Let us fix R0with 1< R0≤+∞. Then there exists a sequence {Kn:n∈N}of compact sets in {1≤ |z|< R0}which have connected complement, such that given a compact set K⊂ {1≤ |z|< R0}with connected complement there exists an m∈Nwith K⊂Km. The preceding lemma is useful in order to “enumerate” the adequate compact sets. In other order of ideas, let R0∈(1,+∞] and let A= [αnν]∞ n,ν=0 be an infinite matrix with complex entries, and consider the following five properties which may or may not be satisfied by A: (a) For all n∈N0, lim sup ν→∞ |αnν|1/ν ≤1 R0 . (b) For all ν∈N0, lim n→∞ αnν = 0. (b’) For all finite subsets F⊂N0, lim inf n→∞ (max ν∈F|αnν|)=0. (c) For every n∈N0, the series ∞ X ν=0 αnν converges, and there exists an α∈ C\ {0}such that lim n→∞ ∞ X ν=0 αnν =α. (c’) For every n∈N0, the series ∞ X ν=0 αnν converges, and some oscillation limit of the sequence (¬¬¬¬¬ ∞ X ν=0 αnν¬¬¬¬¬)∞ n=0 is positive but finite. Observe that (b) implies (b’), that (c) implies (c’), and that (a) implies the first part of (c)–(c’). Note also that the second part of (c’) is equivalent to the existence of a strictly increasing sequence {nj}of natural numbers and some α∈C\ {0} with lim j→∞ ∞ X ν=0 αnjν=α. Finally, we observe that, trivially, any row-finite matrix –and so any triangular matrix– satisfies (a) and the first part of (c)–(c’). We recall that Ais said to be triangular if αnν = 0 for ν > n, while Ais row-finite whenever for each nthere is ν(n) such that αnν = 0 for ν > ν(n). Our main statement, which can be labelled as a “matrix overconvergence generic phenomenon result”, reads as follows. Theorem 2.2. Suppose that 1< R0≤+∞and that A= [αnν]is an infinite matrix which satisfies at least one of the sets of properties [(a),(b),(c0)],[(a),(b0),(c)]. 318 L. BERNAL-GONZ ´ ALEZ, M.C. CALDER ´ ON-MORENO AND W. LUH Let us denote by Mthe subset of all functions f∈H(D)such that the sequence {σn(f, ·)}∞ n=1 of their A-transforms has the following property: For every compact set K⊂ {1≤ |z|< R0}with connected complement and every function g∈A(K)there exists a sequence {nk}with σnk(f, z)→g(z)uniformly on K(k→ ∞). Then Mis residual in H(D). The implication (i)⇒(ii) of the following elementary lemma will be used in the proof of Theorem 2.2. But note that the lemma tells us that property (a) for a matrix Ais sharp in order that the A-transforms are well defined in that theorem. Lemma 2.3. Let be provided R0with 1< R0≤+∞. Assume that {αν}∞ ν=0 is a sequence of complex numbers. Then the following properties are equivalent: (i) lim sup ν→∞ |αν|1/ν ≤1 R0 . (ii) The series ∞ X ν=0 ανsν(f, z)converges uniformly on Kfor every f(z) = ∞ X ν=0 aνzν∈H(D)and every compact subset K⊂ {|z|< R0}. (iii) The series P∞ ν=0 ανsν(f, z)converges for every f∈H(D)and every point zwith 1<|z|< R0. Proof. It is trivial that (ii) implies (iii). Assume now that (i) holds and fix a compact subset K⊂ {|z|< R0}. Then there exists a constant R∈(1, R0) such that |z| ≤ Rfor all z∈K. Fix a function f(z) = P∞ ν=0 aνzν∈H(D). Since R/R0<1, we can select (and fix) an ε > 0 such that β:= ( 1 R0 +ε)(1 + ε)<1. But lim supn→∞(ν|αν|)1/ν ≤1/R0and lim supν→∞ |aν|1/ν ≤1, so (ν+ 1)|αν|< (1 R0+ε)ν·C0and |aν|<(1 + ε)ν·C00 for adequate constants C0,C00 and all ν∈N0. Then |ανsν(f, z)|≤|αν|·¬¬¬¬¬ ν X µ=0 aµzµ¬¬¬¬¬ ≤ |αν| · ν X µ=0 |aµ|Rµ ≤ |αν|·Rν· ν X µ=0 (1 + ε)µC00 ≤(ν+ 1)|αν|Rν(1 + ε)νC00 < C0C00 ·´² 1 R0 +ε³(1 + ε)Rµν =Cβν(z∈K, ν ∈N) UNIVERSAL MATRIX TRANSFORMS OF HOLOMORPHIC FUNCTIONS 319 for some constant C. Then Weierstrass’ M-test yields (ii). Finally, suppose that (iii) is true and that, by way of contradiction, lim sup ν→∞ |αν|1/ν >1/R0. Let us choose the point z0=γ, where max º1,1 lim supν→∞ |αν|1/ν »< γ < R0. Then 1 <|z0|< R and |αν|γν>1 for infinitely many ν∈N. Consider the function f(z) := 1 1−z∈H(D). Then |sν(f, z0)|=|1+γ+γ2+···+γν| ≥ γνfor all ν, therefore |ανsν(f, z0)|>1 for infinitely many ν. Consequently, ∞ X ν=0 ανsν(f, z0) cannot converge, which is a contradiction. £ Corollary 2.4. Under the assumption of Lemma 2.3 we assume that (i) of such lemma is satisfied. We consider the operator T:H(D)→H({|z|< R0})given by (Tf)(z) = P∞ ν=0 ανsν(f, z). Then Tis continuous, if H(D)and H({|z|< R0}) are endowed with the topologies of uniform convergence in compacta. Proof. We observe that by (i) the series P∞ ν=0 ανconverges and (Tf)(z) = ∞ X k=0 "∞ X ν=k αν#f(k)(0) k!zk. The linearity of Tntogether with the Closed Graph Theorem (see [13]) applied to the Fr´echet spaces H(D) and H({|z|< R0}) yield the continuity of T.£ 3. Proof of the main result We assume during the whole proof that properties (a), (b), (c’) are satisfied by the matrix A. The proof under the set of conditions [(a), (b’), (c)] is similar and left to the interested reader. Due to (a), Lemma 2.3 shows that for each n∈Nand each f∈H(D) the A-transform σn(f, z) not only makes sense for z∈K, but also defines a function belonging to A(K), whenever Kis a compact subset of {|z|< R0}. Now, from Mergelyan’s theorem (see [4] or [12]), each g∈A(K) can be uniformly approximated by polynomials on K, where the compact K⊂ {1≤ |z|< R0}has connected complement. Let {Kn}be the sequence given by Lemma 2.1. There is m∈Nwith K⊂Km. Thus, it is not difficult to realize that if the sequence {pj}∞ j=1 is an enumeration of all polynomials with rational real and imaginary parts then M=\ n∈N\ j∈N\ k∈N G(Kn, pj,1 k).(2) 320 L. BERNAL-GONZ ´ ALEZ, M.C. CALDER ´ ON-MORENO AND W. LUH We have denoted G(K, p, ε) := {f∈H(D) : there exists n∈Nsuch that |σn(f, z)−p(z)|< ε for all z∈K}, where Kis a compact subset of {1≤ |z|< R0}with connected complement, pis a polynomial and ε > 0. Fix K,p,εas before. For each n∈Nconsider the mapping Tn:f∈H(D)7→ σn(f, ·)|K∈A(K). We have already shown that Tnis well defined. But note also that every Tnis continuous by Corollary 2.4. On the other hand, we can write G(K, p, ε) = [ n∈N T−1 n(BK(p, ε)) where BK(p, ε) = {g∈A(K) : |g(z)−f(z)|< ε for all z∈K}, the open ball in A(K) with center pand radius ε. Hence G(K, p, ε) is an open subset of H(D), so by (2) Mis a Gδsubset. Since H(D) is a Baire space, it is enough to show that each G(K, p, ε) is dense in H(D). For this, fix a basic open subset of H(D), of the shape D(h, r, δ) = {f∈H(D) : |f(z)−h(z)|< δ for all zwith |z| ≤ r} (h∈H(D), 0 < r < 1, δ > 0). Our goal is to prove that G(K, p, ε)∩D(h, r, δ)6=∅.(3) Due to (c’), there are a sequence n1< n2<··· of positive integers and a value α∈C\ {0}satisfying lim j→∞ ∞ X ν=0 αnjν=α. (4) Since that set {|z| ≤ r} ∪ Kis a compact set with connected complement, Mergelyan’s theorem guarantees the existence of a polynomial fsuch that |f(z)−h(z)|< δ on {|z| ≤ r}(5) and ¬¬¬¬ f(z)−p(z) ᬬ¬¬ <ε 3|α|for all z∈K. (6) Choose R > 0 with K⊂ {|z| ≤ R}. Set d:= degree(f), in such a way that f(z) = Pd ν=0 aνzν. Then |f(z)| ≤ β:= max |t|≤R|f(t)|on Kand, by Cauchy’s UNIVERSAL MATRIX TRANSFORMS OF HOLOMORPHIC FUNCTIONS 321 inequalities, |aνzν| ≤ βfor all ν∈ {0,1,...,d}and all z∈K. Hence, |sν(f, z)−f(z)|=¬¬¬¬¬ d X µ=ν+1 aµzµ¬¬¬¬¬ ≤ d X µ=0 |aµzµ| ≤ (d+ 1) ·β(7) for every z∈Kand every ν∈ {0,1,...,d−1}. On the other hand, sν(f, ·) = f for every ν≥d. In order to get (3), we should verify that f∈G(K, p, ε). By (b) and (4), there exists a positive integer N≥dsatisfying the following properties: ¬¬¬¬¬ α− ∞ X ν=0 αNν¬¬¬¬¬ ≤ε 3(β+ 1),(8) d−1 X ν=0 |αNν| ≤ ε 3(d+ 1)(β+ 1).(9) Therefore, for all z∈K, σN(f, z)−p(z) = ∞ X ν=0 αN ν sν(f, z)−p(z) = d−1 X ν=0 αN ν sν(f, z) + ∞ X ν=d αN ν f(z)−p(z) = d−1 X ν=0 αN ν (sν(f, z)−f(z)) + ∞ X ν=0 αN ν f(z)−p(z) = d−1 X ν=0 αN ν (sν(f, z)−f(z)) + À∞ X ν=0 αN ν −α!f(z) + αf(z)−p(z). From (6), (7), (8), (9) and the triangle inequality we obtain |σN(f, z)−p(z)| ≤ (d+1)βε 3(d+ 1)(β+ 1)+ε 3(β+ 1)·β+|α|· ε 3|α|< ε (z∈K), that is, f∈G(K, p, ε). This and (5) give us (3). The proof is finished. 4. Final Remarks (1) If we consider A= the identity and R0= +∞in Theorem 2.2, then we obtain, as a particular case, the Nestoridis result about overconvergence of Taylor series. (2) Recently, the authors have proved –by using a constructive way– the existence of one function fwhose sequence of A-transforms is universal in the sense of Theorem 2.2, but on the weaker framework of any compact set K⊂∂D,K6=∂D(see [1, Theorem 4]). The matrix Ais this time triangular and satisfies (b)–(c). From this f, they also provided a trigonometric 322 L. BERNAL-GONZ ´ ALEZ, M.C. CALDER ´ ON-MORENO AND W. LUH series P∞ ν=0 aν(cos νt +isin νt) whose sequence of A-transforms {σn(t)} is universal in the following sense: For any two real-valued measurable functions ϕand ψon [0,2π], there exists a sequence {nk}such that Re{σnk(t)} → ϕ(t) Im{σnk(t)} → ψ(t)almost everywhere on [0,2π]. Analogously, we can obtain from our theorem the existence of “many” trigonometric series with the above universal behaviour. (3) Under the hypotheses of Theorem 2.2, “most” functions in H(D) satisfy its statement and, simultaneously, have power series expansions with radius of convergence 1. Indeed, the set Nof functions in H(D) such that ∂Dis a natural boundary is residual (see [6, Section 3]), hence M∩Nis also residual. (4) By using Baire categories, Melas and Nestoridis have recently shown [9, Theorem 3.4] a strong overconvergence-universality result which also covers that in [10]. They even consider a simply connected domain Ω instead of Dand their matrices A(which they called “admissible”) have entries αnν(z) that are holomorphic functions on a certain connected open neighbourhood of C\Ω. Nevertheless, these matrices were row-finite and, even in the case that their entries were just numbers, they were allowed to satisfy conditions stronger than (b’) and (c’). We also point out that the third author had constructed in [7] a function f∈H(D) whose Atransforms exhibited universality on every bounded simply connected domain Gwith G∩{|z| ≤ 1}=∅, where this time the matrix A= [αnν]∞ n,ν=0 was triangular and satisfied (b) and (c) with α= 1. (5) By following the proof of Theorem 2.2 it is not difficult to realize that if conditions (a), (b), (c) are imposed on A, then one would in fact obtain that for any subsequence {mj}of Nthere is a residual set of functions M such that, for every compact set K⊂ {1≤ |z|< R0}with connected complement and every f∈M, the set {σmj(f, ·) : j∈N}is dense in A(K). In the terminology of [5], the sequence {Tn}of operators considered in the proof would be densely hereditarily hypercyclic in this case. (6) Finally, we recall that Toeplitz–Silverman’s theorem (see [11]) asserts that an infinite matrix A= [αnν]∞ n,ν=0 is regular –that is, it preserves convergence and limits of sequences– if and only if (A) limn→∞ αnν = 0 for all ν∈N0, (B) limn→∞ P∞ ν=0 αnν = 1, and UNIVERSAL MATRIX TRANSFORMS OF HOLOMORPHIC FUNCTIONS 323 (C) supnP∞ ν=0 |αnν|<+∞. As for a weaker property, the third author proved in [8] that Ais P-regular –that is, regular for power series– if and only if (A), (B) and (C’) hold, where (C’) is the condition "sup n ∞ X ν=0 |αnν|ρν<+∞for all ρ∈(0,1)#. We observe that (B) is (c) with α= 1. Thus, our matrix in Theorem 2.2 may be far from being regular, even far from being P-regular. In the opposite direction, regular matrices generate dense hereditary hypercyclicity in the sense of the preceding remark. References [1] L. Bernal-Gonz´alez, M.C. Calder´on-Moreno and W. Luh, Universality and summability of trigonometric polynomials and trigonometric series, Per. Math. Hung. 46 (2003), 119–133. [2] L. Bernal-Gonz´alez and A. Montes-Rodr´ıguez, Universal functions for composition operators, Complex Variables 27 (1995), 47–56. [3] C. Chui and M.N. Parnes, Approximation by overconvergence of power series, J. Math. Anal. Appl. 36 (1971), 693–696. [4] D. 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Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill, New York-St. Louis-San Francisco, 1987. [13] W. Rudin, Functional Analysis, 2nd edition, McGraw-Hill, New York, 1991. Received February 19, 2004 Revised version received June 25, 2004 L. Bernal Gonz´ alez and M.C. Calder´ on Moreno, Departamento de An´ alisis Matem´ atico. Facultad de Matem´ aticas, apdo. 1160. Avenida Reina Mercedes. 41080 SEVILLA, SPAIN E-mail address:[email protected]; [email protected]