Spaceability of subsets of the disc algebra

Spaceability of subsets of the disc algebra P.J Gerlach-Mena∗, J. M¨uller Abstract In this paper we analyse the topological and linear structure of different subsets of the disc algebra. Among others, we consider the set of functions in the disc algebra having a Taylor series about 0 which is unboundedly divergent on a given subset of the unit circle of vanishing arc length measure, and the subsets of functions having uniformly bounded or uniformly convergent Taylor series on the unit circle. Mathematics Subject Classification: Primary 30B30, Secondary 30H50. Key words and phrases: Disc algebra, lineability, spaceability, algebrability, unboundedly divergence. 1 Introduction The pathological behaviour of Fourier series of continuous functions has been broadly studied in the past years. Its origin goes back to du Bois-Reymond (1873), who was the first one to exhibit an example of a continuous function on the unit circle T:= {z∈C:|z|= 1}whose Fourier series diverges at a point. This was improved by Kahane and Katznelson (1966, see e.g. [17]) by extending the divergence to arbitrary sets E⊂Tof (arc length) measure zero, and thus complementing the famous Carleson Theorem on almost everywhere convergence of Fourier series of functions in L2([11]). In recent years, these divergence properties have been proved to be topologically generic, that is, not only functions ffulfilling the properties exist, but they hold true for a residual set of functions in the corresponding spaces (see e.g. [16]). Moreover, even linear and algebraic structures can be detected within the set of such functions (see e.g. [2], [4]). We refer to the survey [8] and the book [1] for a wide background on this topic. Our aim is to find linear structures in various subsets of the disc algebra. We recall some standard notation: Let Ddenote the open disc in the complex plane C, so that D=D∪T. For Ka compact space, let C(K) be the Banach space of continuous functions f:K−→ Cendowed with the supremum norm ∥f∥∞= supz∈K|f(z)|. We will write H(D) for the family of holomorphic functions on ∗The author has been partially supported by DAAD Research Grants – Short-Term Grant (57507441), Project Reference 91713397. 1 the unit disc D. Then, each f∈H(D) can be represented as f(z) = P∞ k=0 akzk for z∈D, with akbeing the k-th Taylor coefficient with respect to 0. The disc algebra A(D) is defined as the Banach space of functions f∈H(D) which are continuously extendable up to the boundary T. One can identify A(D) with the space A(D) = {f∈C(D) : fholomorphic in D}with the uniform norm ∥f∥∞= max z∈D |f(z)|= max z∈T|f(z)|= sup z∈D ∞ X k=0 akzk . Fej´er’s Theorem implies that the polynomials form a dense subset of A(D). In the first part we will report on the existence of large topological and algebraic structures of subsets of functions in the disc algebra whose Taylor series about 0 are unboundedly divergent on certain subsets of the unit circle. While having mainly survey character, two results complementing the theory are proved. In the second part we search for linear and algebraic structures in subsets of the space of functions in the disc algebra having uniformly bounded or uniformly convergent Taylor series. In the third part we consider subspaces of the disc algebra consisting of functions whose Taylor series about 0 converges on subsets of the unit circle. We recall some definitions and results that will be used occasionally. If Xis a Baire space, a subset Ais said to be residual if it contains a dense Gδ-subset of X. Given a vector space Xand a subset A, we say that Ais maximal lineable whenever there is a vector space Msuch that M\{0} ⊂ Aand the dimension of Mequals the dimension of X. Moreover, if Xis a topological vector space, then Ais said to be spaceable if there is an infinite dimensional closed vector space Mwith M\ {0} ⊂ A, and dense-lineable if there is a dense infinite dimensional vector space Mwith M\ {0} ⊂ A. If Xis a metrizable separable topological vector space, and Y⊂Xis a vector subspace, then X\Yis dense-lineable if X\Yis lineable (see [10]). Thus, if X\Yis spaceable, then X\Yis also dense-lineable (and maximal lineable). Finally, when Xis a topological vector space contained in some (linear) algebra, then Ais called algebrable if there is an algebra Mso that M\ {0} ⊂ A, and Mis infinitely generated, that is, the cardinality of any system of generators of Mis infinite, and dense-algebrable if the algebra Mcan be chosen to be dense. Remark 1.1. Let Xand Zbe Fr´echet spaces, and T:Z→Xbe a continuous linear operator with range Y=T(Z) non-closed. Then, according to a result of Kitson and Timoney ([18]), the complement X\Yis spaceable. By considering the inclusion map T:Y→X, for a Fr´echet space Y=Xwith Ydensely and continuously embedded in X, it is seen that the complement X\Yis spaceable. Remark 1.2. By assuming the continuum hypothesis, we have that for Xa non-separable F-space, and Ya closed separable subspace of X, the set X\Yis maximal lineable (see [10]). 2 2 Unbounded divergence Given a sequence (wn)n∈Nof complex numbers, we say that (wn)n∈Nis unboundedly divergent if supn∈N|wn|=∞. A strengthened version of the Kahane- Katznelson Theorem says that, for each Fσ-set Eof vanishing (arc length) measure, a residual set of functions in C(T) exists so that the Fourier series of fis unboundedly divergent in each point of E(see e.g. [16]). Moreover, the set of functions in C(T) having unboundedly divergent Fourier series on Eis known to be spaceable and dense-algebrable (see [1, Theorem 6.3.1], [2]). Our aim is to complete the picture by showing that similar results hold for the disc algebra. Thanks to having now a well-elaborated general theory at disposal, it turns out that this is essentially a consequence of the fact that a lemma of Kahane and Katznelson (see [17, Chapter II, Lemma 3.4]), which is the basis of the proof of the Kahane-Katznelson Theorem, holds for algebraic polynomials as well. We write Sn(f, ·) for the n-th partial sum of the Taylor series P∞ k=0 akzkof f, that is, Sn(f, z) = Pn k=0 akzk. Theorem 2.1. Let E⊂Tbe a Fσ-set of measure zero. Then, for a residual set of functions f∈A(D), the sequence (Sn(f, z))nis unboundedly divergent for each z∈E. Theorem 2.2. Let E⊂Tbe of measure zero. Then, the family Aof all functions f∈A(D)such that the partial sums series (Sn(f, z))nis unboundedly divergent for every z∈Eis spaceable and dense-algebrable in A(D). The proofs of the above theorems are based on two general results of Bayart. The first one ([4, Theorem 2]) focuses on topological genericity in Banach spaces: Theorem 2.3 (Bayart, 2005).Let Xbe a Banach space and Eaσ-compact topological space. Suppose that for each t∈Eand each n≥0a linear form φn(·, t)is given such that φn:X×E−→ Cis continuous. For g∈Xand t∈Elet δN(g, t) = sup n>N |φn(g, t)−φN(g, t)|. Suppose that, for all M, N > 0and each compact subset Kof E, there exists g∈Xsuch that ∥g∥X≤1and δN(g, t)> M for every t∈K. Then, for a residual set of functions f∈X, the sequence (φn(f, t))n≥0is unboundedly divergent for each t∈E. Theorem 2.1 is mainly a consequence of Theorem 2.3 combined with the following auxiliary result, which is a variation of the lemma of Kahane and Katznelson mentioned above. The result follows from the corresponding lemma by replacing the anti-holomorphic polynomial φ=φFconstructed in their proof (see [17, Chapter II, Lemma 3.4]) by the holomorphic polynomial QF:= φF. Lemma 2.4. Let F⊂Tbe a union of a finite number of subarcs of T, and denote the measure of Fby δ. There exists a polynomial QFsuch that (i) sup n∈N |Sn(QF, z)|>1 2πlog 1 3δon F. 3 (ii) ∥QF∥∞≤1. As an immediate consequence we have Lemma 2.5. Let E⊂Tbe of measure zero. Then, for all M > 0, there exists a polynomial Qwith ∥Q∥∞≤1and supn∈N|Sn(Q, z)|> M for all z∈E. Proof. Let M > 0 and let δ > 0 be so that log((3δ)−1)>2πM. Since Ehas measure zero, we can find a finite union Fof subarcs of Twith E⊂F, and having measure less than δ. If Q=QFis as in Lemma 2.4, then ∥Q∥∞≤1 and supn∈N|Sn(Q, z)|> M for all z∈E. Proof of Theorem 2.1. Let X=A(D). For all M, N > 0 define g(z) := zN+1Q(z) on D, where Qis the polynomial provided by Lemma 2.5, that is, ∥Q∥∞≤1 and sup n∈N |Sn(Q, z)|> M on E. We have then ∥g∥∞≤1, and sup n>N |Sn(g, z)−SN(g, z)|= sup n>N |Sn(g, z)|= sup m∈N0 |Sm(Q, z)|> M on E. Now, we just apply Theorem 2.3 with X=A(D) and φn(g, ·) = Sn(g, ·), and the proof is complete. Recall that a sequence (un)nin a Banach space Xis called a basic sequence if, for each ubelonging to M= span{un:n∈N}, there exists a sequence (αn)nof scalars such that u=P∞ n=1 αnun. The coefficient functionals u∗ kare defined by u∗ k ∞ X n=1 αnun!=αk, k ∈N. They are continuous on M, and can be extended to Xby the Hahn-Banach theorem. Following a construction provided by Bayart in [4, Theorem 3] and [3, Theorem 3], we obtain for an arbitrary Banach space X⊂L1(T): Theorem 2.6. Let X⊂L1(T)be a Banach space so that the polynomials form a dense subset of X, and let E⊂T. Suppose that ∥znf∥X=∥f∥Xfor all n∈Nand f∈X, and that for all M > 0there exists a polynomial Qsuch that ∥Q∥X≤1and supn∈N|Sn(Q, z)|> M for each z∈E. Moreover, let Abe the family of all functions f∈Xsuch that the sequence (Sn(f, z))nis unboundedly divergent for every z∈E. Then, Ais dense-lineable in X, and if, in addition, there exists a basic sequence (un)n∈N⊂Xsuch that ∥u∗ n∥ ≤ 1for all n∈N, then Ais spaceable in X. 4 Proof of Theorem 2.2. According to Lemma 2.5, for X=A(D), given any M > 0, there exists a polynomial Qwith ∥Q∥∞≤1 and supn∈N|Sn(Q, z)|> M for all z∈E. Moreover, the sequence un(z) = z2nis a basic sequence for A(D) (see e.g. [17, Chapter V, Theorem 1.4]). Now, Theorem 2.6 implies the spaceability of Ain A(D). Finally, using Lemma 2.5 instead of the corresponding (Kahne-Katznelson) lemma for trigonometric polynomials in Section 2.1 of [2], essentially the same proof as of [2, Theorem 2.1] leads to the dense-algebrability of our family A. Various recent work focusses on strengthened notions of unbounded divergence. In [5] (see also [6]) it is shown that for a residual set of functions in C(T), the partial sums of the Fourier series diverge on sets of Hausdorff dimension 1 with a maximal rate of growth. A different kind of maximal divergence can be formulated in terms of universality. A formal series P∞ k=0 fkof functions fk∈C(T) is said to be pointwise universal on a set E⊂Tif, for all Baire class 1 functions h:E→C, a subsequence of the partial sums converges to hpointwise on E. The series is called uniformly universal on Eif Eis closed and the partial sums form a dense set in C(E). In [21] and [7], respectively, for arbitrary countable subsets Eof T, residuality and maximal dense-lineabilty, as well as spaceability of the set of functions in C(T) having pointwise universal Fourier series on Ewere proved. In [15] and [9], similar results for the disc algebra and Taylor series were added. As a consequence of the latter, in [9] also residuality, maximal dense-lineability and spaceability of the set of functions in A(D) having uniformly universal Taylor series on a residual set in the hyperspace of compact sets in Tare proved. Generically, sets in the hyperspace of Tare perfect sets, and, as such, in particular locally uncountable. On the other hand, it is known that sets of uniform universality are severely restricted in that they have to be strongly porous at all points ([9]; cf. also [22]). Combining universality and convergence, in [22] it is shown that for arbitrary countable E⊂Tfunctions in the disc algebra exist with Taylor series being pointwise universal on Eand converging pointwise except for a set of vanishing Hausdorff dimension. For finite E, the convergence can take place locally uniformly on T\E. 3 Uniformly bounded series In this section we consider the space Aub(D) of all functions fin the disc algebra whose partial sums (Sm(f, ·))mare uniformly bounded with respect to the uniform norm, that is Aub(D) = {f∈A(D):(∥Sm(f, ·)∥∞)mis bounded}. We endow this space with the natural metric ∥f∥ub := sup m∈N0 ∥Sm(f, ·)∥∞. 5 Since ∥f∥∞≤ ∥f∥ub for f∈Aub(D), the space Aub(D) is continuously embedded in A(D). Lemma 3.1. (Aub(D),∥·∥ub)is a Banach space. Proof. Let (fn)n⊂Aub(D) be a Cauchy sequence. Then, for all ε > 0, there exists N∈Nsuch that for all r, s ≥Nand m∈N0we have ∥Sm(fr−fs,·)∥∞< ε. Since (fn)nis a Cauchy sequence in A(D), there exists f∈A(D) such that ∥f−fn∥∞→0 as n→+∞. From Cauchy’s estimates, we have |ak|≤∥f∥∞ for k≥0, and then |Sm(f, z)| ≤ m X k=0 |ak| ≤ (m+ 1)∥f∥∞. Hence, for fixed m∈N0, the mapping Tmf=Sm(f, ·) is continuous from A(D) to A(D). Now, for n≥Nand m∈N0 ε > ∥Sm(fr−fN,·)∥∞→ ∥Sm(f−fN,·)∥∞ as r→+∞, hence ∥Sm(f−fn,·)∥∞≤εfor n≥Nand m∈N0. This means that ∥f−fn∥ub ≤ε for n≥N. Finally with M:= supm∥Sm(fN,·)∥∞we obtain that ∥Sm(f, ·)∥∞≤ ∥Sm(f−fN,·)∥∞+∥Sm(fN,·)∥∞< ε +M for all m∈N0, which shows that f∈Aub(D). Since, by definition, for f∈Aub(D) we have uniform boundedness of the sequence partial sums Snfon D, the set Afrom Theorem 2.2 is contained in A(D)\Aub(D). Thus, Theorem 2.1 and Theorem 2.2 imply Theorem 3.2. The set A(D)\Aub(D)is spaceable and dense-algebrable in A(D). As a consequence of the following result of Erd¨os, Herzog and Piranian ([12]), we will show that the space Aub(D) is not separable. Theorem 3.3 (Erd¨os, Herzog and Piranian, 1954).Let E⊂Tbe a Fσ-set of logarithmic measure zero. Then, there exists a function f∈Aub(D)so that Sn(f, ·)diverges on Eand converges on T\E. Theorem 3.4. The space (Aub(D),∥·∥ub)is not separable. Proof. According to Theorem 3.3, there exists a function f∈Aub(D) such that (Sn(f, 1))ndiverges and (Sn(f, z))nconverges for all z∈T\{1}. We choose M > 0 such that for all N∈Nthere are n, m ≥Nwith |Sn(f, 1) −Sm(f, 1)| ≥ M. (1) 6 Now, given ζ∈T, we can consider the functions fζ∈Aub(D) given by fζ(z) := fζz, that is, fζis the rotation of fvia ζ∈T, and so fζhas the same behaviour at the point ζ∈Tas fat the point 1. We consider the family F:= {fζ:ζ∈T}in Aub(D), which is uncountable, and such that for each ζthe sequence (Sn(fζ,·))n converges on T\ {ζ}and satisfies (1) at the point ζ. By way of contradiction, assume that Aub(D) is separable, that is, there exists D⊂Aub(D) countable and dense. We write D:= {gk:k∈N}, and put ε:= M/6. Then, for all ζ∈T, there exists k(ζ)∈Nsuch that ∥fζ−gk(ζ)∥ub = sup m∈N0 ∥Sm(fζ−gk(ζ),·)∥∞< ε. In particular, for all m∈N0and z∈Twe have |Sm(fζ, z)−Sm(gk(ζ), z)|< ε. Since the family Fis uncountable, and Dis countable, there must exist ζ, ζ′∈Twith ζ′=ζ, and such that k(ζ) = k(ζ′) =: k. Then, for z=ζand n, m satisfying (1) we obtain |Sn(gk, ζ)−Sm(gk, ζ)|≥|Sn(fζ, ζ)−Sm(fζ, ζ)|−|Sm(fζ−gk, ζ)| −|Sn(fζ−gk, ζ)| ≥ M−2ε= 2M/3. Thus, the function gk∈Aub(D) has the same property at the point ζ∈Tas the function fζ, that is, the sequence (Sn(gk, ζ))nis divergent. On the other hand, for n, m ∈Nso large that |Sm(fζ′, ζ)−Sm(fζ′, ζ)|< ε, we have that |Sn(gk, ζ)−Sm(gk, ζ)|≤|Sn(gk, ζ)−Sn(fζ′, ζ)|+|Sn(fζ′, ζ)−Sm(fζ′, ζ)| +|Sm(fζ′, ζ)−Sm(gk, ζ)|<3ε=M/2. Hence, the partial sums of gkwould have at the same time a variation greater than 2M/3 and smaller than M/2 at the point ζ∈T, which is clearly a contradiction. Thus, the space Aub(D) is not separable. We denote by Auc(D) the space of all functions in the disc algebra having uniformly convergent Taylor series on T, that is Auc(D) := {f∈A(D) : ∞ X k=0 akzkconverges uniformly on T}. Note that, according to the maximum principle, for all f∈Auc(D) the sequence of partial sums Sn(f, ·) is a uniform Cauchy sequence on D, and therefore converges to funiformly on Das n→ ∞. With the norm ∥ · ∥ub the space Auc(D) is completely metrized: 7 Lemma 3.5. Auc(D)is closed in Aub(D). Proof. Let (fn)nbe a sequence in Auc(D) with ∥fn−f∥ub →0. Then, for all ε > 0, there exists N∈Nsuch that for all n≥Nand all m∈N0, we have ∥fn−f∥∞< ε, and ∥Sm(f−fn,·)∥∞< ε. This implies ∥f−Sm(f, ·)∥∞≤ ∥f−fN∥∞+∥fN−Sm(fN,·)∥∞+∥Sm(fN−f, ·)∥∞ <2ε+∥fN−Sm(fN,·)∥∞ for all m∈N0. Since (Sm(fN,·))mconverges uniformly to fN, we obtain that ∥f−Sm(f, ·)∥∞<3εfor msufficiently large. Lemma 3.6. For all f∈Auc(D)the partial sums Sn(f, ·)converge to fin Auc(D). In particular, Auc(D)is the closure of the polynomials in Aub(D). Proof. Since (Sn(f, ·)) tends to funiformly on D, for each ε > 0 there exists N∈Nsuch that for m>n≥Nwe have ∥Sm(f, ·)−Sn(f, ·)∥∞< ε. The projection property of the Snimplies that sup m∈N0 ∥Sm(f−Sn(f, ·),·)∥∞= sup m∈N0, m>n ∥Sm(f, ·)−Sn(f, ·)∥∞≤ε for all n≥N. Since the space Auc(D) is continuously embedded in A(D), and recalling that Auc(D) is a closed separable subspace of Aub(D), by Remark 1.2 we obtain: Corollary 3.7. The complement Aub(D)\Auc(D)is maximal lineable. Let now Aabs(D) := {f∈A(D) : ∞ X k=0 |ak|<+∞} be the analytic Wiener algebra of all functions in the disc algebra having absolutely convergent Taylor series (also denoted by W+), endowed with the norm ∥f∥abs := ∞ X k=0 |ak|. According to the Denjoy-Lusin theorem (see e.g [24, p. 232]), a function fbelongs to Aabs(D) if (Sn(f, ·))nconverges absolutely on a set of positive measure. Denoting by ℓ1the space of absolutely summable sequences, via the mapping ℓ1∋(an)n∈N07→ P∞ k=0 akzk∈Aabs(D) the space Aabs(D) is isometrically isomorphic to ℓ1, and hence a Banach space. Since ∥f∥ub ≤ ∥f∥abs for f∈Aabs(D), the space Aabs(D) is continuously embedded in Auc(D). 8 Explicit examples of a functions in A(D)\Aabs(D) are given by f(z) = (1 −z)αexp −1 + z 1−z for 0 < α ≤1/2 (see [20, Proposition 5.2]). Hardy’s example f(z) = ∞ X k=2 (−1)k−i kzk log(k) (see [19, p. 68]) gives a function in Auc(D)\Aabs(D). As a consequence of the result from Remark 1.1, Kitson and Timoney obtained that A(D)\Aabs(D) is spaceable. This holds even for the difference between Auc(D) and Aabs(D): Theorem 3.8. The complement Auc(D)\Aabs(D)is spaceable in Auc(D). Proof. As noted above, Aabs(D) is continuously embedded in Auc(D) and, by Hardy’s example, Auc(D)=Aabs(D). Since the polynomials are dense in Auc(D), also Aabs(D) is dense in Auc(D). According to Remark 1.1, the proof is complete. We close this part with a remark concerning conformal mappings in the disc algebra. If f∈H(D) is injective, then Carath´eodory’s Theorem shows that f∈A(D) if and only if f(D) is a bounded simply connected domain with locally connected boundary. By Abel’s Theorem and the Fej´er Tauberian theorem (see e.g. [19, p. 65]), this is also equivalent to f∈Auc(D). Note that fdoes not need to be injective on T, so this subclass of A(D) is not easily detected by the corresponding boundary functions. Moreover, there are functions in A(D) which are injective even on D, and which do not belong to Aabs(D) (see [23]), so Auc(D)\Aabs(D) contains functions which map Dconformally onto a Jordan domain. 4 Uniform convergence on subsets So far we have detected linear structures in the partition Aabs(D)⊂Auc(D)⊂Aub(D)⊂A(D). We are now interested in studying what happens if we reduce the set of uniform convergence. If f∈A(D) has modulus of continuity o(log δ−1) on some closed arc B⊂T, then the local Dini-Lipschitz theorem (see e.g. [24, p. 63]) shows that (Sn(f, ·))nconverges to flocally uniformly on the corresponding open arc. We start with functions in the disc algebra that have only one singularity on T, which we suppose to be the point 1. Then fis smooth on T, and, in particular, (Sn(f, ·))nconverges to flocally uniformly on T\ {1}. If Cδ={z:|arg(z−1)| ≤ δ}is the sector with vertex at 1 and angle 2δ < π, a result of M. Riesz (see e.g. [19, p. 64]) shows that f∈Auc(D) if f∈A(D) 9