Nowhere hölderian functions and Pringsheim singular functions in the disc algebra

NOWHERE H ¨ OLDERIAN FUNCTIONS AND PRINGSHEIM SINGULAR FUNCTIONS IN THE DISC ALGEBRA L. BERNAL-GONZ´ ALEZ, A. BONILLA, J. L´ OPEZ-SALAZAR, AND J.B. SEOANE-SEP´ ULVEDA Abstract. We prove the existence of dense linear subspaces, of infinitely generated subalgebras and of infinite dimensional Banach spaces in the disc algebra all of whose nonzero members are not α-h¨olderian at any point of the unit circle for any α > 0. This completes the recently established result of topological genericity of this kind of functions, as well as the corresponding lineability statements about functions that are nowhere differentiable at the boundary. Topological and algebraic genericity is also studied for the family of boundary-smooth holomorphic functions that are Pringsheim singular at any point of the unit circle. 1. Introduction, Notation and Preliminaries The existence of real functions that are continuous everywhere but differentiable nowhere in a given real interval Iis well known at least from Weierstrass [50]. Using the Baire category theorem, Banach and Mazurkiewicz [7, 42] obtained that, in fact, the set of these functions is residual –that is, its complement is of first category– in the space of continuous functions on Iendowed with the topology of uniform convergence on compacta. Giving a step forward, we have that the more stringent class of continuous nowhere h¨olderian functions on Iis also residual [5]. In a different –but related– order of ideas, the family of smooth functions on Ithat are not analytic at any point of Iis also known to be residual in the space of smooth functions on Iunder its natural topology (see [26,43]). It is our aim in this paper to shed light on the structure of these families of special functions, not only in the topological sense, but also in the algebraic sense. We focus our attention on those periodic functions that can be extended holomorphically on the unit disc in the complex plane. Before going on, let us fix some notation. As usual, the symbols N,N0,Q, R,C,D,Tand cwill denote the set of positive integers, the set N∪ {0}, the field of rational numbers, the real line, the complex plane, the open unit disc {z∈C:|z|<1}, the unit circle {z∈C:|z|= 1}and the 2010 Mathematics Subject Classification. Primary 30H50; Secondary 15A03, 26A27, 26E10, 46E10. Key words and phrases. Nowhere h¨olderian function, Pringsheim singular function, disc algebra, lineability, spaceability, algebrability. 1 2 BERNAL, BONILLA, L´ OPEZ-SALAZAR, AND SEOANE cardinality of the continuum, respectively. If Iis a compact interval of R, then C(I) will represent the Banach space of all real- or complex-valued continuous functions defined on I, endowed with the maximum norm. The space C∞(I) of smooth functions I→K(K=Ror C), that is, the family of all infinitely many differentiable functions on I, when endowed with the topology of uniform convergence of functions and all derivatives, becomes an F-space, that is, a complete metrizable topological vector space. If S⊂C, then Sand ∂S will stand for the closure and the boundary of Sin C, respectively. If Ω is a domain, that is, a nonempty connected open subset of C, then H(Ω) represents, as usual, the space of all holomorphic functions on Ω. By A(D) we will denote the disc algebra, that is, the space of those continuous functions on Dthat belong to H(D). This space becomes a Banach space when endowed with the topology of uniform convergence on D. The symbol A∞(D) will stand for the set of functions f∈H(D) that are smooth on the boundary, that is, such that every derivative f(n)(n∈N0) can be continuously extended to D. This family is, again, an F-space under the topology of uniform convergence of all derivatives on D. If S⊂C,z0∈Sand α∈(0,1], then a function f:S→Cis called α-h¨olderian at z0whenever there are a constant C > 0 and a neighborhood Uof z0in Ssuch that |f(z)−f(z0)| ≤ C|z−z0|αfor all z∈U. Note that 1-h¨olderian equals lipschitzian, that differentiable implies lipschitzian, and that α-h¨olderian for some αimplies continuous, both reverses being not true. A function f:S→Cis called nowhere h¨olderian on Sprovided that, for any z0∈Sand any α∈(0,1], fis not α-h¨olderian at z0. If I⊂Ris an interval and x0∈I, then a smooth function f:I→K (K=Ror C) is called Pringsheim-singular at x0provided that the radius of convergence R(f, x0) of the Taylor series associated to fat x0equals 0 or, equivalently, the sequence {|f(n)(x0)/n!|1/n :n∈N} is not bounded. Note that this is a notion strictly stronger than mere nonanalyticity at x0. By PS(I) we shall denote the set of all functions f∈ C∞(I) that are Pringsheim-singular at any point of I. Again, the family PS(I) turns out to be residual in C∞(I) (see [12,45,48,49]). In a natural way, we say that a function f:T→Cis smooth if the associated periodic function t∈R7→ f(eit)∈Cbelongs to C∞(R), and that such an fis Pringsheim-singular at a point z0=eit0∈Twhenever the function t7→ f(eit) is Pringsheim-singular at t0. Some additional terminology, coming from the new theory of lineability, will be used. For an account of results on lineability, the reader is referred to the survey [20] and the monograph [3], or [2,21,24,31–33] in order to see the many fields in which this theory has made an impact so far. Assume that NOWHERE H ¨ OLDERIAN FUNCTIONS AND PRINGSHEIM SINGULAR FUNCTIONS 3 Xis a vector space and αis a cardinal number. Then a subset A⊂Xis said to be: •lineable if there is an infinite dimensional vector space Msuch that M\ {0} ⊂ A. •α-lineable if there exists a vector space Mwith dim(M) = αand M\ {0} ⊂ A. If, in addition, Xis a topological vector space, then the subset Ais said to be: •spaceable in Xwhenever there is a closed infinite-dimensional vector subspace Mof Xsuch that M\ {0} ⊂ A. •dense-lineable in Xwhenever there is a dense vector subspace Mof Xsatisfying M\ {0} ⊂ A. •α-dense-lineable in Xwhenever there is a dense vector subspace M of Xwith dim(M) = αand M\ {0} ⊂ A. And, provided that Xis a vector space contained in some (linear) algebra, then Ais called: •algebrable if there is an algebra Mso that M\{0} ⊂ Aand Mis infinitely generated, that is, the cardinality of any system of generators of Mis infinite. •strongly α-algebrable if there exists an α-generated free algebra M with M\ {0} ⊂ A. Recall that if Xis contained in a commutative algebra, then a set B⊂Xis a generating set of some free algebra contained in Aif and only if for any N∈N, any nonzero polynomial Pin Nvariables without constant term and any distinct f1, . . . , fN∈B, we have P(f1, . . . , fN)6= 0 and P(f1, . . . , fN)∈A. In 2000, Hencl [37] was able to show that every separable infinite-dimen- sional Banach space is isometrically isomorphic to a space of continuous functions on [0,1] that are, except for the null function, nowhere h¨olderian on [0,1]. In particular, the family of continuous nowhere h¨olderian functions on [0,1] is spaceable in C([0,1]), which improves corresponding results by a number of mathematicians on continuous nowhere differentiable functions (see [30,35,39,46]). In 2007, Bayart and Quarta [11] established the algebrability of the mentioned family, with the additional property that the existent algebra is dense in C([0,1]). The strong c-algebrabilty of the same family was finally proved by Bartoszewicz et al. in [8]. Concerning Pringsheim-singular functions, the c-dense-lineability of PS([0,1]) in C∞([0,1]) is shown in [17] (see [14,23] and [3, Section 1.4] for previous weaker results). The strong calgebrability of the (bigger) sets of smooth nowhere analytic functions and of smooth functions that are Pringsheim-singular at a cofinite subset of [0,1] were respectively proved in [8] and [17] (see also [10,29]). All results in the preceding paragraphs still hold, with essentially the same proofs, if we restrict ourselves to periodic functions that are continuous nowhere h¨olderian or smooth Pringsheim-singular. In fact, respective 4 BERNAL, BONILLA, L´ OPEZ-SALAZAR, AND SEOANE explicit examples of these kinds of functions are known (see, respectively, [38] and [14, Remark 2.2.1]): f(x) = ∞ X n=1 1 n2cos(2nπx), g(x) = ∞ X n=1 b1−n nsin(bnx), where bn:= 22 + cn+ [cn−1+Pn−1 j=1 bn+1−j j],cn:= (n+ 1)!(n+ 1)n+1 and the term inside the square brackets in the expression of bnis defined as 0 if n= 1. Consequently, we can in fact consider functions that are, respectively, continuous or smooth on T. In this order of ideas, recall that not every continuous function fon Tcan be extended to a function F∈A(D) (this takes place if and only if R2π 0f(eiθ)einθ dθ = 0 for all n∈N0). Therefore, the extensions of the previous residuality and lineability results from C(T) to A(D) is not automatic. Recently, Eskenazis [27] has established the residuality of the family of those F∈A(D) such that F|Tis nowhere differentiable on T. Even more, Eskenazis and Makridis [28] have proved that the more stringent class of functions F∈A(D) such that F|Tis nowhere h¨olderian on Tis residual in A(D). In fact, it is shown that, generically, both <f|Tand =f|Tare nowhere h¨olderian on T. (See also [40], where for certain domains Ω and closed sets J⊂∂Ω, the residuality of the set of those F∈H(Ω) such that Fis continuous on Ω and F|Jis nowhere lipschitzian on Jis studied). Turning to an algebraic point of view, the lineability –in its various degrees– of the class of functions Fin the disc algebra whose restriction to Tis nowhere differentiable/lipschitzian has been recently analyzed in [18]. In this paper, we go deeper into the subject and undertake the study of lineability of the family of those functions in the disc algebra whose restriction to the unit circle is nowhere h¨olderian, and of the family of those holomorphic functions on the unit disc that are smooth on the boundary but also Pringsheim singular (P-singular, from now on) at every point of this boundary. This will be performed in Sections 3 and 4, respectively. Section 2 will be devoted to provide a number of auxiliary statements. 2. Some auxiliary results The next lemma tells us essentially that both properties of non-h¨olderian and P-singular are preserved under composition with holomorphic functions. The first part is a complex version of Theorem 4.7 in [8], and can be verified along the same lines. Nevertheless, we furnish a (slightly modified) proof for the sake of completeness. The second part is in the same spirit as [8, Theorem 4.10], but the proof of this must be refined in order to give the conclusion of our lemma. Lemma 2.1. Let us suppose that f:T→Cis a function, z0∈T,Ωis a domain in Csuch that f(T)⊂Ω,ϕ∈H(Ω) is not constant and g:T→C is defined as g=ϕ◦f. We have: NOWHERE H ¨ OLDERIAN FUNCTIONS AND PRINGSHEIM SINGULAR FUNCTIONS 5 (a) If fis not α-h¨olderian at z0for any α∈(0,1], then gis not α-h¨olderian at z0for any α∈(0,1]. (b) If fis smooth on Tand P-singular at z0, then gis also P-singular at the point z0. Proof. (a) Fix α > 0 and k∈N. On the one hand, due to the assumption, there exists a sequence (zm)∞ m=1 ⊂T\ {z0}such that zm→z0and lim m→∞  f(zm)−f(z0) (zm−z0)α k= +∞. Without loss of generality we can assume f(zm)6=f(z0) for all m∈N. The continuity of fat z0entails f(zm)→f(z0) as m→ ∞. On the other hand, since ϕis nonconstant and holomorphic, there is k∈Nwith ϕ(k)(f(z0)) 6= 0. Assume that kis the least natural number with this property. Then lim m→∞  ϕ(f(zm)) −ϕ(f(z0)) (f(zm)−f(z0))k=ϕ(k)(f(z0)) k!>0. Consequently, we get lim m→∞  g(zm)−g(z0) (zm−z0)α= lim m→∞  ϕ(f(zm)) −ϕ(f(z0)) (f(zm)−f(z0))k· f(zm)−f(z0) (zm−z0)α k k = +∞. This means that gis not α-h¨olderian at z0. (b) Here we proceed by improving part of the argument used in the proof of [17, Theorem 4.6]. Firstly, it is evident that gis smooth on T. Next, we invoke the theory of formal power series, which can be found, for instance, in [22, Chap. 1]. Let z0=eit0,f0(t) := f(eit) and g0(t) := g(eit), so that g0=ϕ◦f0. Observe that g0=ψ2◦eϕ◦e f0◦ψ1, where ψ1(t) := t−t0,ψ2(z) := z+g(z0), e f0(t) := f0(t+t0) and eϕ(z) := ϕ(z)−g(z0). Since the fact of being P-singular is invariant under translations (ψ1and ψ2, in this case), we may assume that g0=ϕ◦f0with f0P-singular at t0= 0 and g0(0) = ϕ(f0(0)) = 0. Since ϕis not constant, there must be a first N∈Nsuch that ϕ(N)(f0(0)) 6= 0. By [47, Theorem 10.32], there exists a neighborhood U⊂Ω of f0(0) as well as a one-to-one function h∈H(U) such that ϕ(z) = h(z)Nfor all z∈U. Therefore g0= (h◦f0)Non some open real interval Jcontaining 0. Let G:= h◦f0, which belongs to C∞(J). Hence g0=GNand f0=h−1◦G on J, where h−1is well defined and analytic on a neighborhood of G(0) = 0. If f∗ 0(z) := P∞ n=0 f(n) 0(0) n!znand G∗(z) := P∞ n=1 G(n)(0) n!znare, respectively, the formal Taylor series of f0and Gat the origin, then we get from the analyticity of hthat G∗equals the formal Taylor series associated to h◦f∗ 0 6 BERNAL, BONILLA, L´ OPEZ-SALAZAR, AND SEOANE at the origin. If R(G, 0) = R(G∗,0) were not 0 then, since R(h−1,0) >0, we would have R(f0,0) = R(f∗ 0,0) >0 (see [22, Chap. 1]), so contradicting that f0is P-singular at 0. Therefore R(G, 0) = 0 and Gis P-singular at 0. Finally, since g0=GNon J, we get R(g0,0) = 0 (and we are done). Indeed, suppose that R(g0,0) = R(g∗ 0,0) >0, where g∗ 0represents the formal Taylor series g∗ 0(z) := P∞ n=1 g(n) 0(0) n!znassociated to g0at the origin. Thus, g∗ 0defines a holomorphic function on a neighborhood of 0. Let k be the first n∈Nwith G(n)(0) 6= 0. It follows from g0=GNthat Nk is the first n∈Nsuch that g(n) 0(0) 6= 0. Then g∗ 0(z) = zNkS(z) with Sholomorphic on a neighborhood of 0 and S(0) 6= 0. Therefore there exists a holomorphic function Ψ on a neighborhood Vof 0 with Ψ(z)N=S(z) (see [1, pp. 143-144]). Hence g∗ 0(z) = (zkΨ(z))Nin such neighborhood. Now, g∗ 0= (GN)∗= (G∗)N, and zkΨ(z) is a holomorphic branch of the Nth-root of g∗ 0on V. Therefore, there is an Nth-root αof 1 such that G∗(z) = αzkΨ(z) on V. In particular, R(G, 0) = R(G∗,0) >0, which contradicts the conclusion of the preceding paragraph. Therefore, R(g0,0) = 0 and the lemma is proved.  Remark 2.2. Under the notation and assumptions of the preceding lemma, it can also be proved that if fis smooth on Tand non-analytic at z0, then gis also non-analytic at z0. Anyway, this result will be not invoked along this paper. We shall also make use of the following auxiliary assertion, which is useful to discover dense-lineability from mere lineability. The proof of its part (a) can be found in [19, Theorem 2.3] (see also [3, Section 7.3], [4, Theorem 2.2 and Remark 2.5] and [15, Lemma 2.1]), while the proof of (b) can be extracted from the one of Theorem 2.2 in [4], because in fact the condition A+B⊂Ais not needed in its full strength. Lemma 2.3. Let Xbe a metrizable separable topological vector space. Suppose that Ais a subset of X. We have: (a) If αis an infinite cardinal number such that Ais α-lineable and there is a dense-lineable set B⊂Xsuch that A+B⊂Aand A∩B=∅, then Ais α-dense-lineable in X. (b) If there are an infinite dimensional vector space A0and a denselineable set B⊂Xsuch that A0\{0} ⊂ Aand (A0\{0})+B⊂A, then Ais dense-lineable in X. In order to construct large algebras of strange functions, we will employ the next lemma, whose proof can be found in [16] (see also [3, Section 7.5]). The lemma contains a complex version of a very useful method developed by Balcerzak et al. in [6, Proposition 7] (see also [8, Theorem 1.5 and Section 6] and [9]). By Ewe denote the family of exponential-like functions on C, NOWHERE H ¨ OLDERIAN FUNCTIONS AND PRINGSHEIM SINGULAR FUNCTIONS 7 that is, the functions of the form ϕ(z) = m X j=1 ajebjz for some m∈N, some a1, . . . , am∈C\ {0}and some distinct b1, . . . , bm∈ C\ {0}. Lemma 2.4. Let Ωbe a nonempty set and let Fbe a family of functions from Ωinto C. Assume that there exists a function f: Ω →Csuch that f(Ω) is uncountable and ϕ◦f∈ F for every ϕ∈ E. Then Fis strongly c-algebrable. By using Mergelyan’s approximation theorem (see, for instance, [47, Theorem 20.5]) and Cauchy’s inequalities, one can easily obtain the following well-known result. Lemma 2.5. Let Pdenote the family of restrictions to Dof all polynomials on C. Assume that f(z) := P∞ l=0 alzl∈H(D). We have: (a) The set Pis dense in A(D). (b) The set Pis dense in A∞(D). (c) f∈A∞(D)if and only if lNal∞ l=1 is a bounded sequence for every N∈N. 3. Large spaces of nowhere h¨ olderian functions Let NH(T) denote the family of all functions f∈A(D) such that the restriction f|Tis nowhere h¨olderian on T. Although the non-vacuousness of NH(T) has been already established by Eskenazis and Makridis [28], we here exhibit an explicit example for the convenience of the reader. Let f(z) = ∞ X n=1 z2n n2. Since P∞ n=1 1/n2<∞, the Weierstrass M-test for uniform convergence implies that f∈A(D). Let α∈(0,1], x0∈Rand z0=eiπx0∈T. If x∈R, then feiπx= ∞ X n=1 ei2nπx n2= ∞ X n=1 cos (2nπx) n2+i ∞ X n=1 sin (2nπx) n2. It is known that u(x) = P∞ n=1 cos(2nπx) n2is nowhere h¨olderian (see [38]). Thus, there is a sequence (xm)∞ m=1 in Rsuch that xm→x0as m→ ∞ and |u(xm)−u(x0)|> m |xm−x0|αfor every m∈N. Moreover, lim m→∞  xm−x0 eiπxm−eiπx0= 1 iπeiπx0=1 π. Hence there is m0∈Nsuch that if m≥m0, then  xm−x0 eiπxm−eiπx0>1 2π. 8 BERNAL, BONILLA, L´ OPEZ-SALAZAR, AND SEOANE Therefore, zm=eiπxm∈T,zm→z0as m→ ∞ and |f(zm)−f(z0)| ≥ |< [f(zm)−f(z0)]|=|u(xm)−u(x0)| > m |xm−x0|α=meiπxm−eiπx0α· xm−x0 eiπxm−eiπx0 α >m 2π|zm−z0|α for all m≥m0. This proves that fis nowhere h¨olderian on T. In this section, it will be shown that NH(T) possesses maximal algebraic size, in several senses. Take into account that both card(A(D)) and dim(A(D)) equal c. Theorem 3.1. The set N H(T)is strongly c-algebrable. In particular, it is c-lineable. Proof. Just take any f∈ NH(T) and apply Lemma 2.1(a) together with Lemma 2.4 with Ω = Dand F=N H(T).  Remark 3.2. If one had required mere algebrability for N H(T), the proof would have been even easier: the family {ϕ◦f:ϕentire, ϕ(0) = 0}is an infinitely generated algebra all of whose nonzero members belong to N H(T). Theorem 3.3. The set NH(T)is c-dense-lineable. Proof. According to Lemma 2.5, the set Pof restrictions to Dof all polynomials on Cis dense in A(D). In fact, Pis dense-lineable, because it is a vector space. Consequently, the set of all polynomials with coefficients in Q+iQis dense in A(D) and thus A(D) is a separable Banach space. Let us fix f∈ N H(T), P∈ P and z0∈T. The polynomial Pis differentiable at z0, so it is also α-h¨olderian at z0for any α∈(0,1]. If f+Pwere α-h¨olderian at z0for some α, then so would be f= (f+P)−P, which is absurd. Therefore, f+P∈ NH(T), that is, NH(T) + P ⊂ NH(T). Moreover, of course, NH(T)∩P =∅. Now, the conclusion follows from Theorem 3.1 and Lemma 2.3 as soon as we take α=c,X=A(D), A=NH(T) and B=P. Finally, we show the existence of large Banach spaces inside the disc algebra consisting, except for zero, of nowhere h¨olderian functions on the circle. To this end, let us recall (see, e.g., [41] and the references contained in it) that if v:C→(0,+∞) is a continuous function, then the weighted space of entire functions associated to vis defined as Hv(C) := f∈H(C) : sup z∈C v(z)· |f(z)|<∞. It is known that Hv(C) is a Banach space when it is endowed with the norm kfkv:= sup z∈C v(z)· |f(z)| NOWHERE H ¨ OLDERIAN FUNCTIONS AND PRINGSHEIM SINGULAR FUNCTIONS 9 and that convergence in k·kvimplies uniform convergence on compacta in C. Theorem 3.4. There exists a Banach space Xsatisfying the following properties: (a) X⊂A(D)and X\ {0} ⊂ NH(T). (b) Xis infinite dimensional. (c) The norm topology of Xis stronger than the one inherited from A(D). Proof. Fix any f∈ NH(T), for instance f(z) = P∞ n=1 z2n n2, and set v(z) := e−|z|. Consider the corresponding weighted Banach space Hv(C) and define X:= ϕ◦f:ϕ∈Hv(C), ϕ(0) = 0. Plainly, Xis a vector space and X⊂A(D). If g∈X\ {0}, then there is ϕ∈Hv(C) such that ϕ6= 0, ϕ(0) = 0 and g=ϕ◦f. Then ϕis a nonconstant holomorphic function on C, so g∈ NH(T) according to Lemma 2.1(a). That dim(X) = ∞follows from the facts that the monomials ϕk(z) = zk (k∈N) are linearly independent, belong to Hv(C) and satisfy ϕk(0) = 0. From this one derives that the functions f(z)k(k∈N) belong to Xand are linearly independent. Indeed, if λ1f(z) + · · · +λpfp(z) = 0 on D, then λ1z+· · · +λpzp= 0 on the nonempty open set f(D), so λ1= 0, . . . , λp= 0. Hence (a) and (b) have been proved. If g∈Xand ϕ1◦f=g=ϕ2◦f, where ϕ1, ϕ2∈Hv(C), then ϕ1=ϕ2 on the nonempty open set f(D), so ϕ1=ϕ2on the whole Cby the identity principle. Therefore, the mapping g=ϕ◦f∈X7→ kgkX:= kϕkv is well defined, and it is in fact a norm on X. Since convergence in Hv(C) implies uniform convergence on compacta, it follows that Y:= {ϕ∈Hv(C) : ϕ(0) = 0}is a closed subspace of Hv(C). Therefore, (Y, k·kv) is a Banach space, which in turn implies that k·kXis a complete norm on X. In order to prove (c), consider the supremum norm k·k∞on A(D). Since f(D) is compact, it follows that supz∈f(D)|z|<∞. Let K:= esupz∈f(D)|z|. For each g∈Xthere is ϕ∈Hv(C) with g=ϕ◦fand then kgk∞=kϕ◦fk∞= sup z∈f(D) |ϕ(z)| ≤sup z∈f(D) (K e−|z||ϕ(z)|) = K·sup z∈f(D) e−|z||ϕ(z)| ≤K·sup z∈C e−|z||ϕ(z)|=K· kϕkv=K· kgkX. This completes the proof.  16 BERNAL, BONILLA, L´ OPEZ-SALAZAR, AND SEOANE where K=β γ ·PN j=0 αj. This proves that the topology on Xis stronger than the one inherited from A∞(D), so the proof is finished.  Remarks 4.7. 1. By taking into account that convergence in Hv(C) implies uniform convergence on compacta in Cof derivatives of all orders, we can use the same example of Remark 3.5.1 (with f∈ PS(T) this time) to show that the norm topology of the space Xin Theorem 4.6 is strictly stronger than the one inherited from A∞(D). 2. Let He(D) be the class of functions in H(D) that cannot be continued holomorphically beyond T. In [13] and [16] the dense-lineability and the strong c-algebrability of He(D)∩A∞(D) were proved. But, plainly, PS(T)⊂ {f∈A∞(D) : f0is not analytic at any t∈R} ⊂ He(D)∩A∞(D). Hence Theorems 4.4 and 4.5 above provide two new approaches to get these properties. 2. Despite the guarantee of a large supply of functions in PS(T) given in Proposition 4.2 and Theorems 4.4-4.5-4.6, it might be of some interest to exhibit an explicit member of such special family. Since A((n!nn),T)⊂ PS(T), it is enough to give an example of a function f∈ A((cn),T) for every prescribed sequence (cn)∞ n=1 ⊂(0,+∞). To do this, notice that if we replace each cnby any natural number bigger than cn, then we can assume (cn)∞ n=1 ⊂N. Let us define recursively the sequence (bn)∞ n=1 of natural numbers as b1:= 2 + c1and bk=ck+2+ k−1 X j=1 bk+1−j j for k≥2. It is easy to see that (bk)∞ k=1 is increasing and bk>2 for all k∈N. Define the function f:D→Cby the lacunary power series f(z) = ∞ X k=1 b1−k kzbk. For every fixed N∈Nand every k≥N+1, we have that bN+1−k k≤2N+1−k. The majoring sequence 2N+1−k∞ k=1 tends to 0, so bN kb1−k k∞ k=1 is bounded. This entails f∈A∞(D) by using Lemma 2.5(c). Finally, observe that f0(t) = f(eit) = ∞ X k=1 b1−k keibkt(t∈R) and this function is proved to satisfy |f(n) 0(t)|> cn(n∈N, t ∈R) in [12, Section 2]. Then f∈ A((cn),T). Similarly to the end of Section 3, we conclude with a natural question: Is PS(T)spaceable in A∞(D)? NOWHERE H ¨ OLDERIAN FUNCTIONS AND PRINGSHEIM SINGULAR FUNCTIONS17 Note that PS(T) is not spaceable in A(D). 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Madrid, 28031, Spain E-mail address:[email protected] Instituto de Matem´ atica Interdisciplinar (IMI) Departamento de An´ alisis Matem´ atico Facultad de Ciencias Matem´ aticas Universidad Complutense de Madrid Plaza de Ciencias 3 Madrid, 28040, Spain E-mail address:[email protected]