On Taylor coefficients of entire functions integrable against exponential weights Oscar Blasco and Antonio Galbis of Valencia Abstract In this paper we shall analyze the Taylor coefficients of entire functions p −|z|p | |p−2 integrable against dµp(z)= 2π e z dσ(z) where dσ stands for the Lebesgue measure on the plane and p ∈ IN, as well as the Taylor coefficients of entire functions in some weighted sup-norm spaces. In this paper we shall analyze the Taylor coefficients of entire functions sat- isfying some growth estimates. To be more precise, given p ∈ IN, we will deal with the Banach space B1(p) of entire functions belonging to L1(dµ), where p −|z|p | |p−2 dµ(z)= 2π e z dσ(z) and dσ stands for the Lebesgue measure on p the plane, as well as with the Banach space H(e−|z| )(C) of those entire func- −|z|p | | ∞ tions f such that supz∈C e f(z) < . These spaces have been considered in several contexts by different authors. See [1, 6, 7, 8, 9, 10]. The general question we are going to discuss can be stated as follows: given a function ∞ n −|z|p f(z)= n=0 anz in X(:= B1(p)orH(e )(C)), what can be said on the Taylor coefficients (an)?. Conversely, it is also interesting to ask how a func- tion in X can be recognized by the behaviour of its Taylor coefficients. The paper is organized as follows. In the first section we present a method to de- scribe the boundedness of operators from B1(p) into a general Banach space X by the fact that the X−valued analytic function constructed by the action of the operator on the reproducing kernel Kp belongs to the vector-valued space −|z|p H(e )(C; X). This will allow to identify the dual space of B1(p) with the p weighted sup-norm space H(e−|z| )(C). Then we will discuss a Hardy’s type inequality for Taylor coefficients of functions in B1(p). In the second section we give a complete characterization of the Taylor coefficients for lacunary entire −|z|p functions in both spaces B1(p) and H(e )(C). As an application we obtain a sufficient condition on the Taylor coefficients of a function f in order to enssure −|z|p that it belongs to H(e )(C). In section 3 we find conditions on nk in order to nk get the unconditional convergence of akz to be equivalent to the absolute convergence of the series. −|z|p −|z|p Let us denote by H(e )0(C) the closed subspace of H(e )(C) con- p sisting of those functions f such that e−|z| f(z) vanishes at infinity. Since 1 −|z|p the polynomials are dense in B1(p) and in H(e )0(C) it is natural to ask whether the Taylor series of a function in those spaces necessarily converges in norm. Such a question was raised by D.J.H. Garling and P. Wojtaszczyk [7] for the space B1(2), corresponding to those entire functions which are integrable with respect to a gaussian measure, and it was recently solved in the negative −|z|p by W. Lusky [10] for all the spaces B1(p) and H(e )0(C). Nevertheless our results in Section 2 show that when restricted to a lacunary sequence nk, i.e. nk+1 nk ≥ λ>1 for all k ∈ IN, we have that (z ) is a basic subsequence in B1(p). nk The final part of the paper is devoted to give a necessary and also two sufficient conditions in order to ensure the unconditional convergence of a given Taylor −|z|p series in H(e )0(C). 1 Duality −|z|p In this section we present the Banach spaces B1(p) and H(e )(C) and show ∗ −|z|p that (B1(p)) = H(e )(C). This duality is applied to discuss the sharpness of a Hardy’s type inequality for functions in B1(p). Moreover, as a previous step to get the duality some necessary and sufficient conditions for a function p to belong to H(e−|z| )(C) are given. Definition 1.1 Given a continuous and radial weight v on C and a complex Banach space (X, . ) we define (a) H(v)(C,X):={F : C → X entire function; F := sup v(z) F (z) < ∞}, (b) H(v)0(C,X) is the subspace of H(v)(C,X) consisting of those functions F such that Fv vanishes at infinity. If X is the field of complex numbers we drop it from the notation and write p H(v)(C) or H(v)0(C). We are interested in weights v(z) = exp(−|z | ), p ∈ IN . Definition 1.2 Given a natural number p ∈ IN we denote by B1(p) the space of entire functions f such that p p f := | f(z) | e−|z| | z |p−2 dσ(z) < ∞. 2π C 1 2π We write M∞(f,r) := max{| f(z) |:| z |= r} and M1(f,r):= | ∞ 2π 0 it | ∈ −rp p−1 f(re ) dt. Then, for every f B1(p), we have f = 0 M1(f,r)e pr dr. Lemma 1.1 (a) Let v be a continuous and radial weight on C such that the polynomials are contained in H(v)0(C). Then the polynomials are dense in H(v)0(C). (b) For every p ∈ IN , the polynomials are dense in B1(p). 2 Proof: A proof of part (a) can be found in [3, 1.5(a)]. To prove (b) proceed as in [7, Proposition 5]. − p 1 n r n p Let us first remark that ϕp(r)=r e is an increasing function in [0, ( p ) ] n 1 p ∞ n −| |p and decreasing in [( p ) , + [. What shows that un(z)=z satisfies un H(e z )(C)= n n − n ( ) p e p . This, using the trivial estimate | b | Rn ≤ M∞(g,R), also allows to p n n −|z|p say that if g(z)= bnz ∈ H(e )(C) then | | n bn Γ( p +1) √ ≤ p (1.1) sup C g H(e−|z| ) . n∈IN n +1 Let us start by mentioning a simple condition on (bn) which implies that p g ∈ H(e−|z| )(C). Lemma 1.2 (a) Let p ∈ IN and let (bn) be a sequence such that sup | bn | n∈IN n n −|z|p Γ( +1)< ∞. Then g(z)= bnz ∈ H(e )(C). p n n −|z|p (b) If lim | bn | Γ( +1)=0 then g(z)= bnz ∈ H(e )0(C). n→∞ p Proof: To see (a) it suffices to show that ∞ n r p ≤ Cer Γ( n +1) n=0 p for every r>0. For each n ∈ IN write n = pk + j, k ∈ IN and j =0, 1,...p− 1, and decompose the sum as follows ∞ p−1 rn = ϕ (rp) Γ( n +1) j n=0 p j=0 where ∞ k+ j t p ϕj(t)= j . k=0 Γ(k + p +1) j − j p 1 Since ϕj(t)= j t + ϕj(t)wehave pΓ( p +1) t j − j − t s p 1 ≤ t ϕj(t)=e (ϕj(0) + j e s ds) e (ϕj(0)+1). pΓ( p +1) 0 Adding the values for j =0, 1,...p− 1weget ∞ − n p 1 r p = ϕ (rp) ≤ (1+p)er . Γ( n +1) j n=0 p j=0 3 −|z|p −|z|p (b) Since H(e )0(C) is a closed subspace of H(e )(C) it suffices to N −|z|p show that g = lim bkuk in H(e )(C). But this follows from N→∞ k=0 ∞ ∞ k | | k −rp ≤ | | k r −rp ( bk r )e sup bk Γ( +1) k e k>N p Γ( +1) k=N+1 k=N+1 p k ≤ C sup |bk|Γ( +1) . k>N p p Let us now find some necessary condition for a function to belong to H(e−|z| )(C). ∞ n Lemma 1.3 Let (αn) be a sequence of positive real numbers. If f(z)= αnz p n=0 belongs to H(e−|z| )(C) then m 1 n sup αnΓ( +1)< ∞. ∈ m p m IN n=0 ∞ n ≥ p ≤ r Proof: Since αn 0 then we are assuming that n=0 αnr Ce for every r>0. Hence, multiplying by e−ar (a>1) and integrating over (0, ∞) we get ∞ αn n ≤ C n +1 Γ( +1) . p p a − 1 n=0 a ∈ m+1 For m IN take a = m and then m ∞ 1 m +1 n n 1 n +1 (1 − ) p α Γ( +1)≤ α Γ( + 1)(1 − ) p ≤ Cm. m +1 n p n p m +1 n=0 n=0 1 m +1 − 1 →∞ − p p Using that limm (1 m+1 ) = e we finish the proof. −|z|p In order to get the duality between B1(p) and H(e )(C) let us first give −|z|p a natural pairing on these spaces. If f ∈ B1(p) and g ∈ H(e )(C) we can define p p <f,g>= f(ω)g(ω)e−2|ω| | ω |p−2 dσ(ω). 2π C | |≤ −|z|p Clearly <f,g> f B1(p) g H(e )(C) . Observe that <un,g> = 2n Γ( p +1) n bn 2n +1 for g(z)= bnz . 2 p 2n +1 ∞ 2 p n This leads to the consideration of the following function Kp(z)= n=0 2n z .