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On Taylor Coefficients of Entire Functions Integrable Against 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 .
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