Universal transforms of the geometric series under generalized Riesz methods

Universal transforms of the geometric series under generalized Riesz methods L. Bernal-Gonz´alez, M.C. Calder´on-Moreno∗and W. Luh Dedicated to the memory of Dieter Gaier Abstract In this paper generalized Riesz methods (R, p, M) of summability are considered. We prove that, to each open set O⊂Cwith adequate topological properties and each sequence {Pn} ⊂ Ctending to infinity, we can associate a corresponding P-regular (R, p, M)-method so that the geometric series and a certain trigonometric series become universal in the sense that its (R, p, M)-transforms approximate any member of certain spaces of holomorphic functions or measurable functions. 2000 Mathematics Subject Classification: Primary 30E10. Secondary 40C05, 40G99, 42A10. Key words and phrases: Riesz method, universal function, geometric series, trigonometric series, P-regularity. 1 Introduction Suppose that p:= {pν}∞ ν=0 is a sequence of complex numbers with the property that Pn:= mn X ν=0 pν6= 0 (n∈N0) ∗The first two authors have been partially supported by Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa. 0 for a subsequence M:= {mn}∞ n=0 of N0:= N∪ {0}={0,1,2, ...}. The row-finite matrix A= [αnν] with entries αnν := pν Pn for 0 ≤ν≤mn;αnν := 0 for ν > mn generates a summability method “of weighted mean type”, which occasionally is denoted as a generalized Riesz method (R, p, M) and which was first investigated by Faulstich (see [2]). Such a method (R, p, M) is regular (by the well known Silverman-Toeplitz conditions) if and only if lim n→∞ Pn=∞,sup n 1 |Pn| mn X ν=0 |pν|<∞; it is P-regular if and only if lim n→∞ Pn=∞,sup n 1 |Pn| mn X ν=0 |pν|ρν<∞for all ρ∈(0,1) (see Remark 3.2 (2) below). We recall that if A= [αnν]∞ n,ν=0 is a general infinite matrix with complex entries, then A(or the summability method generated by it) is called regular if it preserves convergence of limits of sequences, that is, given a sequence {sn}with sn→s∈Cthen it is also A-summable to sor, in other words, the sequence σn:= P∞ ν=0 αnνsνof its A-transforms also converges to s. And A is called P-regular (“regular for power series”) whenever for any given power series f(z) = P∞ ν=0 aνzνwith radius of convergence R∈(0,∞) the sequence σn(z) := P∞ ν=0 αnνsν(z) of its A-transforms (where sν(z) = Pν µ=0 aµzµ) converges to f(z) compactly in {z:|z|< R}. The exact conditions for regularity and P-regularity of a matrix Aare due respectively to Silverman and Toeplitz (see for instance [15] or [17, pages 6–7]) and Luh (see [8]). In 1945 Mensˇov [12] proved the existence of a so-called universal trigonometric series ∞ X ν=0 {aνcos ν t +bνsin ν t} with the property that for every Lebesgue measurable function ϕon [0,2π] there exists a subsequence {nk}of the natural numbers such that the corresponding sequence of partial sums snk(t) = nk X ν=0 {aνcos ν t +bνsin ν t} 1 converges to ϕ(t) almost everywhere on [0,2π]. It was shown in [10] (see also [13]) that there exist universal Taylor series ∞ P ν=0 aνzνwith radius of convergence 1 which for z=eit ∈Dbecome universal in the sense of Mensˇov. The trigonometric series ∞ P ν=0 {cos νt+sin νt}or the geometric series ∞ P ν=0 zν obviously cannot have corresponding universal properties. However it is the aim of the present paper to apply (R, p, M)-methods to these series, in such a way that they become universal in the sense that the corresponding transforms approximate any member of certain spaces of holomorphic and measurable functions. Our results strongly generalize those, which were obtained in [3]. The outline of the paper is as follows. Section 2 is auxiliary and in it a topological, crucial property is considered in order to be used later. In Section 3 we present two results on approximation of holomorphic functions. Section 4 is again auxiliary, and in it a general statement on Radon measures is shown; this section is of independent interest. In Section 5 we employ the assertions of the foregoing sections to establish a strong result about approximation of Lebesgue-measurable functions. 2 Sets with an exhausting property Throughout this paper we use the following notations and abbreviations. For an open set O⊂Cwe denote by H(O) as usual the family of all functions which are holomorphic in O. If K⊂Cis a compact set then A(K) stands for the collection of all functions which are continuous on K and holomorphic in the interior K0of K. By =⇒ Awe denote uniform convergence on a set A⊂C, while ≡> Astands for uniform convergence on any compact subset of A. Finally, Dwill represent the open unit disk {z:|z|< 1}. By Mwe denote the collection of all compact sets Kof the complex plane Cwhich have connected complement Kc. Definition 2.1. Suppose that Fis a set in C. Then Fhas the property E 2 (“exhausting property”) if either F=∅, or F6=∅and there exists a sequence {Kn}⊂Mwith Kn⊂Ffor all n∈Nsuch that for any K⊂Fwith K∈ M there exists an n0=n0(K)∈Nwith K⊂Kn0. Any such a sequence {Kn} is called an “exhausting sequence” for F. Examples 2.2. Let G⊂Cbe a simply connected domain, then Ghas the property E. Indeed, if G=Cwe choose S(n):= {z:|z| ≤ n}, and in the case that G6=Clet φbe a conformal mapping of Donto Gand consider the sets S(n):= φ{z:|z| ≤ 1−1 n+ 1}(n∈N). Then in both cases {S(n)}∞ n=1 is an exhausting sequence for G(actually for any compact set K⊂Gthere exists an n0∈Nwith K⊂S(n0)). More generally any domain G⊂Chas the property E. In fact it is well known that any open set G⊂Chas this exhausting property (cf. [9, page 198], [5, chapter 2.2]). The referee has kindly supplied an alternative proof of this assertion, which is a modification of the proof of Lemma 2.1 in [10] (see also [11]): In [10] having a locally finite number of components we succeds to consider a finite subset of Q+iQ. In the present situation the intersection of the component of Gwith a big closed disk is compact and is at a positive distance δfrom a compact set K. By compactness we can find a finite number of open disks with centers in this intersection, all with radii δ 1000, which cover this intersection. We can select an element of Q+iQin each one of these open disks. Then we arrive again to deal with a finite subset of Q+iQ and the approach of [10, Lemma 2.1] applies. The following examples show that closed sets may or may not have the property E. Examples 2.3. 1. If O⊂Cis an open set with infinitely many components then F=Ochas in general not the property E. This can be seen for instance by the example (see [10]) O= ∞ [ n=1 {z:|z−2−n|<2−n−2}. 3 However, if it is supposed that the number of components of Ois locally finite –that is, every compact subset L⊂Conly intersects a finite number N=N(L) of components of O– then it was shown by Melas and Nestoridis [10, Lemma 2.1] that Ochas the property E. 2. In particular, if O⊂Cis an open set with a finite number of components, then F=Ochas the property E. 3. Consequently, if Gis any domain in C(= a nonempty connected open subset of C), then F=Gchas the property E. In this case, we can provide with the following easy proof, which is independent of that of [10]: The result is clear if G=C. Suppose that G6=Cand choose a closed circle S={z:|z−z0| ≤ r} ⊂ G. Let {Ln}be an enumeration of all Jordan domains in Scwhich are bounded by polygons with vertices in points which have rational real and imaginary parts. Then each closure Lnbelongs to M and for any set L∈ M with L⊂Scthere exists an n0with L⊂Ln0. It is easy to see that the sequence {Kn}with Kn:= Ln∩Gcis an exhausting sequence for Gc. 3 Approximation of holomorphic functions We first proof the following results. Theorem 3.1. Let be prescribed: – an open set O⊂Cwith simply connected components and D⊂O, 1/∈O; – a set F⊂Ocwhich has the property E; – a function ϕ∈H(O)with ϕ|D=ϕ0, where ϕ0(z)≡0; – a sequence {Pn} ⊂ C\ {0}with Pn→ ∞. (a) Then there exist sequences {pν} ⊂ Cand {mn} ⊂ N0such that Pn= Pmn ν=0 pν(n∈N)and τn(z) := 1 Pn mn X ν=0 pνzν≡> Oϕ(z) (n→ ∞). 4 (b) If in addition any set K∈ M,K⊂F,1/∈Kand any function f∈A(K)are given then there exists a sequence {nk}with τnk(z)=⇒ Kf(z) (k→ ∞). Proof. 1. Suppose that O=[ ν∈I Gν, where 0 ∈Iand I⊂N0, that the Gν’s are pairwise disjoint simply connected domains (the components of O) and assume D⊂G0. For ν∈Ichoose a conformal mapping φνof Donto Gνand consider the sets S(n) ν:= φν({z:|z| ≤ 1−1 n+ 1}) (n∈N, ν ∈I), Tn:= S(n) 0, Sn:= [ ν∈I 1≤ν≤n S(n) ν(n∈N), which are compact, have connected complement and the sets Tnand Snare pairwise disjoint for each n∈N. Since Fhas the property Eit is not hard to see that also F1:= F\{1} has the property E. Without loss of generality we may assume that F16=φ. Let {K∗ n}be an exhausting sequence for F1and denote by {Π∗ n}an enumeration of all polynomials whose coefficients have rational real and imaginary parts. Finally let {(Kn,Πn)}∞ n=1 be an arrangement of the sets K∗ nand the polynomials Π∗ nin which any combination (K∗ r,Π∗ s) occurs infinitely often. It follows that for each nthe sets Tn, Sn, Kn,{1}are pairwise disjoint. 2. We construct by induction a sequence {Qn}of polynomials and sequences {qn}and {mn}of nonnegative integers. Suppose that Q0(z)≡P0and q0= 0 and assume that for an n∈N the polynomials Q0, ..., Qn−1and the numbers q0, ..., qn−1have already been determined. The degree of the polynomial zqn−1Qn−1(z) will be denoted by mn−1. We choose qn∈Nso great that qn> mn−1and using Runge’s approximation theorem (see [4, Chapter II,3]) we find a polynomial Qnwhich satisfies simultaneously (1) Qn(1) = Pn−Pn−1, 5 (2) max Tn |Qn(z)|<1 n2maxTn|zqn|, (3) max Sn Qn(z)−Pn·ϕ(z)−Pn−1 ν=0 zqνQν(z) zqn <1 maxSn|zqn|, (4) max Kn Qn(z)−Pn·Πn(z)−Pn−1 ν=0 zqνQν(z) zqn <1 maxKn|zqn|. By induction we get {Qn},{qn},{mn}; note that mn> mn−1and qn> qn−1 for all n∈N. 3. The series P∞ ν=0 zqνQν(z) converges by (2) compactly in G0and therefore g(z) := P∞ ν=0 zqνQν(z) is holomorphic in G0. The properties of the qnand mn imply that the polynomials zqn−1Qn−1(z) and zqnQn(z) do not have powers in common. Therefore, if the power series of the function garound the origin is denoted by g(z) = P∞ ν=0 pνzν, we obtain mn X ν=0 pνzν= n X ν=0 zqνQν(z). ¿From (1) we get Pmn ν=0 pν=Pn ν=0 Qν(1) = Pn→ ∞ for n→ ∞ which implies that the power series P∞ ν=0 pνzνhas radius of convergence 1. We obviously have ϕ|G0(z)≡0, which together with (2) gives τn(z) := 1 Pn mn X ν=0 pνzν=1 Pn n X ν=0 zqνQν(z)≡> G0 ϕ|G0(z)≡0. 4. The property (3) implies max Sn |τn(z)−ϕ(z)|= max Sn 1 Pn n X ν=0 zqνQν(z)−ϕ(z) <1 |Pn| and we obtain τn(z)≡> Gν ϕ(z) for all ν∈I, ν > 0. Together with step 3 we have τn(z)≡> Oϕ(z). 6 5. From (4) we get (5) max Kn |τn(z)−Πn(z)|<1 |Pn|. Let now be given a set K∈ M with K⊂F\ {1}=F1and a function f∈A(K). Then by Mergelian’s theorem (see [4, Chapter III,2]) there exists a sequence {sk}with sk→ ∞ and Π∗ sk(z)=⇒ Kf(z). By the exhaustion property of F1there exists an r0with K⊂K∗ r0and we find a sequence {nk} with Πnk(z) = Π∗ sk(z) and Knk=K∗ r0for all k. Together with (5) we get τnk(z)=⇒ Kf(z), which proves the theorem. Remarks 3.2. 1. We consider especially the open set O=D. By the Example 2.3(3) the set F:= Dchas the property E. Therefore Theorem 3.1 generalizes Theorem 1 of [1], where we proved the same approximation properties given in the former theorem but just in the very special case O=D,F=Dc,ϕ= 0, K⊂∂D\ {1}. Several consequences on the approximation of measurable functions by universal trigonometric series are given in [1]. This will also be treated later in our current, more general situation, see Section 5. 2. The sequence p={pν}which was constructed in the proof of Theorem 3.1 obviously satisfies τn(1) = 1 for all n∈N0and τn(z)≡> D0. It is not difficult to show that these conditions are equivalent to those which were mentioned in the introduction for P-regularity (for details we refer to [8]). Therefore the considered (R, p, M)-method is P-regular. 3. However it is easy to see that the method (R, p, M) in general cannot chosen to be regular. Indeed, suppose that the sets Oand Fhave the property that there exists a ς0∈∂D∩Fwith ς06= 1. If (R, p, M) would be regular, then there would exist a constant Cwith 1 |Pn| mn X ν=0 |pν| ≤ Cfor all n∈N. 7 On the other hand by property (b) of Theorem 3.1 we can find a sequence {nk}of natural numbers with τnk(ς0) = 1 Pnk mnk X ν=0 pνςν 0→C+ 1 for k→ ∞ , which is obviously not possible. ¿From Theorem 3.1 the following statement about the universal behaviour of the (R, p, M)-transforms of the geometric series follows very easily. Theorem 3.3. Suppose that O, F and {Pn}are the same as in Theorem 3.1. Let be given a function Φ∈H(O)with Φ|D= Φ0, where Φ0(z)≡1 1−z. (a) Then there exist sequences {pν} ⊂ Cand {mn} ⊂ N0such that Pn= mn X ν=0 pν(n∈N)and σn(z) := 1 Pn mn X ν=0 pν ν X µ=0 zµ≡> OΦ(z). (b) If in addition any set K∈ M,K⊂F,16∈ Kand any function f∈A(K)are given then there exists a sequence {nk}with σnk(z)=⇒ Kf(z). Proof. Let {τn(z)}be the sequence which by Theorem 3.1 exists according to O, F, {Pn}and the function ϕwith ϕ(z) := 0 if z∈G0 1 z−1−z zΦ(z) if z∈O\G0. Then there are sequences {pν}and {mn}such that τn(z) := 1 Pn mn X ν=0 pνzν≡> Oϕ(z). 8 References [1] L. Bernal-Gonz´alez, M.C. Calder´on-Moreno and W. Luh, Universality and summability of trigonometric polynomials and trigonometric series, to appear in Periodica Math. Hungar. [2] K. Faulstich, Summierbarkeit von Potenzreihen durch Riesz-Verfahren mit komplexen Erzeugendenfolgen, Mitt. Math. Sem. Gießen 139 (1979). [3] K. Faulstich, W. Luh and L. Tomm, Universelle Approximation durch Riesz-Transformierte der geometrischen Reihe, Manuscripta Math. 36 (1981), 309–321. [4] D. Gaier, “Lectures on Complex Approximation”, Birkh¨auser, Basel/London/Stuttgart, 1987. [5] K. G. 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Nestoridis, Universal Taylor Series, Ann. Inst. Fourier 46 (1996), 1293–1306. [14] O.A. Nielsen, “An Introduction to Integration and Measure Theory”, J. Wiley and Sons, Inc., New York, 1997. [15] A. Peyerimhoff, ”Lectures on summability”, Lecture Notes in Mathematics, Springer, Berlin-Heidelberg-New York-Tokyo, 1970. [16] W. Rudin, “Real and Complex Analysis”, 3rd ed., McGraw-Hill, New York, 1987. [17] A. Wilansky, “Summability through Functional Analysis”, NordHolland, Amsterdam, 1984. L. BERNAL-GONZ´ ALEZ and M.C. CALDER´ ON-MORENO WOLFGANG LUH DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO UNIVERSIT¨ AT TRIER FACULTAD DE MATEM´ ATICAS, APDO. 1160 FACHBEREICH MATHEMATIK AVENIDA REINA MERCEDES D-54286 TRIER 41080 SEVILLA, SPAIN GERMANY E-mails: lb[email protected], [email protected] E-mail: [email protected] 16