Anti M-Weierstrass function sequences

Depósi o de in es igación de la Uni e sidad de Se illa h ps://idus.us.es/ Es a es la e sión acep ada del a ículo publicado en: This is an accep ed manusc ip o a pape published in: Jou nal o Ma hema ical Analysis and Applica ions (2020): 1 No embe DOI: 10.1016/j.jmaa.2020.124261 Copy igh : El acceso a la e sión publicada del a ículo puede eque i la susc ipción de la e is a. Access o he published e sion may equi e subsc ip ion. “This is an Accep ed Manusc ip o an a icle published by Else ie in [JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS] on [1 No embe 2020], a ailable a : h ps://doi.o g/10.1016/j.jmaa.2020.124261].” ” ANTI M-WEIERSTRASS FUNCTION SEQUENCES MAR´ IA DEL CARMEN CALDER´ ON-MORENO, PABLO JOS´ E GERLACH-MENA, AND JOS´ E ANTONIO PRADO-BASSAS Abs ac . La ge algeb aic s uc u es a e ound inside he space o sequences o con inuous unc ions on a compac in e al ha ing he p ope y ha , he se ies de ined by each sequence con e ges absolu ely and uni o mly on he in e al bu he se ies o he uppe bounds di e ges. So showing ha he e exis many examples sa is ying he conclusion bu no he hypo hesis o he Weie s ass M- es . 1. In oduc ion In any ield o Ma hema ics, whene e he uni o m con e gence o a se ies o unc ions P∞ n=1 n(x) mus be s udied (whe e n:X→K,X6=∅and K= he eal line Ro he complex plane C), he ool we i s hink abou is he well-known Weie s ass M- es . I he e exis s a sequence (Mn)no posi i e numbe s such ha i s se ies is con e gen and | n(x)| ≤ Mn o e e y x∈Xand e e y n∈N(:= he se o all posi i e in ege s), hen he se ies P∞ n=1 n(x) is absolu ely and uni o mly con e gen on X(see, o ins ance, [2, § 9.6]). The con e se is alse in gene al. Conc e ely, he hypo hesis abou he majo an sequence can be d opped wi hou losing he uni o m con e gence o he se ies. Indeed, o e e y n∈Nand e e y x∈[0,1], de ine (1) n(x) = (1 nsin2(2n+1πx) i x∈2−(n+1),2−n 0 elsewhe e. The sequence n(x) canno be majo a ed by any summable sequence o posi i e numbe s, since supx∈[0,1] | n(x)|=1 n; howe e , he se ies P∞ n=1 n(x) is absolu ely and uni o mly con e gen (see [16, Chap e 1, Example 10] o mo e de ails). Such a sequence o unc ions will be called An i M-Weie s ass (see De ini ion 1.1 below). We deno e by C([a, b])N he se o all sequences o con inuous unc ions on a non- degene a e compac in e al [a, b]⊂R. Recall ha i we endow C([a, b])Nwi h he Key wo ds and ph ases. se ies o unc ions, uni o m con e gence, di e gence, spaceabili y, algeb abili y. 2020 Ma hema ics Subjec Classi ica ion. 46B87, 28A20, 40A30, 46A45. The au ho s ha e been pa ially suppo ed by he Plan Andaluz de In es igaci´on de la Jun a de Andaluc´ıa FQM-127 and by MICINN G an PGC2018-098474-B-C21. 1 2 CALDER ´ ON, GERLACH, AND PRADO p oduc opology inhe i ed om (C([a, b]),k·k∞), whe e k k∞= supx∈[a,b]| (x)| is he classical sup emum no m, we ge a comple e me izable and sepa able opo- logical ec o space (see [18, Chap e 7]). De ini ion 1.1. Le ( n)n∈C([a, b])Nbe a sequence o con inuous unc ions on [a, b]. We say ha ( n)nis an An i M-Weie s ass sequence i P∞ n=1 n(x) is absolu ely and uni o mly con e gen on [a, b], bu P∞ n=1 k nk∞di e ges. We will deno e by AMW([a, b]) (o jus AMW, whene e he e is no possibili y o con usion) he amily o An i M-Weie s ass sequences on [a, b]. The aim o his pape is cons uc ing la ge linea s uc u es o An i M-Weie s- ass sequences, and his objec i e lies in he b anch o Lineabili y, a end o esea ch ha has a ac ed he a en ion o many ma hema icians du ing he pas ew decades. We in oduce in he nex pa ag aph he main no ions abou Line- abili y we a e going o deal wi h. We e e he in e es ed eade o [4, 15] o a gene al backg ound. Gi en a ec o space Xand a ( ini e o in ini e) ca dinal numbe κ, we say ha a subse Ao Xis κ-lineable whene e he e is a ec o space Mo dimension κ such ha M {0} ⊂ A; when κ= dim(X), we speak abou maximal-lineabili y. Mo eo e , i Xis a opological ec o space, hen Ais said o be spaceable (dense- lineable, κ-dense-lineable, maximal-dense-lineable, esp.) in X, i he e is a closed in ini e dimensional (a dense, a dense κ-dimensional, a dense dim(X)-dimensional, esp.) ec o space Mwi h M {0} ⊂ A. When, in addi ion, Xis con ained in some (linea ) algeb a, hen Ais called κ-algeb able i he e is an algeb a Mso ha M {0} ⊂ Aand he ca dinali y o any minimal sys em o gene a o s o Mis κ; i he algeb a can be aken o be ee, we say ha Ais s ongly κ-algeb able; and i he ( ee) algeb a is dense in X, we speak abou (s ongly) κ-dense-algeb abili y. I he algeb aic s uc u e o Xis commu a i e, he s ong algeb abili y is equi alen o he exis ence o a gene a ing sys em Bwi h ca d(B) = κo he algeb a Msuch ha o any N∈N, any nonze o polynomial Pin N a iables wi hou cons an e m and any dis inc b1, . . . , bN∈B, we ha e ha P(b1, . . . , bN)∈M {0}. Conce ning lineabili y o se ies, in 2005 Baya [10] showed ha he se o con- inuous unc ions on he uni ci cle Twhose Fou ie se ies di e ges on a se E⊂T o measu e ze o is dense-lineable. A yea la e , A on, P´e ez-Ga c´ıa and Seoane- Sep´ul eda [6] es ablished he dense-algeb abili y. When Eis coun able, esul s on lineabili y o di e gen Fou ie se ies wi h addi ional p ope ies a e ob ained in [13] o [17]. Again Baya [9, 10] showed ha he se o Di ichle se ies (s) = P∞ n=1 ann−s ha a e bounded in he igh hal -plane and di e ge e e ywhe e in he imagi- na y axis is lineable and spaceable. Toge he wi h Qua a [11], hey we e able o es ablish also he algeb abili y. In he se ing o Banach spaces, Aizpu u, P´e ez-Esla a and Seoane-Sep´ul eda in 2006 [1], asse ed ha he se o uncondi ionally con e gen bu no absolu ely ANTI M-WEIERSTRASS FUNCTION SEQUENCES 3 con e gen se ies o any in ini e dimensional Banach space is c-lineable (whe e c deno es he ca dinali y o con inuum). Mo eo e , he se o sequences such ha i s pa ial sums a e bounded bu he se ies di e ges is also c-lineable. I we ocus on he complex plane, Ba oszewicz, G lab and Po eda p o ed in [8] ha he se o nonabsolu ely con e gen complex se ies and he se o di e gen complex se ies wi h bounded pa ial sums a e c-algeb able. Following wi h sequences o eal numbe s, in 2013 Ba oszewicz and G lab [7] showed ha c0 Sp≥1`pis densely s ongly c-algeb able, whe e c0deno es he se o all eal sequences con e ging o ze o, and `pis he se o all p-summable eal sequences. La e , in 2017, A a´ujo e al. [3] s udied he linea s uc u e o he amily o sequences o eal numbe s whose co esponding se ies ail bo h he oo and a io es . Speci ically, hey show ha he se o sequences in `1gene a ing se ies o which he a io o he oo es s ail and he se o sequences in ω( he ec o space o all eal sequences endowed wi h he p oduc opology) gene a ing di e gen se ies o which he a io and he oo es s ail a e c-dense-lineable in `1o ω. In his pape we a e conce ned wi h lineabili y –in i s se e al deg ees– o he am- ily AMW desc ibed in De ini ion 1.1. In Sec ion 2 we shall de ine a special amily o sequences o unc ions ha will be c ucial in he o hcoming cons uc ions and p o ide se e al auxilia esul s abou i . Sec ion 3 is de o ed o a cons uc ion o se e al ec o spaces inside AMW and o ge i s maximal dense lineabili y and spaceabili y. Finally, in Sec ion 4, we conclude he pape by es ablishing, as a consequence o a mo e gene al esul , he s ong c-algeb abili y o ou amily AMW. 2. Concep s and Examples Inspi ed by he Example (1) o An i M-Weie s ass sequence p o ided in he in oduc ion, we can es ablish a e y use ul auxilia y esul . By supp( ) we deno e he suppo o a unc ion : [a, b]→K, ha is, he se o poin s x∈[a, b] whe e does no anish. Le Fbe he amily o all sequences o unc ions (un)n∈C([a, b])Nsuch ha : •The suppo s a e pai wise disjoin , ha is, (2) supp(un)∩supp(um) = ∅, n 6=m. •The sequence (un)nis bo h uni o mly bounded and uni o mly a om ze o, ha is, (3) 0 <in n||un||∞≤sup n ||un||∞<+∞. Lemma 2.1. Le (un)n∈ F and le (an)n⊂R. Then we ha e: (a) The se ies ∞ X n=1 anun(x)con e ges absolu ely on [a, b]. 4 CALDER ´ ON, GERLACH, AND PRADO (b) The se ies ∞ X n=1 anun(x)con e ges uni o mly on [a, b]i and only i (an)n∈ c0. (c) The se ies ∞ X n=1 kanunk∞<+∞i and only i (an)n∈`1. P oo . (a) The absolu e con e gence o he se ies is immedia e, since he disjoin - ness o he suppo s o he un’s implies ha , o a ixed x0∈[a, b], ei he un(x0)=0 o all n∈N, o he e exis s only one n0∈Nsuch ha x0∈supp(un0), and ∞ X n=1 |anun(x0)|=|an0un0(x0)|(<+∞). (b) Fi s ly, assume ha (an)n∈c0. Because (un)n∈ F, hen M:= supn||un||∞∈ (0,+∞). Gi en any ε > 0, he e exis s N∈Nsuch ha |an|<ε M o any n≥N. Thus, as o each x∈[a, b] he e is a mos one n0≥Nsuch ha x∈supp(un0),  ∞ X n=N anun(x) =|an0un0(x)| ≤ M·ε M=ε. Hence, he uni o mly con e gence o he se ies on [a, b] is ob ained. Con e sely, as he se ies is uni o mly con e gen , we ha e anun(x)→0 (n→ ∞) uni o mly on [a, b]. Because (un)n∈ F, hen L:= in n||un||∞>0. Gi en any ε > 0, he e is n0∈Nsuch ha |anun(x)|< ε ·L o any x∈[a, b] and any n≥n0. Bu o any n∈N, he e exis s, by con inui y, a poin xn∈[a, b] such ha |un(xn)|=||un||∞≥L, hence o n≥n0we ge |an|=|anun(xn)| |un(xn)|≤ε·L L=ε, and an→0 (n→ ∞). (c) As (un)n∈ F, we ha e (4) 0 < L := in n||un||∞≤M:= sup n ||un||∞<+∞. Thus, by he compa ison es (an)n∈`1i and only i (||anun||∞)n∈`1, and we ha e (c).  As a consequence o he abo e lemma we ob ain he nex ac . Co olla y 2.2. Fo any sequence o unc ions (un)n∈ F and any sequence o scala s (an)n∈c0 `1, he sequence o con inuous unc ions (anun)nis in AMW. In pa icula , his co olla y will allow us o p o ide a wide ple ho a o sequences exhibi ing his s ange beha iou . In he ollowing example we show ha om any non null unc ion ∈C([a, b]), many sequences in Fcan be cons uc ed. ANTI M-WEIERSTRASS FUNCTION SEQUENCES 5 Example 2.3. Le ∈C([a, b]) {0}. Le Λ := (αn)nbe any sequence o scala s such ha a=α1< α2<· · · < αn−1< αn<· · · −→ b(n→ ∞). Fo each n∈Nwe de ine he unc ion uΛ, n: [a, b]→Rby uΛ, n(x) :=                      (a)·x−α3n−2 α3n−1−α3n−2 i x∈[α3n−2, α3n−1] a+b−a α3n−α3n−1 (x−α3n−1)i x∈[α3n−1, α3n] (b)·α3n+1 −x α3n+1 −α3n i x∈[α3n, α3n+1] 0 o he wise. I is clea ha uΛ, n∈C([a, b]) and supp(uΛ, n)⊂(α3n−2, α3n+1) o each n∈N, so he suppo s o he uΛ, n’s a e pai wise disjoin . In addi ion, ||uΛ, n||∞=|| ||∞∈ (0,+∞) o any n∈N. So, i ially, (uΛ, n)n∈ F. The e o e, o any p e ixed inc easing sequence o scala s Λ as abo e, we ha e de ined an injec i e mapping JΛ:C([a, b]) {0} −→ F 7−→ JΛ( ) := (uΛ, n)n, sa is ying he ollowing p ope ies: (1) uΛ, n(α3n−1) = (a) and uΛ, n(α3n) = (b) o any n∈Nand any ∈ C([a, b]) {0}. (2) supp(uΛ, n)⊂(α3n−2, α3n+1) o any ∈C([a, b]) {0}and any n∈N. (3) Fo any n∈N he e exis s a linea a ine ans o ma ion τnsuch ha τn([α3n−1, α3n]) = [a, b] and uΛ, n= ◦τnon [α3n−1, α3n] o each ∈ C([a, b]) {0}. (4) ||uΛ, n||∞=|| ||∞ o any n∈Nand any ∈C([a, b]) {0}. Obse e ha i we de ine JΛ(0) as he null sequence, we ha e ha JΛis a linea and injec i e mapping om C([a, b]) o F ∪{0}. In pa icula i Cis a linea ec o subspace in C([a, b]) wi h dimension κ, hen JΛ(C) is a linea ec o subspace in F ∪ {0}wi h dimension κ. Fu he mo e, le `∞(C([a, b])) be he ec o space o all uni o mly bounded sequences o con inuous unc ions in [a, b], which is a Banach space when endowed wi h he uni o m sup emum no m ||( n)n||`∞:= sup n || n||∞. Obse e ha F ⊂ `∞(C([a, b])), and o any sequence Λ = (αn)nas abo e and o each unc ion ∈C([a, b]) we ha e JΛ( )∈`∞(C([a, b])) and ||JΛ( )||`∞= sup n ||uΛ, n||∞=|| ||∞. 6 CALDER ´ ON, GERLACH, AND PRADO Thus, we ha e ob ained he ollowing use ul s a emen . P oposi ion 2.4. Le Λ = (αn)n⊂Rwi h a=α1< α2<· · · < αn−1< αn< · · · −→ b(n→ ∞).Then he mapping JΛ:C([a, b]) −→ `∞(C([a, b])) 7−→ JΛ( ) := (uΛ, n)n, is a linea isome y whose ange is con ained in F ∪{0}. In pa icula , he no med spaces (C([a, b]),|| · ||∞)and (JΛ(C([a, b])),|| · ||`∞)a e isome ically isomo phic. 3. Lineabili y and spaceabili y in AMW In his sec ion we show ha AMW is maximal dense lineable bu a he same ime we p o ide many examples o conc e e linea s uc u es in AMW. Theo em 3.1. Le M be a linea subspace such ha M {0} ⊂ c0 `1and dim(M) = κ. Le Ibe a se wi h ca d(I) = κand {(ai n)n}i∈Ibe an algeb aic basis o M. Then o any sequence o unc ions (un)n∈ F, he se M:= {(anun)n: (an)n∈M} is a linea subspace o dimension κ, he amily {(ai nun)n}i∈Iis an algeb aic basis o Mand M {0} ⊂ AMW. P oo . I is clea ha he mapping Φ : (an)n∈c07−→ (anun)n∈C([a, b])Nis linea . Mo eo e , he ac ha un6= 0 o all n∈Nimplies ha Φ is one- o-one. Then all conclusions ollow om he equali y M= Φ(M) and om Co olla y 2.2.  By ollowing a dual way ( ha is, ixing his ime a sequence o c0 `1), we a e able o ob ain subspaces o AMW. Theo em 3.2. Le (an)n∈c0 `1wi h an6= 0 o all n∈N. Le Ube a linea subspace o dimension κ, ha ing an algeb aic basis {(ui n)n}i∈Iwi h ca d(I) = κ, such ha U {0}⊂F. Then U:= {(anun)n: (un)n∈U} is a linea subspace o dimension κ, he amily {(anui n)n}i∈Iis an algeb aic basis o U, and U {0} ⊂ AMW. P oo . I is simila o he p oo o Theo em 3.1, excep ha his ime one conside s he mapping Ψ : (un)n∈U7−→ (anun)n∈C([a, b])N. Then Ψ is linea and injec i e (because an6= 0 o each n∈N), U= Ψ(U), and Co olla y 2.2 applies again.  Recall ha in any sepa able, me izable, comple e opological ec o space (as i is he case o c0,`p,C([a, b]) o C([a, b])N) he maximal dimension o any linea ec o subspace is he dimension o he con inuum c. In pa icula i we conside ANTI M-WEIERSTRASS FUNCTION SEQUENCES 7 in Theo em 3.1 any linea ec o subspace Min c0 `1o dimension c, o ins ance M= span 1 ncn:c∈(0,1)), we ob ain: Co olla y 3.3. The se AMW is maximal lineable in C([a, b])N. We can ob ain he same om Theo em 3.2 jus conside ing he linea subspace in F ∪ {0}gi en by U= span{JΛ(xc) : c∈(0,+∞)}, whe e Λ is any p e ixed sequence o scala s as in Example 2.3. We deno e by c0(C([a, b])) he opological ec o space o all sequences ( n)n∈ C([a, b])Nsuch ha || n||∞→0 (n→ ∞). I is known ha c0(C([a, b])) endowed wi h he opology inhe i ed by ||( n)n||`∞:= supnk nk∞is a sepa able Banach space (indeed, o he sepa abili y jus ake he se o sequences (gn)nsuch ha gn6= 0 o ini ely many e ms which belong o a dense and nume able se o C([a, b]) - o example, polynomials wi h a ional coe icien s-). I is clea ha AMW ⊂ c0(C([a, b])), so i we ake in o accoun he opological s uc u e o c0(C([a, b])), we can ocus ou a en ion on densi y p ope ies enjoyed by AMW. A i s posi i e esul in his di ec ion is gi en by he applica ion o he ollowing lemma due o Be nal [12] (see also [4, Chap e 4], [5] and [14]). Lemma 3.4. Le Xbe a sepa able me izable opological ec o space, A⊂X maximal lineable and B⊂Xdense lineable in Xwi h A∩B=∅. I A+B⊂A hen Ais maximal dense lineable in X. Theo em 3.5. The amily AMW o An i-M Weie s ass sequences is maximal dense lineable in c0(C([a, b])). P oo . F om Co olla y 3.3, he amily AMW is maximal lineable. Now, he amily c00(C([a, b])) := {( n)n∈C([a, b])N: he e is Nsuch ha n= 0 o any n≥N} is a dense linea subspace o c0(C([a, b])) and so, i ially, i is a dense lineable sub- se o c0(C([a, b])). Fo a ixed ( n)n∈ AMW, each sequence (gn)n∈c00(C([a, b])) only modi ies, unde addi ion, a ini e numbe o componen s o ( n)n. So ( n+ gn)n∈ AMW. In o he wo ds, c00(C([a, b])) + AMW ⊂ AMW. Now, an appli- ca ion o Lemma 3.4, wi h X:= c0(C([a, b])), A:= AMW and B:= c00(C([a, b])), gi es us he maximal dense-lineabili y o AMW in c0(C([a, b])).  Finally, we a e also able o ge he spaceabili y o he an i-M Weie s ass amily o sequences. Fo his, he linea isome ic p ope y o he mapping JΛand he ( i ial) spaceabili y o C([a, b]) (see [4], [15]) play an impo an ole. Theo em 3.6. The amily AMW o an i-M Weie s ass sequences is spaceable in c0(C([a, b])). In ac , o any Λ=(αn)nwi h a=α1< α2<· · · < αn<· · · → b(n→ ∞), any sequence (an)n∈c0 `1and any in ini e dimensional closed ec o subspace Co C([a, b]), he se M:= {(anuΛ, n)n: ∈ C} 8 CALDER ´ ON, GERLACH, AND PRADO is an in ini e dimensional closed ec o subspace o (c0(C([a, b])),||·||`∞)such ha M {0} ⊂ AMW. P oo . Le Λ, (an)nand Cas abo e. I is clea ha Mis a linea space. F om P oposi ion 2.4, JΛis a linea isome y, so JΛ(C) is an in ini e dimensional closed ec o space o `∞(C([a, b])) such ha JΛ(C) {0} ⊂ F. Bu (an)n∈c0, so he e exis s L > 0 such ha |an| ≤ L o all n∈N, and o any ∈ C we ha e ||(anuΛ, n)n||`∞= sup n∈N ||anuΛ, n||∞≤L· ||(uΛ, n)n||`∞, and ||anuΛ, n||∞=|an| · || ||∞→0 (n→ ∞). The e o e Mis a closed ec o subspace o (`∞(C([a, b])),||·||`∞) ha is con ained in c0(C([a, b])). Bu c0(C([a, b])) is closed in `∞(C([a, b])), hence Mis a closed ec o subspace o (c0(C([a, b])),|| · ||`∞). O cou se, Mis in ini e dimensional because Cis, he e a e in ini elly many an6= 0 ( ecall (an)n/∈`1) and JΛis a linea isome y such ha uΛ, n= 0 o some n∈Ni and only i = 0 on [a, b]. Finally, as (an)n∈c0 `1and JΛ(C) {0}⊂F, we ge om Co olla y 2.2, M {0} ⊂ AMW. Rema k 3.7. Obse e ha , o any sequence Λ as abo e and any (an)n∈c0 `1 we ha e ha o L:= maxn∈N|an| ∈ (0,+∞), ||λ1(anuΛ, 1 n)n+· · · +λs(anuΛ, s n)n||`∞=||(anuΛ,(λ1 1+···+λs s) n)n||`∞ =L· ||λ1 1+· · · +λs s||∞, whe e λ1, . . . , λs∈Rand 1, . . . , s∈C([a, b]). In pa icula , we can d op he es ic ion an6= 0 in Theo em 3.2 and we can s a e he algeb aic basis o he subspaces in AMW. Speci ically: (1) I { i}i∈I⊂C([a, b]) a e linea ly independen , hen he amily M={(anuΛ, i n)n: i∈I}is linea ly independen and span(M) {0} ⊂ AMW; (2) I span{ i:i∈I}is a closed ec o subspace o C([a, b]) wi h dimension κ, hen he closed ec o subspace M:= span{(anuΛ, i n)n:i∈I}o c0(C([a, b])) has he same dimension κ, and M {0} ⊂ AMW. 4. Algeb abili y o AMW In his sec ion we p o ide se e al ways o gene a e ee algeb as in AMW. Bu be o e his we s a e wo echnical lemmas. Lemma 4.1. Le Ube a ee algeb a in C([a, b]), gene a ed by U:= {ui}i∈I. Then, o any amily P={pi}i∈Io polynomials o deg ee exac ly 1, he se UP:= {pi◦ui}i∈Iis a gene a o sys em o a ee algeb a in C([a, b]).