A Remark on the Uniform Convergence of Some Sequences of Functions

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1 Advaces Pure Mathematcs Publshed Ole July 05 ScRes. A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut de Mathematques et de Sceces Physques (IMSP) Porto-Novo Be Iteratoal Cetre for Theoretcal Physcs (ICTP) Treste Italy Emal: gadegla@yahoo.fr Receved May 05; accepted 3 July 05; publshed 6 July 05 Copyrght 05 by author ad Scetfc Research Publshg Ic. Ths work s lcesed uder the Creatve Commos Attrbuto Iteratoal Lcese (CC BY). Abstract We stress a basc crtero that shows a smple way how a sequece of real-valued fuctos ca coverge uformly whe t s more or less evdet that the sequece coverges uformly away from a fte umber of pots of the closure of ts doma. For fuctos of a real varable ulke most classcal textbooks our crtero avods the search of extrema (by dfferetal calculus) of ther geeral term. Keywords Sequece of Fuctos Uform Covergece Metrc Boudedess. Itroducto Let X be a oempty set f : X be a fucto ad { f } X to. Recall []-[3] that the sequece { f } { f ( x) f ( x) x X} lm sup : 0. + Obvously f { f } coverges to f ( x ) ; that s { } { f } be a sequece of real-valued fuctos from s sad to coverge uformly to f o X f coverges uformly to f o X the for each x X f coverges potwse to f o X the { f } fxed the sequece { f } x coverges potwse to f. It s also obvous that whe X s fte ad coverges uformly to f o X. However ths coverse does t hold geeral for a arbtrary (fte) set X;.e. the potwse covergece may ot mply the uform covergece whe X s a arbtrary (fte) set. How to cte ths paper: Degla G. (05) A Remark o the Uform Covergece of Some Sequeces of Fuctos. Advaces Pure Mathematcs

2 Oe ca observe that the mathematcal lterature there are very few kow results that gve codtos uder whch a potwse covergece mples the uform covergece. Cocerg sequeces of cotuous fuctos defed o a compact set we have the followg facts: Proposto A. (D s Theorem) [4] If K s a compact metrc space : a mootoe sequece of f K a cotuous fucto ad { f } cotuous fuctos from K to that coverges potwse to f o K the { f } coverges uformly to f o K. Proposto B. [5] If E s a Baach space ad { T } s a sequece of bouded lear operators of E that coverges potwse to a bouded lear operator T of E the for every compact set K E { T } coverges uformly to T o K. (For the sake of completeess we gve the proof of ths proposto the Appedx Secto). Therefore our am s to hghlght a ew basc crtero that shows some way how a sequece of real-valued fuctos ca coverge uformly whe t s more or less obvous that the sequece coverges uformly away from a fte umber of pots of the closure of ts doma. I the case of sequeces of fuctos of a real varable our crtero avods ulke most classcal textbooks [3] [6] the search of extrema (by dfferetal calculus) of ther geeral terms. Several examples that satsfy the crtero are gve.. The Ma Result (Remark).. Theorem of fuctos defed from Ω to. Suppose that there exsts a fucto f from Ω to some pots a ak Ω some postve real umbers r r ad a oegatve costat M such that Let ( Ed ) be a metrc space ad Ω be a subset of E. Cosder a sequece { f } k k r f x f x M d xa ; x Ωad for all. (D) Suppose furthermore that for each ε > 0 { f } ( ) coverges uformly to f o \ k B( a ε ) Ω ; where B a ε deotes the ope ball of E cetered at a ad wth radus ε. The the sequece of fuctos { f } coverges uformly to f o Ω. Proof Let ε > 0 be arbtrarly fxed (t may be suffcetly small order to be meagful). The for every atural umber we have r rk sup f( x) f ( x) max Mε + + sup f( x) f ( x). x Ω x Ω d( xa ) ε ; k Thus lm sup sup f r r k x f x + + Mε ε > 0 + x Ω by the uform covergece of { f } o \ k B( a ε ) Ad so.e. Ω. lmsup sup f ( x) f ( x ) 0; + x Ω lm sup f ( x) f ( x ) 0. + x Ω 58

3 .. Observato The boudedess codto (D) of the above theorem ca ot be removed as show by the sequece of fuctos defed from [ 0 ] to as follows: f ( x) x( x) ; x [ 0 ] ; where s equpped wth ts stadard metrc. Ideed { f } coverges uformly to 0 o [ ε ] for each ε ( 0) but wth k ad a 0 there s o postve umber r for whch the codto (D) s satsfed sce f ( x) Ad we ca see that { f } 3. Examples r > 0 sup sup. r 0< x x does ot coverge uformly to 0 o [ 0 ] sce 0 x + + lm sup f ( x ) lm 0. e We gve some examples that llustrate the theorem. () Let ( Ed ) be a fte metrc space ad let a E be fxed. Deote by ϕ the fucto defed from E to by The the sequece of fuctos { } coverges uformly to ϕ o E. ϕ x d xa x E. ϕ defed by ( ) ( ) m { ( )} + d ( x a) d xa + d xa d xa ϕ x x E () Gve a fte metrc space ( ) ) the sequece of fuctos { } u Ed a E defed by ad ( 0 ) ( ) d( xa) d xa u ( x) x E + coverges uformly to 0 o E ) the sequece of fuctos { v } defed by ( ) + we have that v x d xa exp d xa x E coverges uformly to 0 o E. (3) Let ( Ed ) be a fte metrc space ad Ω be a bouded ad fte subset of E let a ad b be two dfferet pots of Ω ad let ad be two fxed postve umbers. defed by ) Cosder the sequece of fuctos { } f The { f } ) Cosder the sequece of fuctos { g } ( ) d( x b) + d( x a) d( x b) d x a f ( x) x Ω. coverges uformly to 0 o Ω. defed by 59

4 The { g } ) Cosder the sequece of fuctos { h } The { h } ( ) d( x b) + d( x a) d( x b) d x a g ( x) x Ω. coverges uformly to 0 o Ω. defed by ( ) h x d x a d x b exp d x a d x b x Ω. coverges uformly to 0 o Ω. (4) I real aalyss we ca recover the facts that each of the followg sequeces coverges uformly to 0 o ther respectve domas: x x ; 0 x 3. x x; 0 x 3. s xcos x; 0 x 3. cos xs x; 0 x 3. x xe ; x 0 3. Justfcatos (Proofs) of the examples () For every we have { d( xa) } d ( x a) m ϕ ( x) ϕ( x) x E. + Therefore o the oe had for each ε > 0 we have ϕ ( x) ϕ( x) x E \ B( a ε) + ε showg that { ϕ } coverges uformly to ϕ o \ ( ) O the other had we have ϕ fulfllg codto (D) of the above theorem. ϕ coverges uformly to ϕ o E. Thus { } () ) O the oe had for each 0 ad so { u } O the other had we have E B a ε. ϕ x x d xa x E ad ε > we have for all x E\ B( a ε ) [ d( xa )] [ + d( xa )] [ + d( xa )] ( ε ) u ( x) u ( x) + coverges uformly to 0 o E\ B ( a ε ). fulfllg codto (D) of the above theorem. coverges uformly to 0 o E. Thus { u } ) The uform covergece of { v } u x d xa x E ad follows that of { u } sce Observe that the uform covergece of { } (3) Note that for all atural umber we have 0 v x u x x E ad. v 0 h g f ad for all wth > : could also be proved usg drectly the above theorem. 530

5 because followg from t e t 0 ad + t ( + t) Therefore t suffces to prove that { } t + t + t e t 0 ad. f coverges uformly to 0 o Ω although each of these three sequeces ca be hadled drectly wth the above theorem. Let δ be the dameter of Ω. The o the oe had for each 0 x Ω\ Ba ε Bb ε ad for all : ad so { f } O the other had we have ε > we have for all ( ) ( ) d( x b + ) δ + d( x a) d( x b) + ε d x a f( x) f( x) coverges uformly to 0 o \ ( Ba ( ε) Bb ( ε) ) Ω. f x d x a d x b x Ω ad showg codto (D) of the above theorem. coverges uformly to 0 o Ω ad we are doe. Thus { f } (4) ) Let us set ψ x x x ; 0 x wth 3. O the oe had we have for every : O the other had we have for every [ ] ψ x x x x 0. ε 0 : ψ x ε x ε ε for all showg that { ψ } coverges uformly to 0 o ( ε ε) Therefore by takg E Ω [ 0] a 0 that { ψ } coverges uformly to 0 o [ 0 ]. ψ x x x; 0 x wth 3. O the oe had we have for every : ) For O the other had we have for every. a r r ad M the above theorem mples [ ] ψ x x x x 0. ε 0 : ψ x ε x ε ε for all showg that { ψ } coverges uformly to 0 o ( ε ε) Therefore by takg E Ω [ 0] 0 mples that { ψ } coverges uformly to 0 o [ 0 ]. ) For ψ ( x) s xcos x; 0 x wth 3. O the oe had we have for every :. a a r r ad M the above theorem 53

6 O the other had we have for every showg that { } ψ ( x) x x x 0. ε 0 4 : ψ ( x) s ε cos ε x ε ε for all ψ coverges uformly to 0 o ε ε sce cos ε <. Therefore by takg E Ω 0 a 0 a r r ad M the above theorem ψ coverges uformly to 0 o 0. v) For ψ ( x) cos xs x; 0 x wth 3. O the oe had we have for every : mples that { } O the other had we have for every ψ ( x) x x x 0. ε 0 4 : ψ ( x) cos ε x ε ε for all showg that { ψ } coverges uformly to 0 o ε ε sce cos ε <. Therefore by takg E Ω 0 a 0 a r r ad M the above theorem mples that { ψ } coverges uformly to 0 o 0. x v) The example of ψ ( x) xe ; x 0 wth 3 s a partcular case of Example ()-) above wth E + d( xy ) x y for all xy + a 0 ad. Refereces [] Godemet R. (004) Aalyss I. Covergece Elemetary Fuctos. Sprger Berl. [] Mukres J. (000) Topology. d Edto. Prtce Hall Ic. Upper Saddle Rver. [3] Ross K.A. (03) Elemetary Aalyss. The Theory of Calculus. Sprger New York. [4] Godemet R. ad Spa P. (005) Aalyss II: Dfferetal ad Itegral Calculus Fourer Seres Holomorphc Fctos. Sprger Berl. [5] Ezzb K. Degla G. ad Ndambomve P. ( Press) Cotrollablty for Some Partal Fuctoal Itegrodfferetal Equatos wth Nolocal Codtos Baach Spaces. Dscussoes Mathematcae Dfferetal Iclusos Cotrol ad Optmzato. [6] Freslo J. Poeau J. Fredo D. ad Mor C. (00) Mathématques. Exercces Icotourables MP. Duod Pars. 53

7 Appedx I ths secto we prove Proposto B for the sake of completeess. Proof of Proposto B Let ε > 0 be gve. By the Uform Boudedess Prcple we have that sup M sup T. The there exst Also { } a a am such that : ε x K j m x B a j. We have that ( M + ) m ε K B a. ( M + ) + + T x aj + T aj x + T( aj) T( aj) T x T x T x T a T a T a T a T x j j j j ε M + T a T a M +. j j It follows that T ( x) T( x) ε T ( a ) T( a ) supx K + max m ad therefore supx K T x T x 0 as +. T <. So let 533