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MATH 401 - NOTES Sequences of functions Pointwise and …

SPRING 2009 MATH 401 - NOTESS equences of functionsPointwise and Uniform ConvergencePreviously, we have studied Sequences ofreal numbers. Now we discusssequences of real-valued functions . By a sequence {fn}of real-valued func-tions onD, we mean a sequence (f1, f2, .. , fn, ..) such that eachfnis afunction having domainDand range a subset Pointwise a subset ofRand let{fn}be a sequence of functionsdefined onD. We say that{fn}converges Pointwise onDiflimn fn(x)exists for each other words, limn fn(x) must be a real number that depends only this case, we writef(x) = limn fn(x)for everyxinDandfis called thepointwise limit of the sequence {fn}.Formal Definition:The sequence {fn}converges Pointwise tofonDif foreveryx Dand for every >0, there exists a natural numberN=N(x, )such that|fn(x) f(x)|< whenevern > :The notationN=N(x, ) means that the natural numberNdependson the choice ofxand.

Applying the sandwich theorem for sequences, we obtain that lim n→∞ fn(x) = 0 for all x in R. Therefore, {fn} converges pointwise to the function f = 0 on R. Example 6. Let {fn} be the sequence of functions defined by fn(x) = cosn(x) for −π/2 ≤ x ≤ π/2. Discuss the pointwise convergence of the sequence.

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Transcription of MATH 401 - NOTES Sequences of functions Pointwise and …

1 SPRING 2009 MATH 401 - NOTESS equences of functionsPointwise and Uniform ConvergencePreviously, we have studied Sequences ofreal numbers. Now we discusssequences of real-valued functions . By a sequence {fn}of real-valued func-tions onD, we mean a sequence (f1, f2, .. , fn, ..) such that eachfnis afunction having domainDand range a subset Pointwise a subset ofRand let{fn}be a sequence of functionsdefined onD. We say that{fn}converges Pointwise onDiflimn fn(x)exists for each other words, limn fn(x) must be a real number that depends only this case, we writef(x) = limn fn(x)for everyxinDandfis called thepointwise limit of the sequence {fn}.Formal Definition:The sequence {fn}converges Pointwise tofonDif foreveryx Dand for every >0, there exists a natural numberN=N(x, )such that|fn(x) f(x)|< whenevern > :The notationN=N(x, ) means that the natural numberNdependson the choice ofxand.

2 Example {fn}be the sequence of functions onRdefined byfn(x) =nx. This sequence does not converge Pointwise onRbecause limn fn(x) = for anyx > {fn}be the sequence of functions onRdefined byfn(x) = sequence converges Pointwise to the zero function onR. Indeed, givenany >0, chooseN >x then|fn(x) 0|=xn<xN< ,forn > NExample the sequence {fn}of functions defined byfn(x) =(x+n)2n2for that{fn}converges :For every real numberx, we have:limn fn(x) = limn (x2n2+2xn+ 1)=x2(limn 1n2)+2x(limn 1n)+1 = 0+0+1 = 1 Thus,{fn}converges Pointwise to the functionf(x) = 1 the sequence {fn}of functions defined byfn(x) =n2xnfor0 x 1. Determine whether{fn}is Pointwise convergent on[0,1].Solution:First of all, we observe thatfn(0) = 0 for everyninN. So thesequence{fn(0)}is constant and converges to zero. Now suppose 0< x <1thenn2xn=n2enln(x) 0 asn.

3 Finally,fn(1) =n2for alln. So,limn fn(1) = . Therefore,{fn}is not Pointwise convergent on [0,1]. Al-though, it is Pointwise convergent on [0,1).Example the sequence {fn}of functions defined byfn(x) =sin(nx+ 3) n+ 1for that{fn}converges :For everyxinR, we have 1 n+ 1 sin(nx+ 3) n+ 1 1 n+ 1 Moreover,limn 1 n+ 1= the sandwich theorem for Sequences , we obtain thatlimn fn(x) = 0 for ,{fn}converges Pointwise to the functionf= 0 {fn}be the sequence of functions defined byfn(x) = cosn(x)for /2 x /2. Discuss the Pointwise convergence of the :For /2 x <0 and for 0< x /2, we have0 cos(x)< follows thatlimn (cos(x))n= 0 forx6= , sincefn(0) = 1 for allninN, one gets limn fn(0) = 1. Therefore,{fn}converges Pointwise to the functionfdefined byf(x) ={0 if 2 x <0 or 0< x 21 ifx= 0 Example the sequence {fn}of functions defined byfn(x) =x3 +nx2for that{fn}converges :Moreover, for every real numberx, we have:limn fn(x) = limn x3 +nx2= ,{fn}converges Pointwise to the zero the sequence of functions defined byfn(x) =nx(1 x)non [0,1].]}

4 Show that{fn}converges Pointwise to the zero :Note thatfn(0) =fn(1) = 0, for alln N. Now suppose0< x <1, thenlimn fn(x) = 0 Therefore, the given sequence converges Pointwise to {fn}be the sequence of functions onRdefined byfn(x) ={n3if0< x 1n1 otherwiseShow that{fn}converges Pointwise to the constant functionf= :For anyxinRthere is a natural numberNsuch thatxdoesnot belong to the interval (0,1/N). The intervals (0,1/n) get smaller asn . We see thatfn(x) = 1 for alln > N. Hence,limn fn(x) = 1 for Uniform a subset ofRand let{fn}be a sequence of realvalued functions defined onD. Then{fn}converges uniformly tofif givenany >0, there exists a natural numberN=N( ) such that|fn(x) f(x)|< for everyn > Nand for :In the above definition the natural numberNdepends only on .Therefore, uniform convergence implies Pointwise convergence.}

5 But the con-verse is false as we can see from the following 10 Let{fn}be the sequence of functions on(0, )defined byfn(x) =nx1 + sequence converges Pointwise to zero. Indeed, (1 +n2x2) n2x2asngets larger and larger. So,limn fn(x) = limn nxn2x2=1xlimn 1n= for any <1/2, we have fn(1n) f(1n) =12 0> .Hence{fn}is not uniformly a subset ofRand let{fn}be a sequence ofcontinuousfunctions onDwhich converges uniformly tofonD. Then its limitfiscontinuous {fn}be the sequence of functions defined byfn(x) =cosn(x)for /2 x /2. Discuss the uniform convergence of the :We know that{fn}converges Pointwise to the functionfdefinedby (see Example 6)f(x) ={0 if 2 x <0 or 0< x 21 ifx= 0 Eachfn(x) = cosn(x) is continuous on [ /2, /2]. But the Pointwise limitis not continuous atx= 0. By the above theorem, we conclude that{fn}does not converge uniformly on [ /2, /2].}

6 Example the sequence {fn}of functions defined byfn(x) =sin(nx+ 3) n+ 1for that{fn}converges uniformly to the zero function :We have seen that{fn}converges Pointwise to the zero functiononRn(see Example 5). Moreover|fn(x) 0|=|sin(nx+ 3)| n+ 1 1 n+ any >, we can findN Nsuch that1 n+ 1< whenevern > follows|fn(x) f(x)|< for everyn > Nand for ,{fn}converges uniformly to the zero function


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