12 - Convergence of the Fourier Series

# 12 - Convergence of the Fourier Series - MATH 220...

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Unformatted text preview: MATH 220: CONVERGENCE OF THE FOURIER SERIES We now discuss convergence of the Fourier series on compact intervals I . ‘Conver- gence’ depends on the notion of convergence we use, such as (i) L 2 : u j → u in L 2 if bardbl u j − u bardbl L 2 → 0 as j → ∞ . (ii) uniform, or C : u j → u uniformly if bardbl u j − u bardbl C = sup x ∈ I | u j ( x ) − u ( x ) | → 0. (iii) uniform with all derivatives, or C ∞ : u j → u in C ∞ if for all non-negative integers k , sup x ∈ I | ∂ k u j ( x ) − ∂ k u ( x ) | → 0. (iv) pointwise: u j → u pointwise if for each x ∈ I , u j ( x ) → u ( x ), i.e. for each x ∈ I , | u j ( x ) − u ( x ) | → 0. Note that pointwise convergence is too weak for most purposes, so e.g. just because u j → u pointwise, it does not follow that integraltext I u j ( x ) dx → integraltext u ( x ) dx . This would follow, however, if one assumes uniform convergence, or indeed L 2 (or L 1 ) convergence, since | integraldisplay I u j ( x ) dx − integraldisplay I u ( x ) dx | = | integraldisplay I ( u j ( x ) − u ( x )) dx | = |( u j − u, 1 )| ≤ bardbl u j − u bardbl L 2 bardbl 1 bardbl L 2 . Note also that uniform convergence implies L 2 convergence since bardbl u j − u bardbl L 2 = parenleftbiggintegraldisplay I | u j − u | 2 dx parenrightbigg 1 / 2 ≤ parenleftbiggintegraldisplay I bardbl u j − u bardbl 2 C dx parenrightbigg 1 / 2 = parenleftbiggintegraldisplay I 1 dx parenrightbigg 1 / 2 bardbl u j − u bardbl C , so if u j → u uniformly, it also converges in L 2 . Uniform convergence also implies pointwise convergence directly from the definition. On the other hand, convergence in C ∞ implies uniform convergence directly from the definition. On the failure of converge side: the uniform limit of a sequence of continuous func- tions is continuous, so in view of the continuity of the complex exponentials, sines and cosines, the various Fourier series cannot converge uniformly unless the limit is contin- uous . On the other hand, even if the limit is continuous, the convergence may not be uniform: understanding conditions, under which it is, is one of our first tasks. There are two issues regarding convergence: whether the series in question converges at all (in whatever sense we are interested in), and second whether it converges to the desired limit, in this case the function whose Fourier series we are considering. The first part is easier to answer: we have already seen that even the generalized Fourier series converges in L 2 (but not necessarily to the function!). Now consider uniform convergence. Recall that the typical way one shows conver- gence of a series is that to show that each term in absolute value is ≤ M n , where M n is a non-negative constant, such that ∑ n M n converges. Similarly, one shows that a series ∑ n u n ( x ) converges uniformly by showing that there are non-negative constants M n such that sup x ∈ I | u n ( x ) | ≤ M n and such that...
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12 - Convergence of the Fourier Series - MATH 220...

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