Fourier Series and Transforms

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Lesley Boone
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1 Fourier Series ad rasorms Orthogoal uctios Fourier Series Discrete Fourier Series Fourier rasorm Chebyshev polyomials Scope: wearetryigto approimate a arbitrary uctio ad obtai basis uctios with appropriate coeiciets.

2 Fourier Series he Problem we are tryig to approimate a uctio by aother uctio g which cosists o a sum over orthogoal uctios Φ weighted by some coeiciets a. g i i a Φ i

3 he Problem... ad we are looig or optimal uctios i a least squares l sese... g... a good choice or the basis uctios Φ are orthogoal uctios. What are orthogoal uctios? wo uctios ad g are said to be orthogoal i the iterval [a,b] i b g d a b a { } g d Mi! How is this related to the more coceivable cocept o orthogoal vectors? Let us loo at the origial deiitio o itegrals: / 3

4 Orthogoal Fuctios b a g d lim i i g i Δ... where a ad b, ad i - i- Δ... I we iterpret i ad g i as the ith compoets o a compoet vector, the this sum correspods directly to a scalar product o vectors. he vaishig o the scalar product is the coditio or orthogoality o vectors or uctios. i gi igi i g i i 4

5 Periodic uctios Let us assume we have a piecewise cotiuous uctio o the orm we wat to approimate this uctio with a liear combiatio o periodic uctios:, cos, si, cos, si,..., cos, si g a + { a cos + b si } 5

6 Orthogoality... are these uctios orthogoal? cos si cos si d d >,, > > cos si d, >... YES, ad these relatios are valid or ay iterval o legth. ow we ow that this is a orthogoal basis, but how ca we obtai the coeiciets or the basis uctios? rom miimisig -g 6

7 7 Fourier coeiciets optimal uctios g are give i { } Mi! g or g a leadig to... with the deiitio o g we get... { } + + d b a a a g a si cos { } + + d b d a b a a g,,...,, si,,...,, cos with si cos

8 Fourier approimatio o... Eample..., leads to the Fourier Serie g 4 cos cos 3 cos ad or <4 g loos lie

9 Fourier approimatio o... aother Eample..., < < leads to the Fourier Serie g cos si.. ad or <, g loos lie

10 Fourier - discrete uctios... what happes i we ow our uctio oly at the poits i i it turs out that i this particular case the coeiciets are give by a cos,,,,... b si,,,3,..... the so-deied Fourier polyomial is the uique iterpolatig uctio to the uctio with m g m { a cos + b si } a cos m a + + m

11 Fourier - collocatio poits... with the importat property that... g m i i... i our previous eamples > - blue ; g - red; i - +

12 Fourier series - covergece > - blue ; g - red; i

13 Fourier series - covergece > - blue ; g - red; i

14 Gibb s pheomeo > - blue ; g - red; i he overshoot or equispaced Fourier iterpolatios is 4% o the step height. 4

15 Chebyshev polyomials We have see that Fourier series are ecellet or iterpolatig ad dieretiatig periodic uctios deied o a regularly spaced grid. I may circumstaces physical pheomea which are ot periodic i space ad occur i a limited area. his quest leads to the use o Chebyshev polyomials. We depart by observig that cosϕ ca be epressed by a polyomial i cosϕ: cos ϕ cos 3ϕ cos 4ϕ cos 4 cos 8 cos 3 4 ϕ ϕ 3 cos ϕ ϕ 8 cos ϕ +... which leads us to the deiitio: 5

16 6 Chebyshev polyomials - deiitio,], [, cos, cos cos ϕ ϕ ϕ... or the Chebyshev polyomials. ote that because o cosϕ they are deied i the iterval [-,] which - however - ca be eteded to R. he irst polyomials are ad,] [ or where

17 Chebyshev polyomials - Graphical he irst te polyomials loo lie [, -].5 _ he -th polyomial has etrema with values or - at cos,,,,3,..., et 7

18 Chebyshev collocatio poits hese etrema are ot equidistat lie the Fourier etrema cos,,,,3,..., et 8

19 Chebyshev polyomials - orthogoality... are the Chebyshev polyomials orthogoal? Chebyshev polyomials are a orthogoal set o uctios i the iterval [-,] with respect to the weight uctio such that / or d / or >,, or... this ca be easily veriied otig that cosϕ, d siϕdϕ cos ϕ, cos ϕ 9

20 Chebyshev polyomials - iterpolatio... we are ow aced with the same problem as with the Fourier series. We wat to approimate a uctio, this time ot a periodical uctio but a uctio which is deied betwee [-,]. We are looig or g c c g +... ad we are aced with the problem, how we ca determie the coeiciets c. Agai we obtai this by idig the etremum miimum { } d g c

21 Chebyshev polyomials - iterpolatio... to obtai... c d,,,,...,... surprisigly these coeiciets ca be calculated with FF techiques, otig that c cosϕcos ϕdϕ,,,,...,... ad the act that cosϕ is a -periodic uctio... c cosϕcos ϕdϕ,,,,...,... which meas that the coeiciets c are the Fourier coeiciets a o the periodic uctio Fϕcos ϕ!

22 Chebyshev - discrete uctios... what happes i we ow our uctio oly at the poits i cos i i this particular case the coeiciets are give by c cos cos, ϕ ϕ... leadig to the polyomial...,,,... / g m c + m c... with the property g m at cos /,,,...,

23 Chebyshev - collocatio poits - > - blue ; g - red; i poits poits

24 Chebyshev - collocatio poits - > - blue ; g - red; i poits poits

25 Chebyshev - collocatio poits - > - blue ; g - red; i poits he iterpolatig uctio g was shited by a small amout to be visible at all!.8 64 poits

26 Chebyshev vs. Fourier - umerical Chebyshev Fourier > - blue ; g - red; i - + his graph speas or itsel! Gibb s pheomeo with Chebyshev? 6

27 Chebyshev vs. Fourier - Gibb s Chebyshev Fourier sig- > - blue ; g - red; i - + Gibb s pheomeo with Chebyshev? YES! 7

28 Chebyshev vs. Fourier - Gibb s Chebyshev Fourier sig- > - blue ; g - red; i - + 8

29 Fourier vs. Chebyshev Fourier Chebyshev i i collocatio poits i cos i periodic uctios domai limited area [-,] cos, si basis uctios cos ϕ, cos ϕ g m + a m + a { a cos + b si } m cos iterpolatig uctio g m c + m c 9

30 Fourier vs. Chebyshev cot d Fourier Chebyshev a b cos si coeiciets c cos ϕ cos ϕ Gibb s pheomeo or discotiuous uctios Eiciet calculatio via FF iiite domai through periodicity some properties limited area calculatios grid desiicatio at boudaries coeiciets via FF ecellet covergece at boudaries Gibb s pheomeo 3

Orthogonal functions - Function Approximation

Orthogonal functions - Function Approximation Orthogol uctios - Fuctio Approimtio - he Problem - Fourier Series - Chebyshev Polyomils he Problem we re tryig to pproimte uctio by other uctio g which cosists o sum over orthogol uctios Φ weighted by