Orthogonal functions - Function Approximation

Transcription

1 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 some coeiciets. g i i Φ i umericl Methods i Geophysics

2 he Problem... d we re looig or optiml uctios i lest squres l sese... g b / { } g d Mi!... good choice or the bsis uctios Φ re orthogol uctios. Wht re orthogol uctios? wo uctios d g re sid to be orthogol i the itervl [,b] i b g d How is this relted to the more coceivble cocept o orthogol vectors? Let us loo t the origil deiitio o itegrls: umericl Methods i Geophysics

3 - Deiitio b g d lim i i g i... where d b, d i - i-... I we iterpret i d g i s the ith compoets o compoet vector, the this sum correspods directly to sclr product o vectors. he vishig o the sclr product is the coditio or orthogolity o vectors or uctios. i g i igi i i g i umericl Methods i Geophysics

4 Periodic uctios Let us ssume we hve piecewise cotiuous uctio o the orm we wt to pproimte this uctio with lier combitio o periodic uctios:, cos, si, cos, cos,..., cos, si g + { cos + b si } umericl Methods i Geophysics

5 Orthogolity o Periodic uctios... re these uctios orthogol? cos si cos si d d >,, > > cos si d, >... YES, d these reltios re vlid or y itervl o legth. ow we ow tht this is orthogol bsis, but how c we obti the coeiciets or the bsis uctios? rom miimisig -g umericl Methods i Geophysics

6 umericl Methods i Geophysics Fourier coeiciets Fourier coeiciets optiml uctios g re give i { } Mi! g or g... with the deiitio o g we get... { } + + d b g si cos ledig to ice eercise { } + + d b d b g,,...,, si,,...,, cos with si cos

7 Fourier pproimtio o... Emple..., leds to the Fourier Serie g 4 cos cos 3 cos d or <4 g loos lie umericl Methods i Geophysics

8 Fourier pproimtio o... other Emple..., < < leds to the Fourier Serie g cos si.. d or <, g loos lie umericl Methods i Geophysics

9 Fourier - discrete uctios... wht hppes i we ow our uctio oly t the poits i i it turs out tht i this prticulr cse the coeiciets re give by cos,,,,... b si,,,3,..... the so-deied Fourier polyomil is the uique iterpoltig uctio to the uctio with m g m { cos + b si } cos m + + m umericl Methods i Geophysics

10 Fourier - colloctio poits... with the importt property tht... g m i i... i our previous emples > - blue ; g - red; i - + umericl Methods i Geophysics

11 Fourier series - covergece 5 > - blue ; g - red; i umericl Methods i Geophysics

12 Fourier series - covergece 5 > - blue ; g - red; i umericl Methods i Geophysics

13 Orthogol uctios - Gibb s pheomeo > - blue ; g - red; i he overshoot or equispced Fourier iterpoltios is 4% o the step height. umericl Methods i Geophysics

14 Chebyshev polyomils We hve see tht Fourier series re ecellet or iterpoltig d dieretitig periodic uctios deied o regulrly spced grid. I my circumstces physicl pheome which re ot periodic i spce d occur i limited re. his quest leds to the use o Chebyshev polyomils. We deprt by observig tht cosϕ c be epressed by polyomil 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 leds us to the deiitio: umericl Methods i Geophysics

15 umericl Methods i Geophysics Chebyshev polyomils - deiitio Chebyshev polyomils - deiitio,], [, cos, cos cos ϕ ϕ ϕ... or the Chebyshev polyomils. ote tht becuse o cosϕ they re deied i the itervl [-,] which - however - c be eteded to R. he irst polyomils re d,] [ or where

16 Chebyshev polyomils - Grphicl he irst te polyomils loo lie [, -].5 _ he -th polyomil hs etrem with vlues or - t cos,,,,3,..., et umericl Methods i Geophysics

17 Chebyshev colloctio poits hese etrem re ot equidistt lie the Fourier etrem cos,,,,3,..., et umericl Methods i Geophysics

18 Chebyshev polyomils - orthogolity... re the Chebyshev polyomils orthogol? Chebyshev polyomils re orthogol set o uctios i the itervl [-,] with respect to the weight uctio such tht / or d / or >,, or... this c be esily veriied otig tht cosϕ, d siϕdϕ cos ϕ, cos ϕ umericl Methods i Geophysics

19 umericl Methods i Geophysics Chebyshev polyomils - iterpoltio Chebyshev polyomils - iterpoltio... we re ow ced with the sme problem s with the Fourier series. We wt to pproimte uctio, this time ot periodicl uctio but uctio which is deied betwee [-,]. We re looig or g c c g +... d we re ced with the problem, how we c determie the coeiciets c. Agi we obti this by idig the etremum miimum { } d g c

20 Chebyshev polyomils - iterpoltio... to obti... c d,,,,...,... surprisigly these coeiciets c be clculted with FF techiques, otig tht c cosϕcos ϕdϕ,,,,...,... d the ct tht cosϕ is -periodic uctio... c cosϕcos ϕdϕ,,,,...,... which mes tht the coeiciets c re the Fourier coeiciets o the periodic uctio Fϕcos ϕ! umericl Methods i Geophysics

21 Chebyshev - discrete uctios... wht hppes i we ow our uctio oly t the poits i cos i this prticulr cse the coeiciets re give by c cos cos, ϕ ϕ... ledig to the polyomil... i,,,... / g m c + m c... with the property g m t cos /,,,..., umericl Methods i Geophysics

22 Chebyshev - colloctio poits - > - blue ; g - red; i poits poits umericl Methods i Geophysics

23 Chebyshev - colloctio poits - > - blue ; g - red; i poits poits umericl Methods i Geophysics

24 Chebyshev - colloctio poits - > - blue ; g - red; i poits he iterpoltig uctio g ws shited by smll mout to be visible t ll!.8 64 poits umericl Methods i Geophysics

25 Chebyshev vs. Fourier - umericl Chebyshev Fourier > - blue ; g - red; i - + his grph spes or itsel! Gibb s pheomeo with Chebyshev? umericl Methods i Geophysics

26 Chebyshev vs. Fourier - Gibb s Chebyshev Fourier sig- > - blue ; g - red; i - + Gibb s pheomeo with Chebyshev? YES! umericl Methods i Geophysics

27 Chebyshev vs. Fourier - Gibb s Chebyshev Fourier sig- > - blue ; g - red; i - + umericl Methods i Geophysics

28 Fourier vs. Chebyshev Fourier Chebyshev i i colloctio poits i cos i periodic uctios domi limited re [-,] cos, si bsis uctios cos ϕ, cos ϕ g m + m + { cos + b si } m cos iterpoltig uctio g m c + m c umericl Methods i Geophysics

29 Fourier vs. Chebyshev cot d Fourier Chebyshev b cos si coeiciets c cos ϕ cos ϕ Gibb s pheomeo or discotiuous uctios Eiciet clcultio vi FF iiite domi through periodicity some properties limited re clcultios grid desiictio t boudries coeiciets vi FF ecellet covergece t boudries Gibb s pheomeo umericl Methods i Geophysics