Orthogonal Functions and Fourier Series. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Transcription

1 Orthogonal Functons and Fourer Seres Fall 21 Don Fussell

2 Vector Spaces Set of ectors Closed under the followng operatons Vector addton: = 3 Scalar multplcaton: s 1 = 2 Lnear combnatons: Scalars come from some feld F e.g. real or complex numbers Lnear ndependence Bass Dmenson n = 1 a = Fall 21 Don Fussell

3 Vector Space Axoms Vector addton s assocate and commutate Vector addton has a (unque) dentty element (the ector) Each ector has an addte nerse So we can defne ector subtracton as addng an nerse Scalar multplcaton has an dentty element (1) Scalar multplcaton dstrbutes oer ector addton and feld addton Multplcatons are compatble (a(b)=(ab)) Fall 21 Don Fussell

4 Coordnate Representaton Pck a bass, order the ectors n t, then all ectors n the space can be represented as sequences of coordnates,.e. coeffcents of the bass ectors, n order. Example: Cartesan 3-space Bass: [ k] Lnear combnaton: x + y + zk Coordnate representaton: [x y z] a [ x1 y1 z1] + b[ x2 y2 z2] = [ ax1 + bx2 ay1 + by2 az1 + bz2] Fall 21 Don Fussell

5 Functons as ectors Need a set of functons closed under lnear combnaton, where Functon addton s defned Scalar multplcaton s defned Example: Quadratc polynomals Monomal (power) bass: [x 2 x 1] Lnear combnaton: ax 2 + bx + c Coordnate representaton: [a b c] Fall 21 Don Fussell

6 Metrc spaces Defne a (dstance) metrc d s nonnegate d s symmetrc V Indscernbles are dentcal The trangle nequalty holds, d(, 2) 1 : d(, ) R s.t., V : d(, ) = d(, ), V : d(, ) = =,, k V : d(, ) + d(, k ) d(, k ) Fall 21 Don Fussell

7 Normed spaces Defne the length or norm of a ector Nonnegate V : Poste defnte = = Symmetrc V, a F : a = The trangle nequalty holds, V : + + Banach spaces normed spaces that are complete (no holes or mssng ponts) Real numbers form a Banach space, but not ratonal numbers Eucldean n-space s Banach a Fall 21 Don Fussell

8 Norms and metrcs Examples of norms: p norm: p=1 manhattan norm p=2 eucldean norm Metrc from norm Norm from metrc f d( & D $ = 1 % x p #! " 1 p 1, 2) = 1 2 d s homogeneous, V, a F : d( a, a ) = a d(, ) d s translaton narant,,t V : d(, ) = d( + t, + t) then = d(, ) Fall 21 Don Fussell

9 Inner product spaces Defne [nner, scalar, dot] product, R (for real spaces) s.t. +, =, + k k, k a, = a,, =,,, = = For complex spaces: =,a = a,,, Induces a norm: =, Fall 21 Don Fussell

10 Some nner products Multplcaton n R Dot product n Eucldean n-space D 1, 2 = 1,2, = 1 For real functons oer doman [a,b] b f, g = f ( x) g( x) dx a For complex functons oer doman [a,b] f, g = f ( x) g( x) dx a Can add nonnegate weght functon b b f, g = w f ( x) g( x) w( x) dx a Fall 21 Don Fussell

11 Hlbert Space An nner product space that s complete wrt the nduced norm s called a Hlbert space Infnte dmensonal Eucldean space Inner product defnes dstances and angles Subset of Banach spaces Fall 21 Don Fussell

12 Orthogonalty Two ectors 1 and 2 are orthogonal f 1, 2 = 1 and 2 are orthonormal f they are orthogonal and 1, 1 = 2, 2 Orthonormal set of ectors, = δ, =1 (Kronecker delta) Fall 21 Don Fussell

13 Examples Lnear polynomals oer [-1,1] (orthogonal) B (x) = 1, B 1 (x) = x 1 x dx = 1 Is x 2 orthogonal to these? Is 3x 2 1 orthogonal to them? (Legendre) 2 Fall 21 Don Fussell

14 Fourer seres Cosne seres C (θ) =1, C 1 (θ) = cos(θ), C n (θ) = cos(nθ) f (θ) = a C (θ) = = C m,c n = cos(mθ)cos(nθ)dθ 1 (cos[(m + n)θ] + cos[(m n)θ]) 2 & 1 = 2(m + n) sn[(m + n)θ] + 1 ( ' 2(m n) for m n ) sn[(m n)θ] + * = Fall 21 Don Fussell

15 Fourer seres = = # 1 2 cos(2nθ) + 1 & # % ( dθ = 1 $ 2' 4n sn(2nθ) + θ & % ( $ 2' 1 2 2cos()dθ = for m = n = Sne seres S (θ) =, S 1 (θ) = sn(θ), S n (θ) = sn(nθ) = π for m = n S m,s n = sn(mθ)sn(nθ)dθ = for m n or m = n = = π for m = n f (θ) = b S (θ) = Fall 21 Don Fussell

16 Fourer seres Complete seres f (θ) = C m,s n = cos(mθ)sn(nθ)dθ = Bass functons are orthogonal but not orthonormal Can obtan a n and b n by proecton a n cos(nθ) + b n sn(nθ) n= f,c k = f (θ)cos(kθ) dθ = cos(kθ) dθ a cos(nθ) + b sn(nθ) n= = a k cos 2 (kθ) dθ = π a k (or a k for k = ) Fall 21 Don Fussell

17 Fourer seres a k = 1 π a = 1 f (θ)cos(kθ) dθ f (θ) dθ Smlarly for b k b k = 1 π f (θ)sn(kθ) dθ Fall 21 Don Fussell

18 Next class: Fourer Transform Topcs: - Dere the Fourer transform from the Fourer seres - What does t mean? Fall 21 Don Fussell