Orthogonal Functions and Fourier SeriesVector SpacesVector Space AxiomsCoordinate RepresentationFunctions as vectorsMetric spacesNormed spacesNorms and metricsInner product spacesSome inner productsHilbert SpaceOrthogonalityExamplesFourier seriesSlide 15Slide 16Slide 17Next class: Fourier TransformUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellOrthogonal Functions and Fourier SeriesUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellVector SpacesSet of vectorsClosed under the following operationsVector addition: v1 + v2 = v3Scalar multiplication: s v1 = v2Linear combinations: Scalars come from some field Fe.g. real or complex numbersLinear independenceBasisDimensionvv =∑=iniia1University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellVector Space AxiomsVector addition is associative and commutativeVector addition has a (unique) identity element (the 0 vector)Each vector has an additive inverseSo we can define vector subtraction as adding an inverseScalar multiplication has an identity element (1)Scalar multiplication distributes over vector addition and field additionMultiplications are compatible (a(bv)=(ab)v)University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellCoordinate RepresentationPick a basis, order the vectors in it, then all vectors in the space can be represented as sequences of coordinates, i.e. coefficients of the basis vectors, in order.Example:Cartesian 3-spaceBasis: [i j k]Linear combination: xi + yj + zkCoordinate representation: [x y z]][][][212121222111bzazbyaybxaxzyxbzyxa +++=+University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFunctions as vectorsNeed a set of functions closed under linear combination, whereFunction addition is definedScalar multiplication is definedExample:Quadratic polynomialsMonomial (power) basis: [x2 x 1]Linear combination: ax2 + bx + cCoordinate representation: [a b c]University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellMetric spacesDefine a (distance) metric s.t.d is nonnegatived is symmetricIndiscernibles are identicalThe triangle inequality holdsR⇒)d(21v,v)d()d(:ijjijiv,vv,vVv,v =∈∀0)d(: ≥∈∀jijiv,vVv,v)d()d()d(:kikjjikjiv,vv,vv,vVv,v,v ≥+∈∀jijijivvv,vVv,v =⇔=∈∀ 0)d(:University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellNormed spacesDefine the length or norm of a vectorNonnegativePositive definiteSymmetricThe triangle inequality holdsBanach spaces – normed spaces that are complete (no holes or missing points)Real numbers form a Banach space, but not rational numbers Euclidean n-space is Banachv0: ≥∈∀ vVv0vv =⇒=0vvVv aaFa =∈∈∀ :,jijijivvvvVv,v +≥+∈∀ :University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellNorms and metricsExamples of norms:p norm:p=1 manhattan normp=2 euclidean normMetric from normNorm from metric ifd is homogeneousd is translation invariantthenppDiix11⎟⎟⎠⎞⎜⎜⎝⎛∑=2121vvv,v −=)d()d()d(:,jijijiv,vv,vVv,v aaaFa =∈∈∀€ ∀vi,vj,t ∈ V : d(vi,vj) = d(vi+ t,vj+ t)),d( 0vv =University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellInner product spacesDefine [inner, scalar, dot] product (for real spaces) s.t.For complex spaces:Induces a norm:vv,v =R⇒jiv,vkjkikjiv,vv,vv,vv +=+jijiv,vvv aa =,ijjiv,vvv =,0, ≥vv0vvv =⇔=0,ijjiv,vvv =,jijiv,vvv aa =,University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellSome inner productsMultiplication in RDot product in Euclidean n-spaceFor real functions over domain [a,b]For complex functions over domain [a,b]Can add nonnegative weight function∫=badxxgxfgf )()(,∫=badxxgxfgf )()(,iDii 2,1,21vvv,v∑==1∫=bawdxxwxgxfgf )()()(,University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellHilbert SpaceAn inner product space that is complete wrt the induced norm is called a Hilbert spaceInfinite dimensional Euclidean spaceInner product defines distances and anglesSubset of Banach spacesUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellOrthogonalityTwo vectors v1 and v2 are orthogonal ifv1 and v2 are orthonormal if they are orthogonal andOrthonormal set of vectors (Kronecker delta)0=21v,v1==2211v,vv,vjiji ,δ=v,vUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellExamplesLinear polynomials over [-1,1] (orthogonal)B0(x) = 1, B1(x) = xIs x2 orthogonal to these?Is orthogonal to them? (Legendre)011=∫−dxx2132+xUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesCosine series€ C0(θ) =1, C1(θ) = cos(θ), Cn(θ) = cos(nθ)€ Cm,Cn= cos(mθ)cos(nθ)dθ02π∫=1202π∫(cos[(m + n)θ] + cos[(m − n)θ])=12(m + n)sin[(m + n)θ] +12(m − n)sin[(m − n)θ] ⎛ ⎝ ⎜ ⎞ ⎠ ⎟02π= 0for m ≠ n ≠ 0€ f (θ) = aii= 0∞∑Ci(θ)University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier series€ =12cos(2nθ) +12 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟dθ =14nsin(2nθ) +θ2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟02π∫02π= π for m = n ≠ 0€ =122cos(0)dθ02π∫= 2π for m = n = 0Sine series€ S0(θ) = 0, S1(θ) = sin(θ), Sn(θ) = sin(nθ)€ Sm,Sn= sin(mθ)sin(nθ)dθ02π∫= 0 for m ≠ n or m = n = 0= π for m = n ≠ 0€ f (θ) = bii= 0∞∑Si(θ)University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier seriesComplete seriesBasis functions are orthogonal but not orthonormalCan obtain an and bn by projection€ f (θ) = ann= 0∞∑cos(nθ) + bnsin(nθ)€ Cm,Sn= cos(mθ)sin(nθ)dθ02π∫= 0€ f ,Ck= f (θ)cos(kθ)02π∫dθ = cos02π∫(kθ) dθ ain= 0∞∑cos(nθ) + bisin(nθ)= akcos202π∫(kθ) dθ = π ak(or 2π akfor k = 0)University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellFourier series€ ak=1πf (θ)cos(kθ)02π∫dθa0=12πf (θ) dθ02π∫Similarly for bk€ bk=1πf (θ)sin(kθ)02π∫dθUniversity of Texas at Austin CS384G - Computer Graphics Fall 2010 Don FussellNext class: Fourier TransformTopics:-Derive the Fourier transform from the Fourier series-What does it
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