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Orthogonal Functions and Fourier Series. Vector Spaces. Set of vectors Operations on vectors and scalars Vector addition: v 1 + v 2 = v 3 Scalar multiplication: s v 1 = v 2 Linear combinations: Closed under these operations Linear independence Basis Dimension. Vector Spaces.

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orthogonal functions and fourier series

Orthogonal Functions and Fourier Series

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

vector spaces
Vector Spaces
  • Set of vectors
  • Operations on vectors and scalars
    • Vector addition: v1 + v2 = v3
    • Scalar multiplication: sv1 = v2
    • Linear combinations:
  • Closed under these operations
  • Linear independence
  • Basis
  • Dimension

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

vector spaces3
Vector Spaces
  • Pick 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-space
    • Basis: [i j k]
    • Linear combination: xi + yj + zk
    • Coordinate representation: [x y z]

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

functions as vectors
Functions as vectors
  • Need a set of functions closed under linear combination, where
    • Function addition is defined
    • Scalar multiplication is defined
  • Example:
    • Quadratic polynomials
    • Monomial (power) basis: [x2x1]
    • Linear combination: ax2 + bx + c
    • Coordinate representation: [a b c]

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

metric spaces
Metric spaces
  • Define a (distance)metric s.t.
    • d is nonnegative
    • d is symmetric
    • Indiscernibles are identical
    • The triangle inequality holds

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

normed spaces
Normed spaces
  • Define the length or norm of a vector
    • Nonnegative
    • Positive definite
    • Symmetric
    • The triangle inequality holds
  • Banach 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 Banach

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

norms and metrics
Norms and metrics
  • Examples of norms:
    • p norm:
      • p=1 manhattan norm
      • p=2 euclidean norm
  • Metric from norm
  • Norm from metric if
    • d is homogeneous
    • d is translation invariant

then

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

inner product spaces
Inner product spaces
  • Define [inner, scalar, dot] product s.t.
  • For complex spaces:
  • Induces a norm:

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

some inner products
Some inner products
  • Multiplication in R
  • Dot product in Euclidean n-space
  • For real functions over domain [a,b]
  • For complex functions over domain [a,b]
  • Can add nonnegative weight function

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

hilbert space
Hilbert Space
  • An inner product space that is complete wrt the induced norm is called a Hilbert space
  • Infinite dimensional Euclidean space
  • Inner product defines distances and angles
  • Subset of Banach spaces

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

orthogonality
Orthogonality
  • Two vectors v1 and v2 are orthogonal if
  • v1 and v2 are orthonormal if they are orthogonal and
  • Orthonormal set of vectors

(Kronecker delta)

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

examples
Examples
  • Linear polynomials over [-1,1] (orthogonal)
    • B0(x) = 1, B1(x) = x
    • Is x2 orthogonal to these?
    • Is orthogonal to them? (Legendre)

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier series
Fourier series
  • Cosine series

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier series14
Fourier series
  • Sine series

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier series15
Fourier series
  • Complete series
  • Basis functions are orthogonal but not orthonormal
  • Can obtain an and bn by projection

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell

fourier series16
Fourier series
  • Similarly for bk

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell