Orthogonal bases and Fourier analysis (2)

11/2/2019 Orthogonal bases and Fourier analysis (2) Orthogonal bases and Fourier analysis Nicolás Guarín-Zapata email: [email protected] github: nicoguaro March, 2019 file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 1/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Inner products Inner products let us extend geometrical notions such as length of a vector or angle between vectors for vector spaces that are more abstract than 2 or .3 It R R also let us define the orthogonality between vectors. Inner product spaces generalize the notion of Euclidean spaces to any dimension. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 2/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Orthogonal basis An orthogonal basis for an inner product spaces V , is a basis for V whose vectors are mutually orthogonal. The angle between vectors (θ) is defined using the inner product as ⟨x, y⟩ θ = arccos( ). ∥x∥ ∥y∥ If they have magnitude 1, then the base is called orthonormal. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 3/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Examples of (discrete) orthogonal basis: Fourier basis 11 1∣ { sin(nx), cos(nx), ∣∀n ∈ N, ∀x ∈ [−π, √π √π √2π ∣ file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 4/31

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11/2/2019 Orthogonal bases and Fourier analysis (2) Examples of (discrete) orthogonal basis: Hermite polynomials n x2 d n −x2 ∣ {(−1) e e , ∣∀n ∈ N, ∀x ∈ [−∞, ∞]} dxn ∣ with orthogonality as ∞ −x2 n ∫ Hm (x)Hn (x)e dx = √πs n!δmn −∞ file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 6/31

11/2/2019 Orthogonal bases and Fourier analysis (2) H_n (x) 50 Hermite (physicists') Polynomials 40 30 n=0 20 n=1 10 n=2 n=3 0 n=4 -10 n=5 -20 -1 0 1 2 3 -30 x -40 -2 file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 7/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Examples of (discrete) orthogonal basis: Chebyshev polynomials They are defined by the recursion relation T0 (x) = 1, T1 (x) = x, Tn+1 = 2xTn (x) − Tn−1 (x), with orthogonality as 1 ⎧0 n≠m n=m=0 dx = ⎨π n=m≠0 ∫ Tm (x)Tn (x) ⎩ π/2 √1 − x2 −1 file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 8/31

11/2/2019 Orthogonal bases and Fourier analysis (2) file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 9/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Fourier analysis: definition From Wikipedia In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 10/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Fourier analysis: scientific applications Fourier analysis has many scientific applications: 11/31 Signal Processing. It may be the best application of Fourier analysis. Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial. Control Theory. The Fourier series of functions in the differential equation often gives some file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2)

11/2/2019 Orthogonal bases and Fourier analysis (2) prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution. Partial Differential equation. We use it to solve higher order partial differential equations by the method of separation of variables. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 12/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Fourier analysis: applications Some examples include: JPG image compression. MP3 sound compression. Image processing to remove periodic or anisotropic artifacts such as jaggies from interlaced video. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 13/31

11/2/2019 Orthogonal bases and Fourier analysis (2) X-ray crystallography to reconstruct a crystal structure from its diffraction pattern. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 14/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Fourier series A Fourier series allow us to represent a (periodic) function as the sum of sine and cosine functions. For a function f(x) defined over [x0, x0 + P ], that is continuous or piecewise continuous, we write ∞ a0 2πnx 2πnx f (x) = + ∑ [an cos( ) + bn sin( )] 2 PP n=1 where the coefficients are obtained computing the inner product with the elements of the base, i.e. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 15/31

11/2/2019 Orthogonal bases and Fourier analysis (2) 2 x0+P a0 = ∫ f (x) dx P x0 2 x0+P 2πnx an = ∫ cos( )f (x) dx P P x0 2 x0+P 2πnx bn = ∫ sin( )f (x) dx P P x0 file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 16/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Fourier series visualisation Square 6 Speed file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 17/31

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11/2/2019 Orthogonal bases and Fourier analysis (2) Orthogonal basis: continuum case A set {ϕ(k, x)} with x and k defined over (a, b), and (c, d) are orthogonal with weight w(x) (w(x) real) if: b ∗′ ′ ∫ w(x)ϕ (k, x)ϕ(k , x) dx = δ(k − k ) , x ∈ (a, b), k a file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 19/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Orthogonal basis: continuum case If the basis is complete we can write a function f(x) as d f (x) = ∫ C(k)ϕ(k, x) dk , c with file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 20/31

11/2/2019 Orthogonal bases and Fourier analysis (2) b C(k) = ∫ f (x)w(x)ϕ(k, x) dx . a C(k) is known as the tranform of f(x). file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 21/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Examples of (continuous) orthogonal basis: Fourier transform When we choose the basis functions { eikx }, we can √2π write a function f(x), that is piecewise continuous and does not grow faster than exponentially, as ∞ f (x) = 1 ikx ∫ F (k)e dx . √2π −∞ file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 22/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Using the orthonormality condition ∞ ∫e i(k−k′ )x ′ dx = 2πδ(k − k ) , −∞ we can write ∞ F (k) = 1 f (x)e −ikx dx . ∫ √2π −∞ file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 23/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Example of Fourier transform We can compute the Fourier transform of a Gaussian function f (x) = e −α2 x2 , x ∈ (−∞, ∞) Using the definition and proceeding with the integral we get file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 24/31

11/2/2019 Orthogonal bases and Fourier analysis (2) ∞ F (k) = 1 −α2 (x2 +ikx/α2 ) ∫ e dx √2π −∞ ∞ = −k/4α2 ∫e −α2 (x+ikx/2α2 )2 dx e √2π −∞ = 1 −k2 /4α2 e. √πα file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 25/31

11/2/2019 Orthogonal bases and Fourier analysis (2) Visualization of Fourier Transform file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 26/31

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11/2/2019 Orthogonal bases and Fourier analysis (2) Visualization of Fourier Transform file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 28/31

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11/2/2019 Orthogonal bases and Fourier analysis (2) References Alonso Sepúlveda Soto. Física matemática. 30/31 Ciencia y Tecnología. Universidad de Antioquia, 2009. Pierre Guilleminot’s. Fourier series visualisation with d3.js., 2016. Wikipedia contributors. “Fourier analysis.” Wikipedia, The Free Encyclopedia Wikipedia contributors. “Hermite polynomials.” Wikipedia, The Free Encyclopedia. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2)

11/2/2019 Orthogonal bases and Fourier analysis (2) Wikipedia contributors. “Fourier series.” Wikipedia, The Free Encyclopedia. Wikipedia contributors. “Chebyshev polynomials.” Wikipedia, The Free Encyclopedia. Wikipedia contributors. “Fourier transform.” Wikipedia, The Free Encyclopedia. file:///C:/Users/Invitado/Downloads/AdvancedMath-master/slides/fourier_analysis.html#(2) 31/31