Lecture 4.5: Generalized Fourier series Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 4.5: Generalized Fourier series Advanced Engineering Mathematics 1 / 7
Last time Definition A Sturm-Liouville equation is a 2nd order ODE of the following form: − ( p ( x ) y ′ ) ′ + q ( x ) y = λ w ( x ) y , where p ( x ), q ( x ), w ( x ) > 0. We are usually interested in solutions y ( x ) on a bounded interval [ a , b ], under some homogeneous BCs: α 2 1 + α 2 α 1 y ( a ) + α 2 y ′ ( a ) = 0 2 > 0 β 2 1 + β 2 β 1 y ( b ) + β 2 y ′ ( b ) = 0 2 > 0 . Together, this BVP is called a Sturm-Liouville (SL) problem. Main theorem Given a Sturm-Liouville problem: (a) The eigenvalues are real and can be ordered so λ 1 < λ 2 < λ 3 < · · · → ∞ . (b) Each eigenvalue λ i has a unique (up to scalars) eigenfunction y i ( x ). � b (c) W.r.t. the inner product � f , g � := a f ( x ) g ( x ) w ( x ) dx , the eigenfunctions form an orthonormal basis on the subspace of functions C ∞ α,β [ a , b ] that satisfy the BCs. M. Macauley (Clemson) Lecture 4.5: Generalized Fourier series Advanced Engineering Mathematics 2 / 7
What this means Main theorem Given a Sturm-Liouville problem: (a) The eigenvalues are real and can be ordered so λ 1 < λ 2 < λ 3 < · · · → ∞ . (b) Each eigenvalue λ i has a unique (up to scalars) eigenfunction y i ( x ). � b (c) W.r.t. the inner product � f , g � := a f ( x ) g ( x ) w ( x ) dx , the eigenfunctions form an orthonormal basis on the subspace of functions C ∞ α,β [ a , b ] that satisfy the BCs. Definition If f ∈ C ∞ α,β [ a , b ], then f can be written uniquely as a linear combination of the eigenfunctions. That is, � b ∞ a f ( x ) y n ( x ) w ( x ) dx where c n = � f , y n � � f ( x ) = c n y n ( x ) , � y n , y n � = . � b a || y n ( x ) || 2 w ( x ) dx n =1 This is called a generalized Fourier series with respect to the orthogonal basis { y n ( x ) } and weighting function w ( x ). M. Macauley (Clemson) Lecture 4.5: Generalized Fourier series Advanced Engineering Mathematics 3 / 7
Example 1 (Dirichlet BCs) − y ′′ = λ y , y (0) = 0, y ( π ) = 0 is an SL problem with: Eigenvalues: λ n = n 2 , n = 1 , 2 , 3 , . . . . Eigenfunctions: y n ( x ) = sin( nx ). The orthogonality of the eigenvectors means that � π � π � 0 if m � = n � y m , y n � := y m ( x ) y n ( x ) w ( x ) dx = sin( mx ) sin( nx ) dx = π/ 2 if m = n . 0 0 Note that this means that || y n || := � y n , y n � 1 / 2 = � π/ 2. Fourier series: any function f ( x ), continuous on [0 , π ] satisfying f (0) = 0, f ( π ) = 0 can be written uniquely as ∞ � f ( x ) = b n sin nx n =1 where � π � π 0 f ( x ) sin nx dx � f , sin nx � = 2 b n = � sin nx , sin nx � = f ( x ) sin nx dx . || sin nx || 2 π 0 M. Macauley (Clemson) Lecture 4.5: Generalized Fourier series Advanced Engineering Mathematics 4 / 7
Example 2 (Neumann BCs) − y ′′ = λ y , y ′ (0) = 0, y ′ ( π ) = 0 is an SL problem with: Eigenvalues: λ n = n 2 , n = 0 , 1 , 2 , 3 , . . . . Eigenfunctions: y n ( x ) = cos( nx ). The orthogonality of the eigenvectors means that � π � π � 0 if m � = n � y m , y n � := y m ( x ) y n ( x ) w ( x ) dx = cos( mx ) cos( nx ) dx = π/ 2 if m = n > 0. 0 0 �� π/ 2 n > 0 Note that this means that || y n || := � y n , y n � 1 / 2 = √ π n = 0 . Fourier series: any function f ( x ), continuous on [0 , π ] satisfying f ′ (0) = 0, f ′ ( π ) = 0 can be written uniquely as ∞ � f ( x ) = a n cos nx n =0 where � π � π 0 f ( x ) cos nx dx � f , cos nx � = 2 a n = � cos nx , cos nx � = f ( x ) cos nx dx . || cos nx || 2 π 0 The same formula holds for a 0 if you let the n = 0 (constant) term be a 0 2 rather than a 0 . M. Macauley (Clemson) Lecture 4.5: Generalized Fourier series Advanced Engineering Mathematics 5 / 7
Example 3 (Mixed BCs) − y ′′ = λ y , y (0) = 0, y ′ ( π ) = 0 is an SL problem with: � 2 , n + 1 � Eigenvalues: λ n = n = 0 , 1 , 2 , . . . . 2 n + 1 � � Eigenfunctions: y n ( x ) = sin x . 2 The orthogonality of the eigenvectors means that � π � 0 if m � = n m + 1 n + 1 � � � � � y m , y n � := sin x sin x w ( x ) dx = 2 2 π/ 2 if m = n . 0 Note that this means that || y n || := � y n , y n � 1 / 2 = � π/ 2. (Generalized?) Fourier series: any function f ( x ), continuous on [0 , π ] satisfying f (0) = 0, f ′ ( π ) = 0 can be written uniquely as ∞ � n + 1 � � f ( x ) = b n sin x 2 n =1 where � π � π n + 1 n + 1 � � � � � f , sin x � 0 f ( x ) sin x dx = 2 2 2 n + 1 b n = = f ( x ) sin � � x dx . n + 1 n + 1 n + 1 2 � � � � � � x || 2 π � sin x , sin x � || sin 0 2 2 2 M. Macauley (Clemson) Lecture 4.5: Generalized Fourier series Advanced Engineering Mathematics 6 / 7
Example 4 (Robin BCs) − y ′′ = λ y , y (0) = 0, y (1) + y ′ (1) = 0 is an SL problem with: Eigenvalues: λ n = ω 2 n , n = 1 , 2 , 3 , . . . [ ω n ’s are the positive roots of y ( x ) = x − tan x ]. Eigenfunctions: y n ( x ) = sin( ω n x ). The orthogonality of the eigenvectors means that � 1 � 1 � 0 if m � = n � y m , y n � := y m ( x ) y n ( x ) w ( x ) dx = sin( ω m x ) sin( ω n x ) dx = ??? if m = n . 0 0 Though there isn’t a nice closed-form solution, we still have || y n || := � y n , y n � 1 / 2 . Generalized Fourier series: any function f ( x ), continuous on [0 , 1] satisfying f (0) = 0, f (1) + f ′ (1) = 0 can be written uniquely as ∞ � f ( x ) = b n sin ω n x n =1 where � 1 � 1 � f , sin ω n x � 0 f ( x ) sin ω n x dx 0 f ( x ) sin ω n x dx b n = � sin ω n x , sin ω n x � = = . � 1 || sin ω n x || 2 0 (sin ω n x ) 2 dx M. Macauley (Clemson) Lecture 4.5: Generalized Fourier series Advanced Engineering Mathematics 7 / 7
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