Fourier Series Fourier Sine Series Fourier Cosine Series Fourier - PowerPoint PPT Presentation

Fourier Series • Fourier Sine Series • Fourier Cosine Series • Fourier Series – Convergence of Fourier Series for 2 T -Periodic Functions – Convergence of Half-Range Expansions: Cosine Series – Convergence of Half-Range Expansions: Sine Series • Sawtooth Wave • Triangular Wave • Parseval’s Identity and Bessel’s Inequality • Complex Fourier Series • Dirichlet Kernel and Convergence

Fourier Sine Series � � nπx } ∞ Definition . Consider the orthogonal system { sin n =1 on [ − T, T ] . A Fourier T sine series with coefficients { b n } ∞ n =1 is the expression ∞ � nπx � � F ( x ) = b n sin T n =1 Theorem . A Fourier sine series F ( x ) is an odd 2 T -periodic function. Theorem . The coefficients { b n } ∞ n =1 in a Fourier sine series F ( x ) are determined by the formulas (inner product on [ − T, T ] ) � � �� nπx F, sin � T �� = 2 � nπx � T b n = F ( x ) sin dx. � � � � T T nπx nπx sin , sin 0 T T

Fourier Cosine Series � � mπx } ∞ Definition . Consider the orthogonal system { cos m =0 on [ − T, T ] . A Fourier T cosine series with coefficients { a m } ∞ m =0 is the expression ∞ � mπx � � F ( x ) = a m cos T m =0 Theorem . A Fourier cosine series F ( x ) is an even 2 T -periodic function. Theorem . The coefficients { a m } ∞ m =0 in a Fourier cosine series F ( x ) are determined by the formulas (inner product on [ − T, T ] ) � T � � �� � �  mπx 2 mπx F, cos 0 F ( x ) cos dx m > 0 ,  T T T �� = a m = � � � � � T mπx mπx cos , cos 1 0 F ( x ) dx m = 0 .  T T T

Fourier Series � � � � mπx nπx } ∞ } ∞ Definition . Consider the orthogonal system { cos m =0 , { sin n =1 , on T T [ − T, T ] . A Fourier series with coefficients { a m } ∞ m =0 , { b n } ∞ n =1 is the expression ∞ ∞ � mπx � nπx � � � � F ( x ) = a m cos + b n sin T T m =0 n =1 Theorem . A Fourier series F ( x ) is a 2 T -periodic function. Theorem . The coefficients { a m } ∞ m =0 , { b n } ∞ n =1 in a Fourier series F ( x ) are determined by the formulas (inner product on [ − T, T ] ) � T � � �� � �  mπx 1 mπx F, cos − T F ( x ) cos dx m > 0 ,  T T T �� = a m = � � � � � T mπx mπx cos , cos 1 − T F ( x ) dx m = 0 .  T T 2 T � � �� nπx F, sin � T �� = 1 � nπx � T b n = F ( x ) sin dx. � � � � T T nπx nπx sin , sin − T T T

Convergence of Fourier Series for 2 T -Periodic Functions The Fourier series of a 2 T -periodic piecewise smooth function f ( x ) is ∞ � nπx � nπx � � �� � a 0 + a n cos + b n sin T T n =1 where � T a 0 = 1 f ( x ) dx, 2 T − T � T a n = 1 � nπx � f ( x ) cos dx, T T − T � T b n = 1 � nπx � f ( x ) sin dx. T T − T The series converges to f ( x ) at points of continuity of f and to f ( x +)+ f ( x − ) otherwise. 2

Convergence of Half-Range Expansions: Cosine Series The Fourier cosine series of a piecewise smooth function f ( x ) on [0 , T ] is the even 2 T - periodic function ∞ � nπx � � a 0 + a n cos T n =1 where � T a 0 = 1 f ( x ) dx, T 0 � T a n = 2 � nπx � f ( x ) cos dx. T T 0 The series converges on 0 < x < T to f ( x ) at points of continuity of f and to f ( x +)+ f ( x − ) otherwise. 2

Convergence of Half-Range Expansions: Sine Series The Fourier sine series of a piecewise smooth function f ( x ) on [0 , T ] is the odd 2 T - periodic function ∞ � nπx � � b n sin T n =1 where � T b n = 2 � nπx � f ( x ) sin dx. T T 0 The series converges on 0 < x < T to f ( x ) at points of continuity of f and to f ( x +)+ f ( x − ) otherwise. 2

Sawtooth Wave Definition . The sawtooth wave is the odd 2 π -periodic function defined on − π ≤ x ≤ π by the formula 1  2 ( π − x ) 0 < x ≤ π,     1 sawtooth( x ) = 2 ( − π − x ) − π ≤ x < 0 ,    0 x = 0 .  Theorem . The sawtooth wave has Fourier sine series ∞ 1 � sawtooth( x ) = n sin nx. n =1

Triangular Wave Definition . The triangular wave is the even 2 π -periodic function defined on − π ≤ x ≤ π by the formula � π − x 0 < x ≤ π, twave( x ) = π + x − π ≤ x ≤ 0 . Theorem . The triangular wave has Fourier cosine series ∞ twave( x ) = π 2 + 4 1 � (2 k + 1) 2 cos(2 k + 1) x. π k =0

ParsevaL’s Identity and Bessel’s Inequality Theorem . (Bessel’s Inequality) � T ∞ � ≤ 1 0 + 1 � a 2 � a 2 n + b 2 | f ( x ) | 2 dx n 2 2 T − T n =1 Theorem . (Parseval’s Identity) � T ∞ 1 0 + 1 � | f ( x ) | 2 dx = a 2 � a 2 n + b 2 � n 2 T 2 − T n =1 Theorem . Parseval’s identity for the sawtooth function implies ∞ π 2 12 = 1 1 � 2 n 2 n =1

Complex Fourier Series Definition . Let f ( x ) be 2 T -periodic and piecewise smooth. The complex Fourier series of f is � T ∞ c n = 1 inπx − inπx � T , c n e f ( x ) e dx T 2 T − T n = −∞ Theorem . The complex series converges to f ( x ) at points of continuity of f and to f ( x +)+ f ( x − ) otherwise. 2 Theorem . (Complex Parseval Identity) � T ∞ 1 � | f ( x ) | 2 dx = | c n | 2 2 T − T n = −∞

Dirichlet Kernel and Convergence Theorem . (Dirichlet Kernel Identity) �� � � n + 1 sin u 1 2 2 + cos u + cos 2 u + · · · + cos nu = � � 1 2 sin 2 u Theorem . (Riemann-Lebesgue) � π For piecewise continuous g ( x ) , lim g ( x ) sin( Nx ) dx = 0 . N →∞ − π Proof : Integration theory implies it suffices to establish the result for smooth g . Integrate by parts to � π n ( g ( − π ) − g ( π ))( − 1) n + 1 obtain 1 − π g ( x ) cos( nx ) dx . Letting n → ∞ implies the result. n Theorem . Let f ( x ) be 2 π -periodic and smooth on the whole real line. Then the Fourier series of f ( x ) converges uniformly to f ( x ) .

Convergence Proof STEP 1 . Let s N ( x ) denote the Fourier series partial sum. Using Dirichlet’s kernel for- mula, we verify the identity � x + π � � f ( x ) − s N ( x ) = 1 sin(( N + 1 / 2) w ) ( f ( x ) − f ( x + w )) dw π 2 sin( w/ 2) x − π STEP 2 . The integrand I is re-written as I = f ( x ) − f ( x + w ) w 2 sin( w/ 2) sin(( N + 1 / 2) w ) . w STEP 3 . The function g ( w ) = f ( x ) − f ( x + w ) w is piecewise continu- w sin( w/ 2) ous. Apply the Riemann-Lebesgue Theorem to complete the proof of the theorem.

Gibbs’ Phenomena Engineering Interpretation : The graph of f ( x ) and the graph of a 0 + � N n =1 ( a n cos nx + b n sin nx ) are identical to pixel resolution, provided N is sufficiently large. Computers can therefore graph f ( x ) using a truncated Fourier series. If f ( x ) is only piecewise smooth, then pointwise convergence is still true, at points of continuity of f , but uniformity of the convergence fails near discontinuities of f and f ′ . Gibbs discovered the fixed-jump artifact, which appears at discontinuities of f .