Acceleration of Fourier Series Charles Moore Kansas State University, U.S.A. International Conference on Scientific Computing October 13, 2011
Introduction Functions with multiple jumps Comments, further results Outline Introduction Fourier series Acceleration Previous work Functions with multiple jumps Notation and lemmas Main results Some examples Comments, further results Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Fourier Series For a function f integrable on [ − π, π ] and n ∈ Z , we define the n th Fourier coefficient by � π f ( n ) := 1 ˆ f ( x ) e − inx dx . 2 π − π Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Fourier Series For a function f integrable on [ − π, π ] and n ∈ Z , we define the n th Fourier coefficient by � π f ( n ) := 1 ˆ f ( x ) e − inx dx . 2 π − π For n = 0 , 1 , 2 , . . . and x ∈ [ − π, π ] , define the n th partial sum of the Fourier series: n � ˆ f ( k ) e ikx . S n f ( x ) := k = − n Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Accelerating Convergence The convergence of Fourier series can be quite slow and often depends on subtle cancellation. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Accelerating Convergence The convergence of Fourier series can be quite slow and often depends on subtle cancellation. This is especially true for functions with jump discontinuities. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Accelerating Convergence The convergence of Fourier series can be quite slow and often depends on subtle cancellation. This is especially true for functions with jump discontinuities. For a sequence { s n } that converges to s , we say that a transformation { t n } accelerates the convergence of { s n } if there exists a positive integer k such that each t n depends only on s 1 , s 2 , ..., s n + k and { t n } converges to s faster than { s n } , that is t n − s s n − s = 0 lim n →∞ Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The δ 2 Transformation Given a numerical sequence s n , the δ 2 process transforms it to the sequence ( s n + 1 − s n )( s n − s n − 1 ) t n := s n − ( s n + 1 − s n ) − ( s n − s n − 1 ) , where we set t n = s n if the denominator of the fraction is zero. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The δ 2 Transformation Given a numerical sequence s n , the δ 2 process transforms it to the sequence ( s n + 1 − s n )( s n − s n − 1 ) t n := s n − ( s n + 1 − s n ) − ( s n − s n − 1 ) , where we set t n = s n if the denominator of the fraction is zero. If a complex series and its δ 2 transform converge, then their sums are equal (Tucker, 1967). Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The δ 2 Transformation Given a numerical sequence s n , the δ 2 process transforms it to the sequence ( s n + 1 − s n )( s n − s n − 1 ) t n := s n − ( s n + 1 − s n ) − ( s n − s n − 1 ) , where we set t n = s n if the denominator of the fraction is zero. If a complex series and its δ 2 transform converge, then their sums are equal (Tucker, 1967). It is well known that the δ 2 process behaves badly for many sequences; it can turn a convergent sequence into one with multiple convergent subsequences. It can also turn divergent sequences into convergent sequences. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Applying this to Fourier Series We consider the possibility of applying the δ 2 transform pointwise to the partial sums S n f ( x ) of an integrable function f on [ − π, π ] . This results in the sequence of functions T n f ( x ) given by: ( S n + 1 f ( x ) − S n f ( x ))( S n f ( x ) − S n − 1 f ( x )) T n f ( x ) := S n f ( x ) − ( S n + 1 f ( x ) − S n f ( x )) − ( S n f ( x ) − S n − 1 f ( x )) , Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Applying this to Fourier Series We consider the possibility of applying the δ 2 transform pointwise to the partial sums S n f ( x ) of an integrable function f on [ − π, π ] . This results in the sequence of functions T n f ( x ) given by: ( S n + 1 f ( x ) − S n f ( x ))( S n f ( x ) − S n − 1 f ( x )) T n f ( x ) := S n f ( x ) − ( S n + 1 f ( x ) − S n f ( x )) − ( S n f ( x ) − S n − 1 f ( x )) , Various authors have done numerical experiments: Smith and Ford, Drummond. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Applying this to Fourier Series We consider the possibility of applying the δ 2 transform pointwise to the partial sums S n f ( x ) of an integrable function f on [ − π, π ] . This results in the sequence of functions T n f ( x ) given by: ( S n + 1 f ( x ) − S n f ( x ))( S n f ( x ) − S n − 1 f ( x )) T n f ( x ) := S n f ( x ) − ( S n + 1 f ( x ) − S n f ( x )) − ( S n f ( x ) − S n − 1 f ( x )) , Various authors have done numerical experiments: Smith and Ford, Drummond. If f ( x ) = − 1 on [ − π, 0 ) and f ( x ) = 1 on [ 0 , π ] , then S n f ( π 2 ) gives the partial sums for the Leibnitz series 1 = f ( π 2 ) = 4 π ( 1 − 1 3 + 1 5 − 1 7 + . . . ) . The convergence of this series is extremely slow but is dramatically accelerated by applying the δ 2 process. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The Transform for Functions With One Jump Theorem (joint with E. Abebe and P . Graber, 2007) (a) Suppose f ∈ C 2 ([ − π, π ]) and that f ( − π ) � = f ( π ) . Then the transformed sequence of partial sums T n f ( x ) diverges at every x of the form x = 2 a π , where a ∈ [ − . 5 , . 5 ] is irrational. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The Transform for Functions With One Jump Theorem (joint with E. Abebe and P . Graber, 2007) (a) Suppose f ∈ C 2 ([ − π, π ]) and that f ( − π ) � = f ( π ) . Then the transformed sequence of partial sums T n f ( x ) diverges at every x of the form x = 2 a π , where a ∈ [ − . 5 , . 5 ] is irrational. (b) Suppose f is as above and let x := 2 j π k where j k is in lowest terms and k is odd. Then T n f ( x ) has three limit points, f ( x ) and α 2 sin 2 ( x / 2 ) f ( x ) ± α sin ( x / 2 )+ 2 β cos ( x / 2 ) , where α = [ f ( π ) − f ( − π )] /π and β = [ f ′ ( π ) − f ′ ( − π )] /π . Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work Fourier series and transformed Fourier series for f ( x ) = x 4 4 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Figure: S 100 f for f ( x ) = x . Figure: T 100 f Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The δ 2 process is same as ε ( n ) in family of transforms ε ( n ) k . 2 Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The δ 2 process is same as ε ( n ) in family of transforms ε ( n ) k . 2 Brezinski (see also Wynn): To S n f add conjugate function � S n f ; create analytic function G n f ( z ) . Apply epsilon algorithm to G n f ( e i θ ) ; take real part. Charles Moore Acceleration of Fourier Series
Introduction Fourier series Functions with multiple jumps Acceleration Comments, further results Previous work The δ 2 process is same as ε ( n ) in family of transforms ε ( n ) k . 2 Brezinski (see also Wynn): To S n f add conjugate function � S n f ; create analytic function G n f ( z ) . Apply epsilon algorithm to G n f ( e i θ ) ; take real part. Brezinski gave numerical experiments to demonstrate that this procedure reduces the Gibbs phenomenon. Charles Moore Acceleration of Fourier Series
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