Fourier series Meromorphic function The convergence of Fourier-Takenaka-Malmquist series Adhemar Bultheel Department of Computer Science K.U.Leuven Approximation and extrapolation of convergent and divergent sequences and series Luminy, September 28, 2009 - October 2, 2009 Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
Fourier series Meromorphic function Survey Motivating applications Series tranformation = possible convergence acceleration The Takenake-Malmquist and Hambo bases Convergence theorems: Exponential decay Numerical experiments Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems The functions 2 π -periodic functions ∼ functions on T = { z ∈ C : | z | = 1 } Suppose f is real then Fourier series � c k z k , z = e i ω , f ( z ) = c − k = c k k ∈ Z ∞ � a ′ k cos k ω + b ′ = k sin k ω k =0 If f ∈ H ( D ), D = { z ∈ C : | z | < 1 } ∞ � c k z k , f ( z ) = | z | < 1 k =0 Mostly symmetric: f ( e i ω ) = � ∞ k =0 a ′ k cos k ω Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems The functions 2 π -periodic functions ∼ functions on T = { z ∈ C : | z | = 1 } Suppose f is real then Fourier series � c k z k , z = e i ω , f ( z ) = c − k = c k k ∈ Z ∞ � a ′ k cos k ω + b ′ = k sin k ω k =0 If f ∈ H ( D ), D = { z ∈ C : | z | < 1 } ∞ � c k z k , f ( z ) = | z | < 1 k =0 Mostly symmetric: f ( e i ω ) = � ∞ k =0 a ′ k cos k ω Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems The functions 2 π -periodic functions ∼ functions on T = { z ∈ C : | z | = 1 } Suppose f is real then Fourier series � c k z k , z = e i ω , f ( z ) = c − k = c k k ∈ Z ∞ � a ′ k cos k ω + b ′ = k sin k ω k =0 If f ∈ H ( D ), D = { z ∈ C : | z | < 1 } ∞ � c k z k , f ( z ) = | z | < 1 k =0 Mostly symmetric: f ( e i ω ) = � ∞ k =0 a ′ k cos k ω Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems The functions 2 π -periodic functions ∼ functions on T = { z ∈ C : | z | = 1 } Suppose f is real then Fourier series � c k z k , z = e i ω , f ( z ) = c − k = c k k ∈ Z ∞ � a ′ k cos k ω + b ′ = k sin k ω k =0 If f ∈ H ( D ), D = { z ∈ C : | z | < 1 } ∞ � c k z k , f ( z ) = | z | < 1 k =0 Mostly symmetric: f ( e i ω ) = � ∞ k =0 a ′ k cos k ω Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Motivating applications Impulse response (transfer function) of a LTI system stable and causal = holomorphic in D ∪ T usually real and rational (meromorphic) Realization theory, model reduction,... Identification of a signal want f stable, causal, minimal phase, real,. . . given the spectrum | f ( z ) | 2 , z ∈ T if c − k = c k then approximating | f ( z ) | 2 is like approximating its real part Lowest possible number of terms is an important issue less delay, lower cost rational is easily implemented with circuit loop Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Motivating applications Impulse response (transfer function) of a LTI system stable and causal = holomorphic in D ∪ T usually real and rational (meromorphic) Realization theory, model reduction,... Identification of a signal want f stable, causal, minimal phase, real,. . . given the spectrum | f ( z ) | 2 , z ∈ T if c − k = c k then approximating | f ( z ) | 2 is like approximating its real part Lowest possible number of terms is an important issue less delay, lower cost rational is easily implemented with circuit loop Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Motivating applications Impulse response (transfer function) of a LTI system stable and causal = holomorphic in D ∪ T usually real and rational (meromorphic) Realization theory, model reduction,... Identification of a signal want f stable, causal, minimal phase, real,. . . given the spectrum | f ( z ) | 2 , z ∈ T if c − k = c k then approximating | f ( z ) | 2 is like approximating its real part Lowest possible number of terms is an important issue less delay, lower cost rational is easily implemented with circuit loop Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Slow convergence possible Convergence of power series may be slow if singularity close to T Consider for example ∞ 1 k z k , � 1 − β z = 1 + β z ∈ T k =1 If | β | = 1 − ε < 1, ε small, then very slow convergence! The error � n � � ≤ | β | n +1 � � k z k � � f ( z ) − β � � 1 − | β | � � k =0 More general, the closest-to-1 pole dictates convergence rate Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Slow convergence possible Convergence of power series may be slow if singularity close to T Consider for example ∞ 1 k z k , � 1 − β z = 1 + β z ∈ T k =1 If | β | = 1 − ε < 1, ε small, then very slow convergence! The error � n � � ≤ | β | n +1 � � k z k � � f ( z ) − β � � 1 − | β | � � k =0 More general, the closest-to-1 pole dictates convergence rate Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Slow convergence possible Convergence of power series may be slow if singularity close to T Consider for example ∞ 1 k z k , � 1 − β z = 1 + β z ∈ T k =1 If | β | = 1 − ε < 1, ε small, then very slow convergence! The error � n � � ≤ | β | n +1 � � k z k � � f ( z ) − β � � 1 − | β | � � k =0 More general, the closest-to-1 pole dictates convergence rate Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Slow convergence possible Convergence of power series may be slow if singularity close to T Consider for example ∞ 1 k z k , � 1 − β z = 1 + β z ∈ T k =1 If | β | = 1 − ε < 1, ε small, then very slow convergence! The error � n � � ≤ | β | n +1 � � k z k � � f ( z ) − β � � 1 − | β | � � k =0 More general, the closest-to-1 pole dictates convergence rate Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Slow convergence possible Convergence of power series may be slow if singularity close to T Consider for example ∞ 1 k z k , � 1 − β z = 1 + β z ∈ T k =1 If | β | = 1 − ε < 1, ε small, then very slow convergence! The error � n � � ≤ | β | n +1 � � k z k � � f ( z ) − β � � 1 − | β | � � k =0 More general, the closest-to-1 pole dictates convergence rate Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Change of basis Suppose f belongs to a separable space with bases { e k } and { e ′ k } . � � c ′ k e ′ f ( z ) = c k e k ( z ) = k ( z ) k k For example ω 2 k � f ( z ) = Re ( z ) = cos ω = (2 k )! k The basis used is { e k ( ω ) = ω k : k = 0 , . . . , ∞} , which gives infinitely many terms. But the basis { cos k ω : k = 0 , . . . , ∞} requires only one term, hence extremely fast convergence. Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
The applications Fourier series Different bases Meromorphic function Convergence Theorems Change of basis Suppose f belongs to a separable space with bases { e k } and { e ′ k } . � � c ′ k e ′ f ( z ) = c k e k ( z ) = k ( z ) k k For example ω 2 k � f ( z ) = Re ( z ) = cos ω = (2 k )! k The basis used is { e k ( ω ) = ω k : k = 0 , . . . , ∞} , which gives infinitely many terms. But the basis { cos k ω : k = 0 , . . . , ∞} requires only one term, hence extremely fast convergence. Adhemar Bultheel The convergence of Fourier-Takenaka-Malmquist series
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