Convergence and Stability of a New Quadrature Rule for Evaluating Hilbert Transform Maria Rosaria Capobianco CNR - National Research Council of Italy Institute for Computational Applications “Mauro Picone”, Naples, Italy. G. Criscuolo Dipartimento di Matematica e Applicazioni, Universit` a degli Studi di Napoli “Federico II”, Naples, Italy. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 1
w nonnegative weight function on the interval [ a, b ] , � b −∞ ≤ a < b ≤ ∞ , 0 < a w ( x ) dx < ∞ . Weighted Hilbert Transform � b f ( t ) f ( t ) � H ( wf ; x ) := t − xw ( t ) dt = lim t − xw ( t ) dt, t ∈ ( a, b ) . ε → 0 + a | t − x |≥ ε SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 2
◮ Causality and generalization of the phaser idea beyond pure alternating current 1 R. Bracewell, The Fourier Transform and its Applications , Electrical and Electronic Engineering Series, McGraw–Hill, New York, 1965. ◮ Boundary Value Problems as Singular Integral Eequations Involving Cauchy Principal Value Integrals 2 S.G. Mikhlin, S. Pr¨ ossdorf, Singular Integral Operators , Springer–Verlag, Berlin, 1986. 3 N.I. Muskhelishvili, Singular Integral Equations , Noordhoff, 1977. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 3
◮ Numerical Evaluation of H ( wf ) when a − ∞ < a < b < ∞ 4 G. Criscuolo, A new algorithm for Cauchy principal value and Hadamard finite–part integrals , J. Comput. Appl. Math. 78 (1997), 255–275. 5 G. Criscuolo, L. Scuderi The numerical evaluation of Cauchy principal value integrals with non–standard weight functions , BIT 38 (1998), 256–274. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 4
◮ Numerical Evaluation of the Weighted Hilbert Transform on the Real Line 6 M.R. Capobianco, G. Criscuolo, R. Giova, Approximation of the Hilbert transform on the real line by an interpolatory process , BIT 41 (2001), 666–682. 7 M.R. Capobianco, G. Criscuolo, R. Giova, A stable and convergent algorithm to evaluate Hilbert transform , Numerical Algorithms 28 (2001), 11–26. 8 S.B. Damelin, K. Diethelm, Interpolatory product quadrature for Cauchy principal value integrals with Freud weights , Numer. Math. 83 (1999), 87–105. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 5
9 S.B. Damelin, K. Diethelm, Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line , J. Funct. Anal. Optim. 22 n.1–2 (2001), 13–54. 10 K. Diethelm, A method for the practical evaluation of the Hilbert transform on the real line , J. Comp. Appl. Math. 112 (1999), 45–53. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 6
Essentially two kinds of quadrature rules of interpolatory type have been proposed Gaussian Rules Product Rules SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 7
Hypothesis on the function f � � | t |→∞ f ( t ) e − t 2 / 2 = 0 W ∞ f ∈ C 0 := LOC ( R ) : lim , 0 where C 0 LOC ( R ) is the set of all locally continuous functions on R and f satisfies a Dini type condition by the Ditzian–Totik modulus of continuity, then H ( wf ) is bounded on R (see Theorem 1.2 in S.B. Damelin, K. Diethelm, Boundedness and uniform numerical approximation of the weighted Hilbert transform on the real line , J. Funct. Anal. Optim. 22 n.1–2 (2001), 13–54.) SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 8
{ p m ( w ) } ∞ m =0 sequence of the orthonormal Hermite polynomials associated with the weight w ( t ) = e − t 2 , p m ( w ; t ) = γ m t m + · · · , γ m > 0 −∞ < t m,m < t m,m − 1 < · · · < t m, 2 < t m, 1 < + ∞ √ t m, 1 = − t m,m < 2 m + 1 For any x ∈ R , m ∈ N we denote by t m,c the zero of p m ( w ) closest to x , defined by | t m,c − x | = 1 ≤ k ≤ m | t m,k − x | . min SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 9
When x is equidistant between two zeros, i.e. x = ( t m,k + t m,k +1 ) / 2 for some k ∈ { 1 , 2 , · · · , m − 1 } , it makes no difference for the subsequent analysis to define t m,c = t m,k or t m,c = t m,k +1 . � + ∞ � + ∞ e − t 2 f ( t ) − f ( x ) e − t 2 dt + f ( x ) H ( wf ; x ) = t − xdx t − x −∞ −∞ and approximate the first integral by interpolating the function f − f ( x ) e 1 − x , e 1 ( t ) = t, on the set of nodes { t m,k , k = 1 , 2 , · · · , m, k � = c } , all of which are far from the singularity x . SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 10
The Lagrange polynomial L m − 1 ( g ) which interpolates the function g at these points may written as m ℓ m,k ( w ; t ) t m,k − t m,c � L m − 1 ( g ; t ) = g ( t m,k ) , t − t m,c k =1 ,k � = c where p m ( w ; t ) ℓ m,k ( w ; t ) = m ( w ; t m,k )( t − t m,k ) , k = 1 , 2 , · · · , m. p ′ � + ∞ ℓ m,k ( w ; t ) e − t 2 dt, t − t m,c −∞ SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 11
can be computed by the Gaussian rule with respect to the Hermite weight � + ∞ p ′ p ′ ℓ m,k ( w ; t ) 1 � m ( w ; t m,k ) m ( w ; t m,c ) � e − t 2 dt = λ m,k + λ m,c , p ′ t − t m,c m ( w ; t m,k ) t m,k − t m,c t m,c − t m,k −∞ where λ m,k = λ m,k ( w ) , k = 1 , 2 , · · · , m , are the Cotes numbers with respect to the Hermite weight. Since γ m 1 p ′ m ( w ; t m,j ) = λ m,j p m − 1 ( w ; t m,j ) , j = 1 , 2 , · · · , m, γ m − 1 we get � + ∞ ℓ m,k ( w ; t ) λ m,k � 1 − p m − 1 ( w ; t m,k ) � e − t 2 dt = , k = 1 , 2 , · · · , m, t − t m,c t m,k − t m,c p m − 1 ( w ; t m,c ) −∞ SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 12
and � + ∞ m λ m,k � 1 − p m − 1 ( w ; t m,k ) � g ( t ) e − t 2 dt = � g ( t m,k )+ R m ( w ; g ) . t m,k − t m,c p m − 1 ( w ; t m,c ) −∞ k =1 ,k � = c Finally, we arrive at the formula H ( wf ; x ) = H m ( w ; f ; x ) + R H m ( w ; f ; x ) , where SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 13
� f ( t m,k ) − f ( x ) m � λ m,k 1 − p m − 1 ( w ; t m,k ) � H m ( w ; f ; x ) = + t m,k − t m,c p m − 1 ( w ; t m,c ) t m,k − x k =1 ,k � = c � + ∞ e − t 2 f ( x ) t − xdt, −∞ and � � w ; f − f ( x ) R H m ( w ; f ; x ) = R m , e 1 − x is the error. The quadrature rule has degree of exactness at least m − 1 , i.e. R H m ( w ; f ) ≡ 0 whenever f is a polynomial of degree m − 1 . SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 14
If, as required sometimes in applications, we want to use only the values of the function f at the interpolation points, then it is convenient to apply () rewritten as m � H m ( w ; f ; x ) = A m ( x ) f ( x ) + A m,k ( x ) f ( t m,k ) , k =1 ,k � = c where m λ m,k � 1 − p m − 1 ( w ; t m,k ) � � A m ( x ) = H ( w ; x ) − , t m,k − x p m − 1 ( w ; t m,c ) k =1 ,k � = c and � � λ m,k 1 − p m − 1 ( w ; t m,k ) A m,k ( x ) = , k = 1 , 2 , · · · , m, k � = c. t m,k − x p m − 1 ( w ; t m,c ) SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 15
We define the Amplification Factor m � K m ( w ; x ) := | A m ( x ) | w − 1 / 2 ( x )+ | A m,k ( x ) | w − 1 / 2 ( t m,k ) , x ∈ R , k =1 ,k � = c THEOREM Let w ( t ) = e − t 2 . Then √ log m, if | x | ≤ ̺ 2 m, 0 < ̺ < 1 , K m ( w ; x ) ≤ C m 1 / 6 log m, if | x | ≤ 2 t m, 1 − t m, 2 , with some constant C independent of m and x . SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 16
For any function � � � f ∈ C 0 √ w := f continuous on R and | t |→∞ f ( t ) lim w ( t ) = 0 , we define the norm √ w := � f √ w � ∞ = max � � f � C 0 t ∈ R | f ( t ) w ( t ) | . We set P ∈ P n � ( f − p ) √ w � ∞ , E m ( f ) √ w, ∞ := inf for any f ∈ C 0 √ w , and where P n denotes the set of the polynomials of degree at most m . Denoting by ω ( f, t ) √ w, ∞ the Ditzian–Lubinsky weighted modulus of smoothness, we can state the following result. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14, 2011 17
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