Convergence and Stability of a New Quadrature Rule for Evaluating Hilbert Transform
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1 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, Università degli Studi di Napoli Federico II, Naples, Italy. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
2 w nonnegative weight function on the interval [a,b], a < b, 0 < b a w(x)dx <. Weighted Hilbert Transform H(wf;x) := b a f(t) w(t)dt = lim t x ε 0 + t x ε f(t) t x w(t)dt, t (a,b). SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
3 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, Boundary Value Problems as Singular Integral Eequations Involving Cauchy Principal Value Integrals 2 S.G. Mikhlin, S. Prössdorf, Singular Integral Operators, Springer Verlag, Berlin, N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
4 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), G. Criscuolo, L. Scuderi The numerical evaluation of Cauchy principal value integrals with non standard weight functions, BIT 38 (1998), SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
5 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), M.R. Capobianco, G. Criscuolo, R. Giova, A stable and convergent algorithm to evaluate Hilbert transform, Numerical Algorithms 28 (2001), S.B. Damelin, K. Diethelm, Interpolatory product quadrature for Cauchy principal value integrals with Freud weights, Numer. Math. 83 (1999), SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
6 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), K. Diethelm, A method for the practical evaluation of the Hilbert transform on the real line, J. Comp. Appl. Math. 112 (1999), SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
7 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,
8 W 0 := Hypothesis on the function f { f C 0 LOC(R) : } 2 lim /2 t f(t)e t = 0, where CLOC 0 (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), ) SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
9 {p m (w)} m=0 sequence of the orthonormal Hermite polynomials associated with the weight w(t) = e t2, 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 < 2m + 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 = min 1 k m t m,k x. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
10 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. H(wf;x) = + f(t) f(x) t x e t2 dt + f(x) + e t2 t x dx 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,
11 The Lagrange polynomial L m 1 (g) which interpolates the function g at these points may written as L m 1 (g;t) = m k=1,k c l m,k (w;t) t m,k t m,c t t m,c g(t m,k ), where l m,k (w;t) = p m (w;t) p m(w;t m,k )(t t m,k ), k = 1, 2,,m. + l m,k (w;t) t t m,c e t 2 dt, SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
12 can be computed by the Gaussian rule with respect to the Hermite weight + l m,k (w;t) t t m,c e t 2 1 dt = p m(w;t m,k ) { p λ m(w;t m,k ) p } m,k + λ m(w;t m,c ) m,c, 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 p m(w;t m,j ) = γ m γ m 1 1 λ m,j p m 1 (w;t m,j ), j = 1, 2,,m, we get + l m,k (w;t) t t m,c e t 2 λ m,k dt = t m,k t m,c { 1 p } m 1(w;t m,k ), k = 1, 2,,m, p m 1 (w;t m,c ) SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
13 and + g(t)e t2 dt = m k=1,k c λ m,k t m,k t m,c { 1 p } m 1(w;t m,k ) g(t m,k )+R m (w;g). p m 1 (w;t m,c ) Finally, we arrive at the formula where H(wf;x) = H m (w;f; x) + R H m(w;f;x), SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
14 H m (w;f; x) = m k=1,k c λ m,k t m,k t m,c { 1 p } m 1(w;t m,k ) f(tm,k ) f(x) + p m 1 (w;t m,c ) t m,k x and is the error. f(x) + e t2 t x dt, ( Rm(w;f;x) H = R m w; f f(x) ) e 1 x, 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,
15 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 H m (w;f;x) = A m (x)f(x) + m k=1,k c A m,k (x)f(t m,k ), where A m (x) = H(w;x) m k=1,k c λ m,k t m,k x { 1 p } m 1(w;t m,k ), p m 1 (w;t m,c ) and A m,k (x) = λ m,k t m,k x { 1 p } m 1(w;t m,k ), k = 1,2,,m, k c. p m 1 (w;t m,c ) SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
16 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 t2. Then K m (w;x) C log m, if x 2m, 0 < < 1, m 1/6 log m, if x 2t 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,
17 For any function f C 0 w := we define the norm { } f continuous on R and lim f(t) w(t) = 0, t f C 0 w := f w = max t R f(t) w(t). We set E m (f) w, := inf (f p) w, P P n 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,
18 THEOREM Assume that f C 0 w satisfies the condition R H m (w;f; x) C 1 0 t 1 ω(f, t) w, dt <. log m E m 1 (f) w, + 1/ m 0 ω(f,t) w, t dt, if x 2m, 0 < < 1, m 1/6 log m E m 1 (f) w, + 1/ m 0 if x 2t m,1 t m,2, for some constant C independent of f, m and x. ω(f,t) w, t dt, SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
19 E m (f) w, const ω(f, m 1/2 ) w,, m 0, w(t) = e t2, Corollary Assume that w(t) = e t2 and f C 0 w satisfies the condition ω(f, t) w, = O(t α ), α > 0. Then R H m(w;f; x) C m α/2 log m, x 2m, 0 < < 1, for some constant C independent of f, m and x. Further, if α > 1/3, then R H m (w;f; x) C m α/2+1/6, x 2t m,1 t m,2, with some constant C independent of f, m and x. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
20 REMARK We have assumed x 2t m,1 t m,2. If x > 2t m,1 t m,2 then x is sufficiently far from all the nodes t m,k, k = 1,2,,m. So that the Gaussian rule converges and does not present the numerical cancelation problem. Thus, the quadrature rule we have introduced in this paper is meaningful exactly when x 2t m,1 t m,2. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
21 Amplification Factor K m (w;x) Table 1: x=0 m K m K m gaussian d d d d d+15 SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
22 Table 2: x= m K m K m gaussian d d d d d+4 SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
23 Table 3: x=0.001 m K m K m gaussian SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
24 Table 4: x=0.01 m K m K m gaussian SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
25 Table 5: x=0.1 m K m K m gaussian SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
26 Table 6: x=1. m K m K m gaussian SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
27 In the first example the exact solution H(wf;x) is given and we have computed it by using the computer algebra package Mathematica. Figure shows the plots of the exact solution, our solution and the solution evaluated with the help of the Matlab routine Hilbert. H(wf;x) = + exp(t)t 4 exp( t 2 ) dt = t x = 1 8 exp(1 4 ) π(7 + 6x + 4x 2 + 8x 3 ) + x 4 exp( 1 4 )H(w;x 1 2 ) where f(t) = exp(t)t 4. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
28 Comparison Between Exact, Matlab and Our Solution True solution Matlab solution our solution SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
29 Also in the second example the exact solution H(wf;x) is given and we have computed it by using the computer algebra package Mathematica. Figure shows the plots of the exact solution, our solution and the solution evaluated with the help of the Matlab routine Hilbert. H(wf;x) = + t 2 5 exp( t 2 ) dt = t x = xγ( 3 4 ) + iexp( x2 )( 1 x 2)1 4 x 3 (π ( 1) 1 4 Γ( 3 4 )Γ(1 4, x2 )) where f(t) = t 5 2. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
30 Comparison Between Exact, Matlab and Our Solution Matlab solution True solution Our solution SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
31 In the last example again the exact solution H(wf;x) is given and we have computed it by using the computer algebra package Mathematica. Figure shows the plots of the exact solution, our solution and the solution evaluated with the help of the Matlab routine Hilbert. H(wf;x) = + exp( t 2 ) (1 + t 2 )(t x) dt = = H(w;x) + eπx(erf(1) 1) 1 + x 2 where f(t) = 1 1+t 2. SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
32 Comparison Between Exact, Matlab and Our Solution Matlab solution True solution Our solution SC2011, International Conference on Scientific Computing,S. Margherita di Pula, Sardinia, Italy, October 10-14,
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