Optimal minimax algorithm for integrating fast oscillatory functions in two dimensions

Engineering Computations (Swansea, Wales)

Volumn 21, Issue 8, 2004, Pages 834-847

  Zotsenko, K N ^a   Melnik, R V N ^b  

^a Electronic Data Systems Australia   (Australia)

^b WILFRID LAURIER UNIVERSITY   (Canada)

Author keywords

Approximation theory; Interpolation; Oscillations

Indexed keywords

APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; FAST FOURIER TRANSFORMS; FUNCTIONS; IMAGE PROCESSING; INTEGRAL EQUATIONS; INTERPOLATION; OSCILLATIONS; SIGNAL PROCESSING;

ENGINEERING APPLICATIONS; IMAGE RECOGNITION; OPTIMAL MINIMAX ALGORITHM; OSCILLATORY FUNCTIONS;

ALGORITHMS;

EID: 10444288730     PISSN: 02644401     EISSN: None     Source Type: Journal    
DOI: 10.1108/02644400410554344     Document Type: Article

References (39)

1. Ahlberg, J.H., Nilson, E.N. and Walsh, J.L. (1967), The Theory of Splines and Their Applications, Academic Press, New York, NY.
2. Alaylioglu, A. (1983), "Numerical evaluation of finite Fourier integrals", J. Comput. Appl. Math., Vol. 9, pp. 305-13.
3. Berezovskii, A.I. and Ivanov, V.V. (1977), "On optimal-by-accuracy uniform spline approximation", Soviet Mathematics (Iz. VUZ), Vol. 10, pp. 14-24.
4. Berezovskii, A.I. and Nechiporenko, N.E. (1991), "Optimal accuracy approximation of functions and their derivatives", Journal of Mathematical Sciences, Vol. 54, pp. 799-812.
5. Blahut, R.E. (1987), Fast Algorithms for Digital Signal Processing, Addison-Wesley, Reading, MA.
6. Brachman, R.W.I. and Moore, I.D. (1999), "Numerical integration during Fourier integral analysis", Int. J. Numer. Anal. Meth. Geomech., Vol. 23, pp. 1309-18.
7. Chan, S.C. and Ho, K.L. (1991), "A new two-dimensional fast cosine transform", IEEE Trans. Signal Process., Vol. 39 No. 2, pp. 481-5.
8. Cools, R. (1997), "Constructing cubature formulae: the science behind the art", Acta Numerica, Cambridge University Press, Cambridge, MA, pp. 1-54.
9. Davies, K.T.R. (1989), "Complex-plane methods for evaluating integrals with highly oscillatory integrands", J. Comput. Phys., Vol. 80, pp. 498-505.
10. Drachman, B. and Ross, J. (1994), "Approximation of certain functions given by integrals with highly oscillatory integrands", IEEE Transactions on Antennas and Propagation, Vol. 42 No. 9, pp. 1355-6.
11. Einarson, B. (1968), "Numerical calculation of Fourier integrals with cubic splines", BIT Numerical Mathematics, Vol. 8 No. 3, pp. 279-86.
12. Ersoy, O.K. (1997), Fourier-Related Transforms, Fast Algorithms, and Applications, Prentice-Hall, Englewood Cliffs, NJ.
13. Escobar, F.H., Jongkittinarukorn, K. and Civan, F. (1997), "Cubature solution of the Poisson equation", Commun. Numer. Meth. Eng., Vol. 13, pp. 453-65.
14. Fettis, H.E. and Pexton, R.L. (1982), "More on the calculation of oscillatory integrals", J. Comput. Phys., Vol. 47, pp. 473-6.
15. Filon, L.N. (1928), "On a quadrature formula for trigonometric integrals", Proc. Roy. Soc. Edinburgh, Vol. 49, pp. 38-47.
16. Haider, Q. and Liu, L.C. (1992), "Fourier and Bessel transformations of highly oscillatory functions", J. Phys. A: Math. Gen., Vol. 25, pp. 6755-60.
17. Heinrich, S., Hickernell, F.J. and Yue, R.X. (2003), "Optimal quadrature for Haar wavelet spaces", Mathematics of Computation, Vol. 73, pp. 259-77.
18. Hickernell, F.J. (2003), "My dream quadrature rule", Journal of Complexity, Vol. 19, pp. 420-7.
19. Hopkins, H.H. (1989), "Numerical evaluation of a class of double integrals of oscillatory functions", IMA J. of Numerical Analysis, Vol. 9, pp. 61-80.
20. Johnston, P.R. and Elliott, D. (2000), "Error estimation of quadrature rules for evaluating singular integrals in boundary element problems", Int. J. Numer. Meth. Eng., Vol. 48, pp. 949-62.
21. Korneichuk, N.P., Ligun, A.A. and Babenko, V.F. (1996), External Properties of Polynomials and Splines, Nova Science Publishers, New York, NY.
22. Levin, D. (1996), "Fast integration of rapidly oscillatory functions", J. Comput. Appl. Math., Vol. 67, pp. 95-101.
23. Melnik, K.N. and Melnik, R.V.N. (1998), "On computational aspects of certain optimal digital signal processing algorithms", in Noye, J., Teubner, M. and Gill, A. (Eds), Proc. of Computational Technique and Applications Conference: CTAC97, World Scientific, Singapore, pp. 433-40.
24. Melnik, K.N. and Melnik, R.V.N. (1999), "Optimal-by-order quadrature formulae for fast oscillatory functions with inaccurately given a priori information", J. Comp. Appl. Math., Vol. 110, pp. 45-72.
25. Melnik, K.N. and Melnik, R.V.N. (2001), "Optimal cubature formulae and recovery of fast-oscillatory functions from an interpolational class", BIT Numerical Mathematics, Vol. 41 No. 4.
26. Morozov, V.A. (1984), Methods of Solving Incorrectly Posed Problems, Springer-Verlag, Berlin.
27. Mysovskih, I.P. (1992), Interpolatorische Kubaturformulen, Institut fur Geometrie und Praktische Matematik der RWTH Aachen, Bericht 74.
28. Nikolskii, S.M. (1964), Quadrature Formulae, International Monographs on Advanced Mathematics and Physics, Hindustan Pub. Corp, Delhi, 29.
29. Nurnberger, G. (1989), Approximation by Spline Functions, Springer-Verlag, Berlin.
30. Pieper, W.M. (1999), "Recursive Gauss integration", Commun. Numer. Meth. Eng., Vol. 15, pp. 77-90.
31. Sukharev, A. (1992), Minimax Models in the Theory of Numerical Methods, Kluwer, Dordrecht.
32. Tikhonov, A.N., Leonov, A.S, and Yagola, A.G. (1996), Nonlinear Ill-Posed Problems, Chapman & Hall, London.
33. Tikhonov, A.N. et al. (1995), Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht.
34. Traub, J.F. and Wozniakowski, H. (1980), A General Theory of Optimal Algorithms, Academic Press, New York, NY.
35. Zadiraka, V.K. and Abatov, N.T. (1991), "Optimally exact algorithms for solutions of a certain numerical integration problem", Ukr. Mathematical Journal, Vol. 43, pp. 43-54.
36. L,N, Ukr", Mathematical Journal, Vol. 38 No. 2, pp. 233-7.
37. Zadiraka, V.K. and Melnikova, S.S. (1993), Digital Signal Processing, Naukova Dumka, Kiev.
38. Zheludev, V.A. (1995), "Processing of periodic signals using spline-wavelets", Radioelectronics and Communications Systems, Vol. 38 No. 3, p. 1.
39. Zhileikin, Ya.M. and Kukarkin, A.B. (1995), "A fast Fourier-Bessel transformation algorithm", Computational Mathematics and Mathematical Physics, Vol. 35 No. 7, p. 901.