Spectral approximation of convolution operators. (English) Zbl 1396.65158
Summary: We develop a unified framework for constructing matrix approximations to the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on \([-1, 1]\). The numerically stable algorithms we propose exploit recurrence relations and symmetric properties satisfied by the entries of these convolution matrices. Laguerre-based convolution matrices that approximate Volterra convolution operators defined by functions on \([0, \infty)\) are also discussed for the sake of completeness.
MSC:
| 65R10 | Numerical methods for integral transforms |
| 44A35 | Convolution as an integral transform |
| 42A85 | Convolution, factorization for one variable harmonic analysis |
| 47A58 | Linear operator approximation theory |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 41A10 | Approximation by polynomials |
| 65Q30 | Numerical aspects of recurrence relations |
Keywords:
convolution; Volterra convolution integral; operator approximation; orthogonal polynomial; pectral method; Chebyshev polynomial; Legendre polynomial; Gegenbauer polynomial; ultraspherical polynomial; Jacobi polynomial; Laguerre polynomialReferences:
| [1] | K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, UK, 1997. · Zbl 0899.65077 |
| [2] | J. L. Aurentz and L. N. Trefethen, Chopping a Chebyshev series, ACM Trans. Math. Software, 43 (2017), pp. 33:1–33:21. · Zbl 1380.65032 |
| [3] | Z. Battles and L. N. Trefethen, An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25 (2004), pp. 1743–1770. · Zbl 1057.65003 |
| [4] | H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, Elsevier, Dordrecht, the Netherlands, 1986. · Zbl 0611.65092 |
| [5] | S. B. Damelin and W. Miller, Jr., The Mathematics of Signal Processing, Cambridge University Press, Cambridge, UK, 2012. · Zbl 1257.94001 |
| [6] | A. Dixit, L. Jiu, V. H. Moll, and C. Vignat, The finite Fourier transform of classical polynomials, J. Aust. Math. Soc., 98 (2015), pp. 145–160. · Zbl 1396.33021 |
| [7] | T. A. Driscoll, N. Hale, and L. N. Trefethen, eds., Chebfun Guide, Pafnuty Publications, Oxford, UK, 2014. |
| [8] | D. G. Duffy, Green’s Functions with Applications, CRC Press, Boca Raton, FL, 2015. · Zbl 1343.35002 |
| [9] | W. Feller, On the integral equation of renewal theory, Ann. Math. Statist., 12 (1941), pp. 243–267. · Zbl 0026.23001 |
| [10] | D. Forsyth and J. Ponce, Computer Vision: A Modern Approach, Prentice-Hall, Englewood Cliffs, NJ, 2011. |
| [11] | T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numer., 8 (1999), pp. 47–106. · Zbl 0940.65108 |
| [12] | N. Hale and A. Townsend, An algorithm for the convolution of Legendre series, SIAM J. Sci. Comput., 36 (2014), pp. A1207–A1220. · Zbl 1296.33010 |
| [13] | R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, 2000. · Zbl 0998.26002 |
| [14] | R. V. Hogg, J. McKean, and A. T. Craig, Introduction to Mathematical Statistics, 7th ed., Pearson Education, New York, 2013. |
| [16] | P. Linz, Analytical and Numerical Methods for Volterra Equations, Stud. Appl. Numer. Math. 7, SIAM, Philadelphia, 1985. · Zbl 0566.65094 |
| [17] | A. F. Loureiro and K. Xu, Volterra-type convolution of classical polynomials, preprint, , 2018. |
| [18] | J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, CRC Press, Boca Raton, FL, 2003. · Zbl 1015.33001 |
| [19] | F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., NIST Digital Library of Mathematical Functions, Release 1.0.18, (2018). |
| [20] | S. Olver and A. Townsend, A fast and well-conditioned spectral method, SIAM Rev., 55 (2013), pp. 462–489. · Zbl 1273.65182 |
| [21] | J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, New York, 2011. · Zbl 1227.65117 |
| [22] | E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer, New York, 2013. |
| [23] | I. Stakgold and M. J. Holst, Green’s Functions and Boundary Value Problems, John Wiley & Sons, New York, 2011. · Zbl 1221.35001 |
| [24] | G. W. Stewart, Afternotes Goes to Graduate School, SIAM, Philadelphia, PA, 1998. · Zbl 0898.65001 |
| [25] | G. Szegö, Orthogonal Polynomials, AMS, Providence, RI, 1939. |
| [26] | L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, 2013. · Zbl 1264.41001 |
| [27] | C. Wagner, T. Hüttl, and P. Sagaut, Large-Eddy Simulation for Acoustics, Cambridge University Press, Cambridge, UK, 2007. |
| [28] | H. Wang, On the optimal estimates and comparison of Gegenbauer expansion coefficients, SIAM J. Numer. Anal., 54 (2016), pp. 1557–1581. · Zbl 1342.41039 |
| [29] | H. Wang and S. Xiang, On the convergence rates of Legendre approximation, Math. Comput., 81 (2012), pp. 861–877. · Zbl 1242.41016 |
| [30] | K. Xu, A. P. Austin, and K. Wei, A fast algorithm for the convolution of functions with compact support using Fourier extensions, SIAM J. Sci. Comput., 39 (2017), pp. A3089–A3106. · Zbl 1379.65096 |
| [31] | X. Zhao, L.-L. Wang, and Z. Xie, Sharp error bounds for Jacobi expansions and Gegenbauer–Gauss quadrature of analytic functions, SIAM J. Numer. Anal., 51 (2013), pp. 1443–1469. · Zbl 1276.65017 |
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