Abstract
For the filtering of peaks in periodic signals, we specify polynomial filters that are optimally localized in space. The space localization of functions f having an expansion in terms of orthogonal polynomials is thereby measured by a generalized mean value ε(f). Solving an optimization problem including the functional ε(f), we determine those polynomials out of a polynomial space that are optimally localized. We give explicit formulas for these optimally space localized polynomials and determine in the case of the Jacobi polynomials the relation of the functional ε(f) to the position variance of a well-known uncertainty principle. Further, we will consider the Hermite polynomials as an example on how to get optimally space localized polynomials in a non-compact setting. Finally, we investigate how the obtained optimal polynomials can be applied as filters in signal processing.
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Aptekarev, A., Dehesa, J.S., Martinez-Finkelshtein, A.: Asymptotics of orthogonal polynomial’s entropy. J. Comput. Appl. Math. 233(6), 1355–1365 (2010)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)
Erb, W.: Uncertainty principles on compact Riemannian manifolds. Appl. Comput. Harmon. Anal. 29(2), 182–197 (2010)
Erb, W.: Uncertainty principles on Riemannian manifolds. Dissertation, Logos Verlag, Berlin (2010). Available at http://mediatum2.ub.tum.de/doc/976465
Erb, W., Toókos, F.: Applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems. Appl. Math. Comput. 217(9), 4771–4780 (2011)
Filbir, F., Mhaskar, H.N., Prestin, J.: On a filter for exponentially localized kernels based on Jacobi polynomials. J. Approx. Theory 160(1–2), 256–280 (2009)
Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–233 (1997)
Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, Oxford (2004)
Goh, S.S., Goodman, T.N.T.: Uncertainty principles and asymptotic behavior. Appl. Comput. Harmon. Anal. 16(1), 69–89 (2004)
Gröchenig, K.: Uncertainty principles for time-frequency representations. In: Feichtinger, H., Strohmer, T. (eds.) Advances in Gabor Analysis, Applied and Numerical Harmonic Analysis, pp. 11–30. Birkhäuser, Basel (2003)
Harris, F.J.: On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66(1), 51–83 (1978)
Ismail, M.E.: The variation of zeros of certain orthogonal polynomials. Adv. Appl. Math. 8, 111–118 (1987)
Ismail, M.E.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)
Ivanov, K., Petrushev, P., Xu, Y.: Sub-exponentially localized kernels and frames induced by orthogonal expansions. Math. Z. 264, 361–397 (2010)
Laín Fernández, N.: Optimally space-localized band-limited wavelets on \(\mathbb{S}^{q-1}\). J. Comput. Appl. Math. 199(1), 68–79 (2007)
Landau, H.J.: An overview of time and frequency limiting. In: Price, J. (ed.) Fourier Techniques and Applications, pp. 201–220. Plenum, New York (1985)
Lasser, R.: Introduction to Fourier Series. Marcel Dekker, New York (1996)
Li, Z., Liu, L.: Uncertainty principles for Jacobi expansions. J. Math. Anal. Appl. 286(2), 652–663 (2003)
Mhaskar, H.N.: Introduction to the Theory of Weighted Polynomial Approximation. World Scientific, Singapore (1996)
Mhaskar, H.N., Narcowich, F.J., Prestin, J., Ward, J.D.: Polynomial frames on the sphere. Adv. Comput. Math. 13(4), 387–403 (2000)
Mhaskar, H.N., Prestin, J.: Polynomial frames: a fast tour. In: Chui, C.K., Schumaker, L.L, Neamtu, M. (eds.) Approximation Theory XI. Gatlinburg, 2004, pp. 287–318. Nashboro Press, Brentwood (2005)
Nguyen, N., Huang, H., Oraintara, S., Vo, A.: Gaborlocal: peak detection in mass spectrum by Gabor filters and Gaussian local maxima. In: Comput. Syst. Bioinformatics Conf., vol. 7, pp. 85–96 (2008)
Prestin, J., Quak, E., Rauhut, H., Selig, K.: On the connection of uncertainty principles for functions on the circle and on the real line. J. Fourier Anal. Appl. 9(4), 387–409 (2003)
Rauhut, H.: Best time localized trigonometric polynomials and wavelets. Adv. Comput. Math. 22(1), 1–20 (2005)
Rösler, M., Voit, M.: An uncertainty principle for ultraspherical expansions. J. Math. Anal. Appl. 209, 624–634 (1997)
Selig, K.K.: Uncertainty principles revisited. Electron. Trans. Numer. Anal. 14, 164–176 (2002)
Slepian, D.: Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25, 379–393 (1983)
Szegö, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1939)
Yang, C., He, Z., Yu, W.: Comparison of public peak detection algorithms for MALDI mass spectrometry data analysis. BMC Bioinform. 10, 4 (2009) Available at http://www.biomedcentral.com/1471-2105/10/4
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Communicated by Michael Frazier.
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Erb, W. Optimally Space Localized Polynomials with Applications in Signal Processing. J Fourier Anal Appl 18, 45–66 (2012). https://doi.org/10.1007/s00041-011-9184-3
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DOI: https://doi.org/10.1007/s00041-011-9184-3