Abstract
In this paper we provide information about the asymptotic properties of polynomial filters which approximate the ideal filter. In particular, we study this problem in the special case of polynomial halfband filters. Specifically we estimate the error between a polynomial filter and an ideal filter and show that the error decays exponentially fast. For the special case of polynomial halfband filters, our n-th root asymptotic error estimates are sharp.
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Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)
Lorentz, G.G.: Approximation of Functions, 2nd edn. Chelsea, New York (1986)
Micchelli, C.A.: Optimal estimation of linear operators from inaccurate data: a second look. Numer. Algorithms 5, 375–390 (1993)
Natanson, I.P.: Constructive Function Theory. Approximation in Mean, vol. II. Frederick Ungar, New York (1965)
Saff, E.B., Varga, R.S.: On incomplete polynomials. In: Collatz, L., Meinardus, G., Werner, H. (eds.) Numerische Methoden der Approximationstheorie. ISNM, vol. 42, pp. 281–298. Birkhäuser, Basel (1978)
Schoenberg, I.J.: The perfect B-spline and a time-optimal control problem. Isr. J. Math. 10, 261–274 (1971)
Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press (1966)
Strang, G., Ngyen, T.: Wavelets and Filter Banks, revised edn. Wellesley-Cambridge (1997)
Szegö, G.: Orthogonal Polynomials, 4th edn. Number 23 in Colloquium Publications. American Mathematical Society (1978)
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Communicated by Yuesheng Xu.
C.A. Micchelli was partially supported by the NSF grant DMS 0712827 and AFOSR FA9550-09-0511.
J. Wang was partially supported by the NSF grant DMS 0712925.
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Micchelli, C.A., Wang, J. & Wang, Y. On an asymptotic analysis of polynomial approximation to halfband filters. Adv Comput Math 38, 601–622 (2013). https://doi.org/10.1007/s10444-011-9251-y
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DOI: https://doi.org/10.1007/s10444-011-9251-y