Abstract
Every s×s matrix A yields a composition map acting on polynomials on R s. Specifically, for every polynomial p we define the mapping C A by the formula (C A p)(x):=p(Ax), x∈R s. For each nonnegative integer n, homogeneous polynomials of degree n form an invariant subspace for C A . We let A (n) be the matrix representation of C A relative to the monomial basis and call A (n) a binomial matrix. This paper studies the asymptotic behavior of A (n) as n→∞. The special case of 2×2 matrices A with the property that A 2=I corresponds to discrete Taylor series and motivated our original interest in binomial matrices.
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References
A.N. Akansu, R.A. Haddad and H. Caglar, The binomial QMF-wavelet transform for multiresolution signal decomposition, IEEE Trans. Signal Process. 41 (1993) 13–19.
C. Cabrelli, C. Heil and U.Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998) 5–52.
C. Cabrelli, C. Heil and U. Molter, Accuracy of several multidimensional refinable distributions, J. Fourier Anal. Appl., to appear.
A.S. Cavaretta and C.A. Micchelli, Pyramid patches provide potential polynomial paradigms, in: Mathematical Methods in Computer Aided Geometric Design, Vol. 2, eds. T. Lyche and L.L. Schumaker (Academic Press, New York, 1992) pp. 69–100.
W. Dahmen and C.A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52 (1983) 217–234.
R.A. Haddad, A class of orthogonal nonrecursive binomial filters, IEEE Trans. Audio Electroacoust. 19 (1971) 296–304.
R.A. Haddad and A.N. Akansu, A new orthogonal transform for signal coding, IEEE Trans. Acoust. Speech Signal Process. 36 (1988) 1404–1411.
X.C. Li and R. Wong, A uniform asymptotic expansion for Krawtchouk polynomials, J. Approx. Theory 106 (2000) 155–184.
F. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, New York, 1977).
C.A. Micchelli, N-width for holomorphic functions on the unit ball in n-space, in: Methods of Functional Analysis in Approximation Theory, eds. C.A. Micchelli, D.V. Pai and B.V. Limaye, Bombay, 1985 (Birkhäuser, Basel/Boston, MA, 1986) pp. 195–204.
C.A. Micchelli, Mathematical Aspects of Geometric Modeling, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 65 (SIAM, Philadelphia, PA, 1995).
G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society Colloqium Publications, Vol. 23 (Amer. Math. Soc., Providence, RI, 1975).
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Boyd, G., Micchelli, C.A., Strang, G. et al. Binomial Matrices. Advances in Computational Mathematics 14, 379–391 (2001). https://doi.org/10.1023/A:1012207124894
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DOI: https://doi.org/10.1023/A:1012207124894