Abstract
We consider infinite products of the form f(ξ=∏k=1 ∞ m k(2-kξ), where {m k} is an arbitrary sequence of trigonometric polynomials of degree at most n with uniformly bounded norms such that m k(0)= 1 for all k. We show that f(ξ) can decrease at infinity not faster than O(ξ-n) and present conditions under which this maximal decay is attained. This result can be applied to the theory of nonstationary wavelets and nonstationary subdivision schemes. In particular, it restricts the smoothness of nonstationary wavelets by the length of their support. This also generalizes well-known similar results obtained for stable sequences of polynomials (when all m k coincide). By means of several examples, we show that by weakening the boundedness conditions one can achieve exponential decay.
Similar content being viewed by others
REFERENCES
G. A. Derfel, "A probabilistic method for a class of functional-differential equations," Ukrain. Math. J., 41 (1989), no.10, 1137-1141.
G. A. Derfel, N. Dyn, and D. Levin, "Generalized refinement equations and subdivision processes," J. Approximation Theory, 80 (1995), 272-297.
V. Yu. Protasov, "Refinement equations with nonnegative coefficients," J. Fourier Anal. Appl., 6 (2000), no. 6, 55-78.
D. Cavaretta, W. Dahmen, and C. Micchelli, "Stationary subdivision," Mem. Amer. Math. Soc., 93 (1991), 1-186.
N. Dyn, J. A. Gregory, and D. Levin, "Analysis of linear binary subdivision schemes for curve design," Constructive Approximation, 7 (1991), 127-147.
P. Erdös, "On the smoothness properties of Bernoulli convolutions," Amer. J. Math., 62 (1940), 180-186.
Y. Peres and B. Solomyak, "Absolute continuity of Bernoulli convolution, a simple proof," Math. Res. Letters, 3 (1996), no. 2, 231-239.
B. Reznick, "Some binary partition functions," in: Analytic Number Theory (Allerton Park II, 1989 ), pp. 451-477.
C. K. Chui, An Introduction to Wavelets, Acad. Press Inc., 1992.
N. Dyn and D. Levin, "Interpolatory subdivision schemes for the generation of curves and surfaces," in: Multivariate Approximation and Interpolation (Duisburg, 1989), Birkhäuser, Basel, 1990, pp. 91-106.
A. Ron, "Smooth refinable functions provide good approximation," SIAM J. Math. Anal. Appl., 28 (1997), 731-748.
I. Daubechies, "Orthonormal bases of wavelets with compact support," Comm. Pure Appl. Math., 41 (1988), 909-996.
A. Cohen and I. Daubechies, "A new technique to estimate the regularity of refinable functions," Rev. Mat. Iberoamericana, 12 (1996), no. 2, 527-591.
A. Ron and Z. Shen, The Sobolev Regularity of Refinable Functions, Preprint, 1997.
I. Daubechies and J. Lagarias, "Two-scale difference equations. I. Global regularity of solutions," SIAM J. Math. Anal. Appl., 22 (1991), 1388-1410.
I. Daubechies and J. Lagarias, "Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals," SIAM J. Math. Anal. Appl., 23 (1992), 1031-1079.
L. L. Schumaker, Spline Functions: Basic Theory, John Wiley, New York, 1981.
M. Z. Berkolaiko and I. Ya. Novikov, "On infinitely differentiable pre-wavelets with compact support," Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 326 (1992), no. 6, 615-618.
C. de Boor, R. DeVore, and A. Ron, "On the construction of multivariate pre-wavelets," Constructive Approximation, 2 (1993), no. 3, 123-166.
I. Ya. Novikov, Wavelet Bases in Function Spaces [in Russian], Doctorate thesis in the physico-mathematical sciences, Voronezh State University, Voronezh, 2000.
A. Cohen and N. Dyn, "Nonstationary subdivision schemes and multiresolution analysis," SIAM J. Math. Anal. Appl., 27 (1996), 1745-1769.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Protasov, V.Y. On the Decay of Infinite Products of Trigonometric Polynomials. Mathematical Notes 72, 819–832 (2002). https://doi.org/10.1023/A:1021442030017
Issue Date:
DOI: https://doi.org/10.1023/A:1021442030017