On the Decay of Infinite Products of Trigonometric Polynomials

V. Yu. Protasov

Matematičeskie zametki, Tome 72 (2002) no. 6, pp. 892-908.

Résumé

We consider infinite products of the form $f(\xi )=\prod _{k=1}^\infty m_k(2^{-k}\xi )$, 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 $f(\xi )$ can decrease at infinity not faster than $O(\xi ^{-n})$ 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.

Bibliographie

[1] Derfel G. A., “A probabilistic method for a class of functional-differential equations”, Ukrain. Math. J., 41:10 (1989), 1137–1141 | DOI | MR | Zbl

[2] Derfel G. A., Dyn N., Levin D., “Generalized refinement equations and subdivision processes”, J. Approximation Theory, 80 (1995), 272–297 | DOI | MR | Zbl

[3] Protasov V., “Refinement equations with nonnegative coefficients”, J. Fourier Anal. Appl., 6:6 (2000), 55–78 | DOI | MR | Zbl

[4] Cavaretta D., Dahmen W., Micchelli C., “Stationary subdivision”, Mem. Amer. Math. Soc., 93, 1991, 1–186 | MR

[5] Dyn N., Gregory J. A., Levin D., “Analysis of linear binary subdivision schemes for curve design”, Constructive Approximation, 7 (1991), 127–147 | DOI | MR | Zbl

[6] Erdös P., “On the smoothness properties of Bernoulli convolutions”, Amer. J. Math., 62 (1940), 180–186 | DOI | MR | Zbl

[7] Peres Y., Solomyak B., “Absolute continuity of Bernoulli convolution, a simple proof”, Math. Res. Letters, 3:2 (1996), 231–239 | MR | Zbl

[8] Reznick B., “Some binary partition functions”, Analytic Number Theory (Allerton Park, IL, 1989), 451–477 | Zbl

[9] Chui C. K., An Introduction to Wavelets, Acad. Press Inc., 1992

[10] Dyn N., Levin D., “Interpolatory subdivision schemes for the generation of curves and surfaces”, Multivariate Approximation and Interpolation (Duisburg, 1989), Birkhäuser, Basel, 1990, 91–106 | MR

[11] Ron A., “Smooth refinable functions provide good approximation”, SIAM J. Math. Anal. Appl., 28 (1997), 731–748 | DOI | MR | Zbl

[12] Daubechies I., “Orthonormal bases of wavelets with compact support”, Comm. Pure Appl. Math., 41 (1988), 909–996 | DOI | MR | Zbl

[13] Cohen A., Daubechies I., “A new technique to estimate the regularity of refinable functions”, Rev. Mat. Iberoamericana, 12:2 (1996), 527–591 | MR | Zbl

[14] Ron A., Shen Z., The Sobolev Regularity of Refinable Functions, Preprint, 1997

[15] Daubechies I., Lagarias J., “Two-scale difference equations. I: Global regularity of solutions”, SIAM J. Math. Anal. Appl., 22 (1991), 1388–1410 | DOI | MR | Zbl

[16] Daubechies I., Lagarias J., “Two-scale difference equations. II: Local regularity, infinite products of matrices and fractals”, SIAM J. Math. Anal. Appl., 23 (1992), 1031–1079 | DOI | MR | Zbl

[17] Schumaker L. L., Spline Functions: Basic Theory, John Wiley, New York, 1981 | Zbl

[18] Berkolaiko M. Z., Novikov I. Ya., “O beskonechno gladkikh pochti-vspleskakh s kompaktnym nositelem”, Dokl. RAN, 326:6 (1992), 615–618 | MR

[19] de Boor C., DeVore R., Ron A., “On the construction of multivariate pre-wavelets”, Constructive Approximation, 2:3 (1993), 123–166 | DOI | Zbl

[20] Novikov I. Ya., Bazisy vspleskov funktsionalnykh prostranstv, Diss. ... dokt. fiz.-matem. nauk, Voronezhskii gosudarstvennyi universitet, Voronezh, 2000 | Zbl

[21] Cohen A., Dyn N., “Nonstationary subdivision schemes and multiresolution analysis”, SIAM J. Math. Anal. Appl., 27 (1996), 1745–1769 | DOI | MR | Zbl