## Rocky Mountain Journal of Mathematics

### Invariant sets for QMF functions

#### Abstract

A quadrature mirror filter (QMF) function can be considered as the transition function for a Markov process on the unit interval. The QMF functions that generate scaling functions for multiresolution analyses are then distinguished by properties of their invariant sets. By characterizing these sets, we answer in the affirmative a question raised by Gundy.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2559-2571.

Dates
First available in Project Euclid: 30 December 2018

https://projecteuclid.org/euclid.rmjm/1546138821

Digital Object Identifier
doi:10.1216/RMJ-2018-48-8-2559

Mathematical Reviews number (MathSciNet)
MR3894993

Zentralblatt MATH identifier
06999274

#### Citation

Jonsson, Adam. Invariant sets for QMF functions. Rocky Mountain J. Math. 48 (2018), no. 8, 2559--2571. doi:10.1216/RMJ-2018-48-8-2559. https://projecteuclid.org/euclid.rmjm/1546138821

#### References

• L.H.Y. Chen, A short note on the conditional Borel-Cantelli lemma, Ann. Probab. 6 (1978), 699–700.
• A. Cohen, Ondelettes, analyses multirésolutions et filtres miroirs en quadrature, Ann. Inst. Poincare 7 (1990), 439–459.
• J.-P. Conze and A. Raugi, Fonctions harmoniques pour un opérateur de transition et applications, Bull. Soc. Math. France 118 (1990), 273–310.
• I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909–996.
• V. Dobrić, R.F. Gundy and P. Hitczenko, Characterizations of orthonormal scale functions: a probabilistic approach, J. Geom. Anal. 10 (2000), 417–434.
• R.F. Gundy, Low-pass filters, martingales, and multiresolution analyses, Appl. Comp. Harmon. Anal. 9 (2000), 204–219.
• ––––, Probability, ergodic theory, and low-pass filters, in Topics in harmonic analysis and ergodic theory, Contemp. Math. 444 (2007), 53–87.
• ––––, Tilings, scaling functions, and a Markov process, Not. Amer. Math. Soc. 57 (2010), 1094–1104.
• R.F. Gundy and A.L. Jonsson, Scaling functions on $\mathbb{R}^2$ for dilations of determinant $\pm2$, Appl. Comp. Harmon. Anal. 29 (2010), 49–62.
• E. Hernández and G. Weiss, A first course on wavelets, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1996.
• A.L. Jonsson, On $g$-functions for countable state subshifts, Arch. Math. 109 (2017), 373–381.
• M. Keane, Strongly mixing $g$-measures, Invent. Math. 16 (1972), 309–324.
• W. Krieger, On $g$-functions for subshifts, in Dynamics & stochastics, IMS Lect. Notes Mono. 48 (2006), 306–316.
• W.M. Lawton, Tight frames of compactly supported affine wavelets, J. Math. Phys. 31 (1990), 1898–1901.
• D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995.
• M. Papadakis, H. Šikić and G. Weiss, The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts, J. Fourier Anal. Appl. 5 (1999), 495–521.