Su, Juan; Dong, Xinhan On dimension extension of a class of iterative equations. (English) Zbl 1275.39009 Abstr. Appl. Anal. 2013, Article ID 180184, 7 p. (2013). Summary: This investigation aims at studying some special properties (convergence, polynomial preservation order, and orthogonal symmetry) of a class of \(r\)-dimension iterative equations, whose state variables are described by the following nonlinear iterative equation: \(\phi^n(x) = T(\phi^{n - 1}(x)) := \sum^m_{j=0} H_j \phi^{n - 1}(2x - k)\). The obtained results in this paper are complementary to some published results. As an application, we construct an orthogonal symmetric multiwavelet with additional vanishing moments. Two examples are also arranged to demonstrate the correctness and effectiveness of the main results. MSC: 39B12 Iteration theory, iterative and composite equations 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems Keywords:convergence; polynomial preservation order; orthogonal symmetry; iterative equation; orthogonal symmetric multiwavelet × Cite Format Result Cite Review PDF Full Text: DOI OA License References: [1] Wen, F.; Yang, X., Skewness of return distribution and coefficient of risk premium, Journal of Systems Science & Complexity, 22, 3, 360-371 (2009) · doi:10.1007/s11424-009-9170-x [2] Wen, F.; Li, Z.; Xie, C.; David, S., Study on the fractal and chaotic features of the Shanghai composite index, Fractals-Complex Geometry Patterns and Scaling in Nature and Society, 20, 133-140 (2012) · Zbl 1250.91084 [3] Han, B.; Mo, Q., Multiwavelet frames from refinable function vectors, Advances in Computational Mathematics, 18, 2-4, 211-245 (2003) · Zbl 1059.42030 · doi:10.1023/A:1021360312348 [4] Chui, C. K.; Lian, J., Construction of orthonormal multi-wavelets with additional vanishing moments, Advances in Computational Mathematics, 24, 1-4, 239-262 (2006) · Zbl 1115.65133 · doi:10.1007/s10444-004-7610-7 [5] Plonka, G., Necessary and Sufficient Conditions for Orthonormality of Scaling Vectors (1997), University Rostock · Zbl 0903.42017 [6] Lawton, W. M., Necessary and sufficient conditions for constructing orthonormal wavelet bases, Journal of Mathematical Physics, 32, 1, 57-61 (1991) · Zbl 0757.46012 · doi:10.1063/1.529093 [7] Jiang, Q., On the design of multifilter banks and orthonromal multiwavelet bases, IEEE Transactions on Signal Processing, 46, 3292-3303 (1998) [8] Plonka, G., Approximation order provided by refinable function vectors, Constructive Approximation, 13, 2, 221-244 (1997) · Zbl 0870.41015 · doi:10.1007/s003659900039 [9] Lian, J.-A., On the order of polynomial reproduction for multi-scaling functions, Applied and Computational Harmonic Analysis, 3, 4, 358-365 (1996) · Zbl 0858.42026 · doi:10.1006/acha.1996.0027 [10] Jiang, Q., Symmetric paraunitary matrix extension and parametrization of symmetric orthogonal multifilter banks, SIAM Journal on Matrix Analysis and Applications, 23, 1, 167-186 (2001) · Zbl 0992.42022 · doi:10.1137/S0895479800372924 [11] Shen, Z., Refinable function vectors, SIAM Journal on Mathematical Analysis, 29, 1, 235-250 (1998) · Zbl 0913.42028 · doi:10.1137/S0036141096302688 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.