ИСТИНА

Various methods of determining the regularity of wavelets and of subdivisions have been presented in the literature for the last two decades: the brut force method, the method of invariant cycles, the method applying the trace-class operators, etc. The matrix approach is one of the most effective among them. It provides a very refine analysis of local and global regularity by means of the joint spectral characteristics of transition matrices (constructed by coefficients of the equation). The joint spectral radius (Rota, Strang, 1960) is responsible for convergence and regularity in the spaces $C^k$, while the $p$-radius (Lau, Wang, 1995) is in the spaces $L_p\, , \, p \in [1, +\infty)$ and in the Sobolev spaces $W^k_p$. The Lyapunov exponent (Furstenberg, Kesten, 1960) indicates the local regularity almost everywhere, while the lower spectral radius (Gurvits, 1995; M.~Dogruel, U.~Ozguner, 1995) expresses the maximal local regularity (see~[6] for an overview). The matrix method originated with Daubechies and Lagarias~[2] and Collela and Heil~[1]. This is the only approach that can theoretically give a sharp value of the exponents of regularity. However, its main disadvantage is the complexity of computation of the joint spectral characteristics. This problem is proved to be computationally hard~(Blondel, Tsitsiklis, 2000). That is why the precise values of the H\"older exponent of Daubechies wavelets were computed only for small orders (for D2 and D3 in~[2]). Then Gripenberg~[3] applying special tricks obtained the precise values for D4, ..., D8. In all the cases, the minimal regularity attained at dyadic points. In~2013 we presented a general method~[4] of exact computation of the joint spectral radius that enabled us to obtain the exact values for D9, ... , D20 and to disprove the conjecture that the minimal regularity always attained at dyadic points. This property fails for D10, D15, and D16. The new approach can be extended to evaluating the Lyapunov exponent and to computing the regularity of non-stationary subdivisions and wavelets. This is done by applying modern tools of convex analysis and of convex optimization. Some of those results are based on recent joint works with N.Guglielmi and R.Jungers. Let us also mention an alternative method form~[5], which allowed the authors to make a regularity analysis of the four-point subdivision scheme. 1. D. Collela, C. Heil, Characterization of scaling functions: I. Continuous solutions, SIAM J. Matrix Anal. Appl., 15 (1994), 496–518. 2. I. Daubechies, J.C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), 227-263. 3. G. Gripenberg, Computing the joint spectral radius, Linear Alg. Appl., 234 (1996), 43-60. 4. N. Guglielmi, V.Yu. Protasov, Exact computation of joint spectral characteristics of matrices, Found. Comput. Math., 13(1) (2013), 37-97. 4. C. M\"oller, U. Reif, A tree-based approach to joint spectral radius determination, Linear Alg. Appl., 563 (2014), 154-170. 5. I. Ya. Novikov, V. Yu. Protasov, M. A. Skopina, Wavelet Theory, Amer. Math. Soc, Providence, R.I., 2011.