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Primitivity of Matrix Families: Separating Cases by Criterion

This study focuses on the problem of distributing power random series and separating two cases using a criterion. It explores the concept of primitive matrix families and their primitivity criterion. The distribution of power random series is analyzed using blocking sets and the absolute continuity condition. Various examples and applications are discussed.

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Primitivity of Matrix Families: Separating Cases by Criterion

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  1. Vladimir Protasov (Moscow State University) Primitivity of matrix families and the problem of distribution of power random series

  2. Random power series x How to separate these two cases by a criterion ?

  3. A continuous monotone functionis absolutely continuous iff it has a summable derivative 0 1 1 0 A continuous monotone function is purely singular iff Any continuous monotone function can be decomposed in a unique way as A ``typical’’ monotone function is of mixed type, i.e.

  4. The ``simplest case’’. Bernoulli convolutions. -1 1 0 -1 0 1 ? -1 0 1

  5. The opposite case. The ``dual’’ problem. In 1995 G.Derfel, N.Dyn, A.Levin formulated a ``dual’’ problem: DDL problem Erdos problem Nothing is changed if we take t = 1/n, n >1 is an integer. Derfel, Dyn and Levin applied this problem to algorithms of extrapolation of functions by its values at integer points.

  6. The density of distribution satisfies the refinement equation The distribution is absolutely continuous if and only if How to check by the coefficients whether Theorem 1 (G.Derfel, N.Dyn, A.Levin, 1995, Y.Wang, 1995) If P {h is even} = P {h is odd} = ½, then is absolutely continuous. This condition is not necessary ! Example 6. • = 0 , 1 , 2 , 3 P = 1/6 , 1/3 , 1/6 , 1/3 P {h is even} = 1/3, P {h is odd} = 2/3 However,exists and, moreover, continuous.

  7. Refinement equations in the construction of wavelets To construct a system of compactly supported wavelets one needs to solve a refinement equation is a sequence of complex numbers sutisfyingsome constraints. This is a usual difference equation, but with the double contraction of the argument

  8. What is known about refinement equations ? then If it possesses a compactly supported solution such that , then there exists a compactly supported solution, which is unique, Conversely, if up to multiplication by a constant and has its support on [0, N]. But only in the sense of distributions ! 0 N The refinable functions are never infinitely smooth

  9. Special examples of refinement equations Example 1. Trivial: 0 1 Example 2. 0 2 Example 3. 0 3 The solution is unstable! Asmallperturbationof the coefficients may lead to the loss of absolute continuity: Example 4. The same with

  10. Cavaretta, Dahmen and Micchelli(1991) Classified all refinable splines with integral nodes. Lawton, Lee andShen (1995) Classified all refinable splines. For any N there are finitely many refinable splines of order N Berg and Plonka (2000), Hirn (2008) Protasov (2005) Classified all piecewise-smooth refinable functions. Thus, all piecewise-smooth refinable functions are splines. All of them are spanned by integral translates ofthe B-spline.

  11. A “ typical ” refinable functionand wavelet function Example 5 (very smooth) (very irregular) 0 3 (breaks at all dyadic points) is the exponent of regularity (the Holder exponent) is not differentiable Nevertheless, it is differentiable almost everywhere the local exponent of regularity at the point x Fractal nature of refinable functions. Varying local regularity

  12. How to compute the regularity of refinable functions ? The joint spectral radius of linear operators How to check if I.Daubechies, D.Lagarias, 1991 A.Cavaretta, W.Dahmen, C.Micchelli, 1991 C.Heil, D.Strang, 1994 R.Q.Jia, 1995, K.S.Lau, J.Wang, 1995 Y.Wang, 1996 Example.

  13. The concept of primitive families. Definition. A family of matrices is called primitive if there exists at least one positive product.

  14. Perron-Frobenius theorem (1912) A matrix A is not primitive if it is either reducible or one of the following equivalent conditions is satisfied:

  15. A criterion of primitivity (conjectured in 2010)

  16. Now we can solve the problem by a criterion of absolute continuity The criterion is formulated in terms of roots of the characteristic function on a binary tree. Binary tree T: 1/2 a 3/4 1/4 1/8 7/8 5/8 3/8 1/16 5/16 15/16 11/16 9/16 13/16 3/16 7/16 a/2 0.5 + a/2

  17. 1/2 3/4 1/4 1/8 7/8 5/8 3/8 1/16 5/16 15/16 11/16 9/16 13/16 3/16 7/16 Definition 1. Examples. The sets A = {1/2},A = {1/4, 3/4}are blocking A = {1/8 , 5/16 , 13/16 , 3/4 }is not (it is non-symmetric). Theorem.. The distributionis absolutely continuous if and only if there is a blocking set A that consists of roots of the polynomial m(z), i.e., m(A) = 0. This condition can be checked within finite time.

  18. Example 7 (the case of Theorem 1 of DDL). P {h is even} = P {h is odd} = ½ iff A = {root} Example 8 (the case of Example 6). • = 0 , 1 , 2 , 3 P = 1/6 , 1/3 , 1/6 , 1/3 1/2 A = {1/2} 3/4 A = {1/4 , 3/4} 1/4 1/8 7/8 5/8 3/8 The criterion is sharp, each case can be realized.

  19. What is the set of all sets of probabilities for which the distribution is absolutely continuous ? A = {1/2} m(1/2) =0 m(1/4) =0 m(3/4) =0 A = {1/4, 3/4} Thank you !

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