Computing the Hessenberg matrix associated with a self-similar measure

We introduce in this paper a method to calculate the Hessenberg matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi matrix associated with a sum of measures from the Jacobi matrices of each of the measures. We apply this method to approximate the Hessenberg matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures. Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg matrix.

[1]  G. Mantica.  On computing Jacobi matrices associated with recurrent and Möbius iterated function systems , 2000 .

[2]  On the `Mandelbrot set' for a pair of linear maps and complex Bernoulli convolutions , 2002, math/0209229.

[3]  W. Gautschi.  On Generating Orthogonal Polynomials , 1982 .

[4]  David I. Schneider,et al.  Modern mathematics and its applications , 1980 .

[5]  A. Wintner,et al.  Distribution functions and the Riemann zeta function , 1935 .

[6]  P. Jorgensen,et al.  Iterated function systems, moments, and transformations of infinite matrices , 2008 .

[7]  M. A. Sastre,et al.  A fixed point theorem for moment matrices of self-similar measures , 2007 .

[8]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[9]  A. Atzmon.  A moment problem for positive measures on the unit disc. , 1975 .

[10]  S. Yau.  Mathematics and its applications , 2002 .

[11]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[12]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[13]  G. Mantica.  A stable stieltjes technique for computing orthogonal polynomials and jacobi matrices associated with a class of singular measures , 1996 .

[14]  P. Mattila.  Geometry of Sets and Measures in Euclidean Spaces: Introduction , 1995 .

[15]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[16]  Carmen Escribano,et al.  Moments of infinite convolutions of symmetric Bernoulli distributions , 2003 .

[17]  Gene H. Golub,et al.  Jacobi matrices for sums of weight functions , 1992 .

[18]  On the moment problem in the bounded case , 1993 .

[19]  S. Krantz.  Fractal geometry , 1989 .

[20]  J. Conway.  The theory of subnormal operators , 1991 .