Clark measures and a theorem of Ritt
DOI:
https://doi.org/10.7146/math.scand.a-104444Abstract
We determine when a finite Blaschke product $B$ can be written, in a non-trivial way, as a composition of two finite Blaschke products (Ritt's problem) in terms of the Clark measure for $B$. Our tools involve the numerical range of compressed shift operators and the geometry of certain polygons circumscribing the numerical range of the relevant operator. As a consequence of our results, we can determine, in terms of Clark measures, when two finite Blaschke products commute.
References
Basallote, M., Contreras, M. D., and Hernández-Mancera, C., Commuting finite Blaschke products with no fixed points in the unit disk, J. Math. Anal. Appl. 359 (2009), no. 2, 547–555. https://doi.org/10.1016/j.jmaa.2009.06.010
Chalendar, I. and Mortini, R., When do finite Blaschke products commute?, Bull. Austral. Math. Soc. 64 (2001), no. 2, 189–200. https://doi.org/10.1017/S0004972700039861
Chalendar, I., Gorkin, P., and Partington, J. R., Numerical ranges of restricted shifts and unitary dilations, Oper. Matrices 3 (2009), no. 2, 271–281. https://doi.org/10.7153/oam-03-17
Cima, J. A., Matheson, A. L., and Ross, W. T., The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. https://doi.org/10.1090/surv/125
Courtney, D. and Sarason, D., A mini-max problem for self-adjoint Toeplitz matrices, Math. Scand. 110 (2012), no. 1, 82–98. https://doi.org/10.7146/math.scand.a-15198
Cowen, C. C., Finite Blaschke products as compositions of other finite Blaschke products, preprint arXiv:1207.4010 [math.CV], 2012.
Daepp, U., Gorkin, P., and Mortini, R., Ellipses and finite Blaschke products, Amer. Math. Monthly 109 (2002), no. 9, 785–795. https://doi.org/10.2307/3072367
Daepp, U., Gorkin, P., Shaffer, A., Sokolowsky, B., and Voss, K., Decomposing finite Blaschke products, J. Math. Anal. Appl. 426 (2015), no. 2, 1201–1216. https://doi.org/10.1016/j.jmaa.2015.01.039
Duren, P. L., Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.
Gau, H.-L. and Wu, P. Y., Numerical range of $S(phi )$, Linear and Multilinear Algebra 45 (1998), no. 1, 49–73. https://doi.org/10.1080/03081089808818577
Gau, H.-L. and Wu, P. Y., Numerical range circumscribed by two polygons, Linear Algebra Appl. 382 (2004), 155–170. https://doi.org/10.1016/j.laa.2003.12.003
Gorkin, P. and Rhoades, R. C., Boundary interpolation by finite Blaschke products, Constr. Approx. 27 (2008), no. 1, 75–98. https://doi.org/10.1007/s00365-006-0646-3
Gustafson, K. E. and Rao, D. K. M., Numerical range: The field of values of linear operators and matrices, Universitext, Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4613-8498-4
Halmos, P. R., Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 125–134.
Jones, W. B. and Ruscheweyh, S., Blaschke product interpolation and its application to the design of digital filters, Constr. Approx. 3 (1987), no. 4, 405–409. https://doi.org/10.1007/BF01890578
Marden, M., Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966.
Ng, T. W. and Tsang, C. Y., Polynomials versus finite Blaschke products, in “Blaschke products and their applications”, Fields Inst. Commun., vol. 65, Springer, New York, 2013, pp. 249--273. https://doi.org/10.1007/978-1-4614-5341-3_14
Ng, T. W. and Tsang, C. Y., Chebyshev-Blaschke products: solutions to certain approximation problems and differential equations, J. Comput. Appl. Math. 277 (2015), 106–114. https://doi.org/10.1016/j.cam.2014.08.028
Ritt, J. F., Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), no. 1, 51–66. https://doi.org/10.2307/1988911
Semmler, G. and Wegert, E., Boundary interpolation with Blaschke products of minimal degree, Comput. Methods Funct. Theory 6 (2006), no. 2, 493–511. https://doi.org/10.1007/BF03321626
