# On a vector potential-theory equilibrium problem with the Angelesco matrix

- 17 Downloads

## Abstract

Vector logarithmic-potential equilibrium problems with the Angelesco interaction matrix are considered. Solutions to two-dimensional problems in the class of measures and in the class of charges are studied. It is proved that in the case of two arbitrary real intervals, a solution to the problem in the class of charges exists and is unique. The Cauchy transforms of the components of the equilibrium charge are algebraic functions whose degree can take values 2, 3, 4, and 6 depending on the arrangement of the intervals. A constructive method for finding the vector equilibrium charge in an explicit form is presented, which is based on the uniformization of an algebraic curve. An explicit form of the vector equilibrium measure is found under some constraints on the arrangement of the intervals.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Angelesco, “Sur deux extensions des fractions continues algébriques,” C. R. Acad. Sci. Paris
**168**, 262–265 (1919).MATHGoogle Scholar - 2.A. I. Aptekarev, “Asymptotics of simultaneously orthogonal polynomials in the Angelesco case,” Mat. Sb.
**136**(1), 56–84 (1988) [Math. USSR, Sb.**64**(1), 57–84 (1989)].MATHGoogle Scholar - 3.A. I. Aptekarev, “Asymptotics of Hermite–Padé approximants for two functions with branch points,” Dokl. Akad. Nauk
**422**(4), 443–445 (2008) [Dokl. Math.**78**(2), 717–719 (2008)].MATHGoogle Scholar - 4.A. I. Aptekarev, “The Mhaskar–Saff variational principle and location of the shocks of certain hyperbolic equations,” in
*Modern Trends in Constructive Function Theory*(Am. Math. Soc., Providence, RI, 2016), Contemp. Math. 661, pp. 167–186.CrossRefGoogle Scholar - 5.A. I. Aptekarev and V. A. Kalyagin, “Asymptotic behavior of an nth degree root of polynomials of simultaneous orthogonality, and algebraic functions,” Preprint No. 60 (Keldysh Inst. Appl. Math., Moscow, 1986).Google Scholar
- 6.A. I. Aptekarev, V. A. Kalyagin, V. G. Lysov, and D. N. Toulyakov, “Equilibrium of vector potentials and uniformization of the algebraic curves of genus 0,” J. Comput. Appl. Math.
**233**(3), 602–616 (2009).MathSciNetCrossRefMATHGoogle Scholar - 7.A. I. Aptekarev and V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite–Padé approximants,” Mat. Sb.
**201**(2), 29–78 (2010) [Sb. Math.**201**, 183–234 (2010)].MathSciNetCrossRefMATHGoogle Scholar - 8.A. I. Aptekarev, V. G. Lysov, and D. N. Tulyakov, “Three-sheeted Riemann surfaces of genus 0 with fixed projections of the branch points,” Preprint No. 13 (Keldysh Inst. Appl. Math., Moscow, 2007).Google Scholar
- 9.A. I. Aptekarev, V. G. Lysov, and D. N. Tulyakov, “Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials,” Mat. Sb.
**202**(2), 3–56 (2011) [Sb. Math.**202**, 155–206 (2011)].MathSciNetCrossRefMATHGoogle Scholar - 10.A. I. Aptekarev and D. N. Tulyakov, “Nuttall’s Abelian integral on the Riemann surface of the cube root of a polynomial of degree 3,” Izv. Ross. Akad. Nauk, Ser. Mat.
**80**(6), 5–42 (2016) [Izv. Math.**80**, 997–1034 (2016)].MathSciNetCrossRefMATHGoogle Scholar - 11.A. I. Aptekarev, D. N. Toulyakov, and W. Van Assche, “Hyperelliptic uniformization of algebraic curves of the third order,” J. Comput. Appl. Math.
**284**, 38–49 (2015).MathSciNetCrossRefMATHGoogle Scholar - 12.A.I Aptekarev, W. Van Assche, and M. L. Yattselev, “Hermite–Padé approximants for a pair of Cauchy transforms with overlapping symmetric supports,” Commun. Pure Appl. Math.
**70**(3), 444–510 (2017).CrossRefMATHGoogle Scholar - 13.B. Beckermann, V. Kalyagin, A. C. Matos, and F. Wielonsky, “Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses,” Constr. Approx.
**37**(1), 101–134 (2013).MathSciNetCrossRefMATHGoogle Scholar - 14.V. I. Buslaev and S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padé polynomials,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk
**290**, 272–279 (2015) [Proc. Steklov Inst. Math.**290**, 256–263 (2015)].MATHGoogle Scholar - 15.A. A. Gonchar and E. A. Rakhmanov, “On convergence of simultaneous Padé approximants for systems of functions of Markov type,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR
**157**, 31–48 (1981) [Proc. Steklov Inst. Math.**157**, 31–50 (1983)].MATHGoogle Scholar - 16.A. A. Gonchar and E. A. Rakhmanov, “On the equilibrium problem for vector potentials,” Usp. Mat. Nauk
**40**(4), 155–156 (1985) [Russ. Math. Surv.**40**(4), 183–184 (1985)].MathSciNetMATHGoogle Scholar - 17.A. A. Gonchar, E. A. Rakhmanov, and V. N. Sorokin, “Hermite–Padé approximants for systems of Markov-type functions,” Mat. Sb.
**188**(5), 33–58 (1997) [Sb. Math.**188**, 671–696 (1997)].MathSciNetCrossRefMATHGoogle Scholar - 18.V. A. Kalyagin, “On a class of polynomials defined by two orthogonality relations,” Mat. Sb.
**110**(4), 609–627 (1979) [Math. USSR, Sb.**38**(4), 563–580 (1981)].MathSciNetGoogle Scholar - 19.A. V. Komlov, N. G. Kruzhilin, R. V. Palvelev, and S. P. Suetin, “Convergence of Shafer quadratic approximants,” Usp. Mat. Nauk
**71**(2), 205–206 (2016) [Russ. Math. Surv.**71**, 373–375 (2016)].CrossRefMATHGoogle Scholar - 20.A. V. Komlov and S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials,” Usp. Mat. Nauk
**70**(6), 211–212 (2015) [Russ. Math. Surv.**70**, 1179–1181 (2015)].CrossRefMATHGoogle Scholar - 21.M. A. Lapik, “Families of vector measures which are equilibrium measures in an external field,” Mat. Sb.
**206**(2), 41–56 (2015) [Sb. Math.**206**, 211–224 (2015)].MathSciNetCrossRefMATHGoogle Scholar - 22.M. A. Lapik, “The extremal functional for vector extremal logarithmic potential problems with external field and Angelesco matrix of interaction,” Preprint No. 83 (Keldysh Inst. Appl. Math., Moscow, 2015).Google Scholar
- 23.E. M. Nikishin, “On simultaneous Padé approximants,” Mat. Sb.
**113**(4), 499–519 (1980) [Math. USSR, Sb.**41**(4), 409–425 (1982)].MathSciNetMATHGoogle Scholar - 24.E. M. Nikishin and V. N. Sorokin,
*Rational Approximations and Orthogonality*(Nauka, Moscow, 1988; Am. Math. Soc., Providence, RI, 1991).MATHGoogle Scholar - 25.J. Nuttall, “Asymptotics of diagonal Hermite–Padé polynomials,” J. Approx. Theory
**42**(4), 299–386 (1984).MathSciNetCrossRefMATHGoogle Scholar - 26.E. A. Rakhmanov, “The asymptotics of Hermite–Padé polynomials for two Markov-type functions,” Mat. Sb.
**202**(1), 133–140 (2011) [Sb. Math.**202**, 127–134 (2011)].MathSciNetCrossRefMATHGoogle Scholar - 27.E. A. Rakhmanov, “The Gonchar–Stahl
*ρ*^{2}-theorem and associated directions in the theory of rational approximations of analytic functions,” Mat. Sb.**207**(9), 57–90 (2016) [Sb. Math.**207**, 1236–1266 (2016)].MathSciNetCrossRefGoogle Scholar - 28.V. N. Sorokin, “On multiple orthogonal polynomials for discrete Meixner measures,” Mat. Sb.
**201**(10), 137–160 (2010) [Sb. Math.**201**, 1539–1561 (2010)].CrossRefMATHGoogle Scholar - 29.V. N. Sorokin, “On multiple orthogonal polynomials for three Meixner measures,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk
**298**, 315–337 (2017) [Proc. Steklov Inst. Math.**298**, 294–316 (2017)].MathSciNetGoogle Scholar - 30.S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation,” Usp. Mat. Nauk
**70**(5), 121–174 (2015) [Russ. Math. Surv.**70**, 901–951 (2015)].MathSciNetCrossRefMATHGoogle Scholar - 31.M. L. Yattselev, “Strong asymptotics of Hermite–Padé approximants for Angelesco systems,” Can. J. Math.
**68**(5), 1159–1201 (2016).MathSciNetCrossRefMATHGoogle Scholar