Abstract
A classical system of algebraic equations is treated as a finite power moment problem in C and investigated on this base. Being originated from the algebraic theory of binary forms, this system is closely related to an extraordinary number of different subjects in the classical and modern analysis. A survey of these relations is presented.
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References
N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd, 1965.
B.C. Berndt and S. Bhargava, “Ramanujan for lowbrows,” Amer. Math. Monthly 100 (1993), 614-656.
M. Buchmann and A. Pinkus, “On a recovery problem,” Ann. of Num. Math. 4 (1997), 129-142.
P. Delsarte, J.M. Goethals, and J.J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6 (1977), 363-388.
W.J. Ellison, “Waring's problem,” Amer. Math. Monthly 78 (1971), 10-36.
F.R. Gantmacher, The Theory of Matrices, v.2, Chelsea Publ. Co., 1960.
C.D. Godsil, Algebraic Combinatorics, Chapman & Hall, 1993.
U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, Univ. of Calif. Press, 1958.
F. Hausdorff, “Zur Hilbertschen Losung des Waringschen Problems,” Math. Ann. 67 (1909), 301-305.
D. Hilbert, “Beweis fur die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem),” Math. Ann. 67 (1909), 281-300.
I.S. Iohvidov, Hankel and Toeplitz Matrices and Forms, Nauka, 1974 (in Russian).
W.B. Jones and W.J. Thron, Continued Fractions, Addison-Wesley, 1980.
V. Krechmar, Problems in Algebra, Nauka, 1964 (in Russian).
J.P.S. Kung, “Canonical forms for binary forms of even degree. Invariant theory,” Lecture Notes in Math. 1278 (1987), 52-61, Springer Berlin.
J.P.S. Kung, “Canonical forms of binary forms: Variations on a theme of Sylvester. Invariant theory and tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl. 19, (1990) 46-58, Springer, New York.
J.P.S. Kung and G.-C. Rota, “The invariant theory of binary forms,” Bulletin of AMS 10 (1984), 27-85.
B. Ya. Levin, “Lectures on entire functions,” AMS Transl. of Math. Monographs 150 (1996).
Yu. I. Lyubich, “On the boundary spectrum of contractions in Minkowski spaces,” Sib. Math. J. 11 (1970), 271-279.
Yu. Lyubich and L.N. Vaserstein, “Isometric embeddings between classical Banach spaces, cubature formulas and spherical designs,” Geom. Dedicata 47 (1993), 327-362.
V.D. Milman, “A few observations on the connections between local theory and some other fields,” Springer Lecture Notes Math. 1317 (1988), 283-289.
S. Ramanujan, “Note on a set of simultaneous equations,” Journ. Indian Math. Soc. 4 (1912), 94-96.
B. Reznick, “Sums of even powers of real linear forms,” Memoirs of AMS 463 (1992).
B. Reznick, “Some constructions of spherical 5-designs,” Linear Algebra Appl. 226/228 (1995), 163-196.
B. Reznick, “Homogeneous polynomial solutions to constant coefficient PDE's,” Adv. Math. 117 (1996), 179-192.
J.J. Seidel, “Isometric embeddings and geometric designs,” Discrete Math. 136(1–3) (1994), 281-293.
A. Schinzel, “On a decomposition of polynomials in several variables, II,” Colloq. Math. 92 (2002), 67-79.
E. Schmidt, “Zum Hilbertschen Beweise des Waringschen Theorems,” Math. Ann. 74 (1913), 271-274.
E. Stridsberg, “Sur la démonstration de M.Hilbert du théoremè de Waring,” Math. Ann. 72 (1912), 145-152.
J.J. Sylvester, “On a remarkable discovery in the theory of canonical forms and of hyperdeterminants,” Phil. Magazine 2 (1851), 391-410.
G. Szegö, Orthogonal Polynomials, Coll. Publ. of AMS, 1959.
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Lyubich, Y.I. The Sylvester-Ramanujan System of Equations and The Complex Power Moment Problem. The Ramanujan Journal 8, 23–45 (2004). https://doi.org/10.1023/B:RAMA.0000027196.19661.b7
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DOI: https://doi.org/10.1023/B:RAMA.0000027196.19661.b7