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Moment problems and polynomial approximation. (English) Zbl 0877.44003

The author in this very lucid and elegant article gives a brief survey of the moment problems and polynomial approximation. He describes the moment problem starting from Stieltjes till present times. The article is concluded with an example which is very illustrative of various conditions for determinacy of Carleman’s condition.

MSC:

44A60 Moment problems
44-03 History of integral transforms
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References:

[1] Akhiezer, N.I. . - The classical moment problem, Oliver and Boyd, Edinburgh1965 . · Zbl 0173.41001
[2] Al-Salam, W.A. and Carlitz, L. . - Some orthogonal q-polynomials . Math. Nachr., 30 ( 1965), pp. 47-61. · Zbl 0135.27802
[3] Baillaud, B. and Bourget, H. . - Correspondance d’Hermite et de Stieltjes I, II, Gauthier-Villars, Paris1905.
[4] Berg, C. . - Critical exponents for measures. in: Open problems, ed. W. Van Assche , J. Comput. Appl. Math., 48 (1993 ), p. 227.
[5] Berg, C. . - L2-approximation with respect to rotation invariant measures, Suppl. Rend. Circ. Mat. Palermo. Serie II, 33 (1993). pp. 211-217. · Zbl 0840.41004
[6] Berg, C. . - Indeterminate moment problems and the theory of entire functions, J. Comput. Appl. Math., 65 (1995), pp. 27-55. · Zbl 0853.44005
[7] Berg, C. . - Recent results about moment problems, in: Probability measures on groups and related structures XI, World Scientific , Singapore1995. · Zbl 0919.60005
[8] Berg, C. and Christensen, J.P.R. .- Density questions in the classical theory of moments, Ann. Inst. Fourier, 31, n° 3 (1981), pp. 99-114. · Zbl 0437.42007
[9] Berg, C. and Christensen, J.P.R. .- Exposants critiques dans le problème des moments, C.R. Acad. Sci. Paris Série I, 296 ( 1983), pp. 661-663. · Zbl 0531.28008
[10] Berg, C. and Duran, A.J. .- The index of determinacy for measures and the l2-norm of orthonormal polynomials, Trans. Amer. Math. Soc., 347 (1995), pp. 2795-2811. · Zbl 0863.42019
[11] Berg, C. and Duran, A.J. .- When does a discrete differential perturbation of a sequence of orthonormal polynomials belong to l2? J. Funct. Anal., 136 ( 1996), pp. 127-153. · Zbl 0867.46019
[12] Berg, C. and Duran, A.J. .- Orthogonal polynomials, L2-spaces and entire functions, Math. Scand. (to appear). · Zbl 0879.42015
[13] Berg, C. and Thill, M.- Rotation invariant moment problems , Acta Math.167 ( 1991), pp. 207-227. · Zbl 0744.44006
[14] Berg, C. and Thill, M. . - A density index for the Stieltjes moment problem, in: Orthogonal Polynomials and their Applications, ed. C. Brezinski, L. Gori and A. Ronveaux, IMACS Annals on computing and applied mathematics, 9 (1991), pp. 185-188. · Zbl 0830.44006
[15] Berg, C. and Valent, G. - The Nevalinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods and Applications of Analysis, 1 (1994), pp. 169-209. · Zbl 0966.44500
[16] De Branges, L. .- Hilbert spaces of entire functions , Prentice-Hall, Englewood Cliffs, N.J., 1968. · Zbl 0157.43301
[17] Buchwalter, H. and Cassier, G. .- Mesures canoniques dans le problème classique des moments, Ann. Inst. Fourier, 34 (1984), pp. 45-52. · Zbl 0525.28002
[18] Cassier, G. .- Mesures canoniques dans le problème classique des moments, C.R. Acad. Sci., 296, Série I (1984), pp. 717-719. · Zbl 0523.44009
[19] Cassier, G. .- Problème des moments sur un compact de Rn et représentation de polynômes à plusieurs variables, J. Funct. Analysis, 58 (1984), pp. 254-266. · Zbl 0556.44006
[20] Chihara, T.S. .- On indeterminate Hamburger moment problems, Pac. J. Math., 27 (1968), pp. 475-484. · Zbl 0167.41901
[21] Chihara, T.S. .- Indeterminate symmetric moment problems, Math. Anal. Appl., 85 (1982), pp. 331-346. · Zbl 0485.44004
[22] Choquet, G. .- Le problème des moments, Séminaire d’initiation à l’analyse (1962), Paris.
[23] Fuglede, B. .- The multidimensional moment problem , Expo. Math., 1 ( 1983), pp. 47-65. · Zbl 0514.44006
[24] Hamburger, H. .- Über eine Erweiterung des Stieltjesschen Momentenproblems, Math. Ann., 81 (1920), pp. 235-319.; 82 (1921 ), pp. 120-164. 168-187. · JFM 47.0427.04
[25] HAVIN (V. P.) et al. (editors) .- Linear and complex analysis problem book. 199 research problems , , 1043, Springer, Berlin - Heidelberg - New York1984. · Zbl 0545.30038
[26] Ismail, M.E.H. .- A queueing model and a set of orthogonal polynomials, J. Math. Anal. Appl., 108 (1985), pp. 575-594. · Zbl 0579.60093
[27] Ismail, M.E.H. and Masson, D.R. .- Q-Hermite polynomials, biorthogonal rational functions and Q-beta integrals, Trans. Amer. Math. Soc., 346 (1994), pp. 63-116. · Zbl 0812.33012
[28] Kjeldsen, T.H. .- The early history of the moment problem, Historia Math., 20 (1993), pp. 19-44. · Zbl 0769.01008
[29] Koosis, P. .- The logarithmic integral I, Cambridge University Press, Cambridge1988. · Zbl 0665.30038
[30] Naimark, M.A. .- Extremal spectral functions of a symmetric operator, Izv. Akad. Nauk. SSSR ser. matem., 11 (1947), pp. 327-344; Dokl. Akad. Nauk. SSSR, 54 (1946 ), pp. 7-9. · Zbl 0032.21501
[31] Nevanlinna, R. .- Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentenproblem, Ann. Acad. Scient. Fennicae, (A), 18, n° 5 (1922), pp. 1-53. · JFM 48.1226.02
[32] Petersen, L.C. .- On the relation between the multidimensional moment problem and the one-dimensional moment problem, Math. Scand., 51 (1982), pp. 361-366. · Zbl 0514.44007
[33] Riesz, M. .- Sur le problème des moment. Troisième Note, Arkiv för matematik, astronomi och fysik, 17, n° 16 (1923), pp. 1-52. · JFM 49.0195.01
[34] Riesz, M. .- Sur le problème des moments et le théorème de Parseval correspondant, Acta Litt. Ac. Sci. Szeged , 1 (1923), pp. 209-225. · JFM 49.0708.02
[35] Schmüdgen, K. .- On determinacy notions for the two dimensional moment problem, Ark. Mat., 29 (1991), pp. 277-284. · Zbl 0762.44004
[36] Schmüdgen, K. .- The K-moment problem for compact semi-algebraic sets, Math. Ann., 289 (1991), pp. 203-206. · Zbl 0744.44008
[37] Sodin, K. . - A note on the Hall-Mergelyan theme , Mathematical Physics, Analysis and Geometry (To appear). · Zbl 0872.44004
[38] Stieltjes, T.J. .- Recherches sur les fractions continues , Annales de la Faculté des Sciences de Toulouse, 8 (1894), pp. 1-122; 9 (1895), pp. 5-47. · JFM 25.0326.01
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