Abstract
In this paper we consider interpolatory quadrature formulae with multiple nodes, which have the maximal trigonometric degree of exactness. Our approach is based on a procedure given by Ghizzeti and Ossicini (Quadrature formulae, Academie-Verlag, Berlin, 1970). We introduce and consider the so-called σ-orthogonal trigonometric polynomials of semi-integer degree and give a numerical method for their construction. Also, some numerical examples are included.
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Bojanov, B.D.: Oscillating polynomials of least L 1-norm. In: Hämmerlin, G. (ed.) Numerical Integration, ISNM 57, pp. 25–33. Birkhäuser, Basel (1982)
Cvetković A.S., Milovanović G.V.: The Mathematica Package “OrthogonalPolynomials”. Facta Univ. Ser. Math. Inform. 19, 17–36 (2004)
Dryanov D.P.: Quadrature formulae with free nodes for periodic functions. Numer. Math. 67, 441–464 (1994)
Du J., Han H., Jin G.: On trigonometric and paratrigonometric Hermite interpolation. J. Approx. Theory 131, 74–99 (2004)
Engels H.: Numerical Quadrature and Cubature. Academic Press, London (1980)
Gautschi W., Milovanović G.V.: S-orthogonality and construction of Gauss-Turán type quadrature formulae. J. Comput. Appl. Math. 86, 205–218 (1997)
Ghizzeti A., Ossicini A.: Quadrature Formulae. Academie-Verlag, Berlin (1970)
Milovanović G.V., Cvetković A.S., Stanić M.P.: Trigonometric orthogonal systems and quadrature formulae. Comput. Math. Appl. 56(11), 2915–2931 (2008)
Milovanović G.V., Cvetković A.S., Stanić M.P.: Explicit formulas for five-term recurrence coefficients of orthogonal trigonometric polynomials of semi-integer degree. Appl. Math. Comput. 198(2), 559–573 (2008)
Milovanović, G.V., Cvetković, A.S., Stanić, M.P.: Trigonometric orthogonal systems and quadrature formulae with maximal trigonometric degree of exactness. In: Boyanov, T. et al. (eds.) Numerical Methods and Applications. 6th International Conference, NMA 2006, Borovets, Bulgaria, August 20–24, 2006. Revised papers. Lecture Notes in Computer Science, vol. 4310, pp. 402–409, Springer, Berlin (2007)
Milovanović, G.V., Spalević, M.M.: Construction of Chakalov-Popoviciu’s type quadrature formulae. Rend. Circ. Mat. Palermo Ser. II, Suppl. 52, 625–636 (1998)
Milovanović G.V., Spalević M.M., Cvetković A.S.: Calculation of Gaussian type quadratures with multiple nodes. Math. Comput. Modell. 39, 325–347 (2004)
Mirković, B.: Theory of Measures and Integrals. Naučna knjiga, Beograd (1990) (in Serbian)
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. In: Classics in Applied Mathematics, vol. 30. SIAM, Philadelphia (2000). Reprint of the 1970 original
Shi Y.G.: A kind of extremal problem of integration on an arbitrary measure. Acta Sci. Math. (Szeged) 65, 567–575 (1999)
Shi Y.G., Xu G.: Construction of σ-orthogonal polynomials and Gaussian quadrature formulas. Adv. Comput. Math. 27(1), 79–94 (2007)
Turetzkii, A.H.: On quadrature formulae that are exact for trigonometric polynomials. East J. Approx. 11, 337–359 (2005) (translation in English from Uchenye Zapiski, Vypusk 1(149), Seria math. Theory of Functions, Collection of papers, Izdatel’stvo Belgosuniversiteta imeni V.I. Lenina, Minsk, pp. 31–54 (1959))
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The authors were supported in part by the Serbian Ministry of Science and Technological Development (Project: Orthogonal Systems and Applications, grant number #144004) and the Swiss National Science Foundation (SCOPES Joint Research Project No. IB7320-111079 “New Methods for Quadrature”).
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Milovanović, G.V., Cvetković, A.S. & Stanić, M.P. Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness. Numer. Math. 112, 425–448 (2009). https://doi.org/10.1007/s00211-009-0219-5
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DOI: https://doi.org/10.1007/s00211-009-0219-5