Milovanovic G.  

Generalized quadrature processes

 In this lecture we introduce and discuss a few generalized quadrature processes of Gaussian type. The first class of such formulas contains two types nonstandard Gaussian quadratures: (a) interval quadratures (cf. Bojanov & Petrov [Numer. Math. 87 (2001), 625-643; 95 (2003), 53-62], [SIAM J. Numer. Anal. 43 (2005), 787-795] and Milovanović & Cvetković [Numer. Math. 99 (2004), 141-162; 102 (2006), 523-542], [J. Comput. Appl. Math. 182 (2005), 433-446]); (b) Gaussian quadratures based on operator values, in particular with the average Steklov operator and some kind of difference operators (cf. Milovanović & C vetković [Adv. Comput. Math. 32 (2010), 431-486]). The second class of generalized quadratures processes is related to Gaussian quadratures using (only) function derivatives (cf. Milovanović & Cvetković [IMA J. Numer. Anal. 31 (2011), 358-377]). Also, we consider a class of generalized Birkhoff-Young quadratures, including a characterization and an unexpected connection with multiple orthogonal polynomials (cf. Milovanović [Stud. Univ. Babeş-Bolyai Math. 56 (2011), 449 – 464]).