Fourier Transform of Orthogonal Polynomials over the Triangle with Four Parameters

Year 2022, , 314 - 320, 30.12.2022

Esra Güldoğan

https://doi.org/10.47000/tjmcs.1063098

Abstract

In this paper, some new families of orthogonal functions in two variables produced by using Fourier transform of bivariate orthogonal polynomials and their orthogonality relations obtained from Parseval identity are introduced.

Keywords

Bivariate orthogonal functions, Koornwinder polynomials, Jacobi polynomials, hypergeometric functions, Fourier transform, Parseval identity

References

• Agahanov, S.A., A method of constructing orthogonal polynomials of two variables for a certain class of weight functions, Vestnik Leningrad Univ., 20(1965), 5–10.
• Aktaş, R., Altın, A., Taşdelen, F., A note on a family of two-variable polynomials, Journal of Computational and Applied Mathematics, 235(16)(2011), 4825–4833.
• Aktaş, R., A note on parameter derivatives of the Jacobi polynomials on the triangle, Appl. Math. Comp., 247(2014), 368–372 .
• Aktaş, R., On parameter derivatives of a family of polynomials in two variables, Applied Mathematics and Computation, 256(2015), 769–777.
• Aktaş, A., Area, I., Güldoğan, E., A new family of orthogonal polynomials in three variables, Journal of Inequalities and Applications, 2020(1)(2020), 170.
• Davies, B., Integral Transforms and Their Applications, Applied Mathematical Sciences, Springer-Verlag, New York-Heidelberg, 25, 1978.
• Dunkl, C.F., Xu, Y., Orthogonal Polynomials of Several Variables, Cambridge Univ. Press, New York, 2001.
• Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G., Tables of Integral Transforms, McGraw-Hill, 2, (1954).
• Fernandez, L., Perez, T.E., Pinar, M.A., On Koornwinder classical orthogonal polynomials in two variables, J. Comput. Appl. Math., 236(2012), 3817–3826.
• Güldoğan, E., Aktaş, R., Area, I., Some classes of special functions using Fourier transforms of some two-variable orthogonal polynomials, Integral Transform. Spec. Funct., 31(2020), 437–470.
• Koelink, H.T., On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124(3)(1996), 887–898.
• Koeph, W., Masjed-Jamei, M., Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials, Proc. Amer. Math. Soc., 135(11)(2007), 3599–3606.
• Koornwinder, T.H., Two Variable Analogues of the Classical Orthogonal Polynomials, In: Askey, R.A., ed. Theory and Application of Special Functions. Proceedings of an Advanced Seminar, 435–495, The University of Wisconsin-Madison, Academic Press, New York, 1975.
• Koornwinder, T.H., Group Theoretic Interpretations of Askey’s Scheme of Hypergeometric Orthogonal Polynomials, Orthogonal Polynomials and Their Applications, Springer, 1988, 46–72.
• Koornwinder, T.H., Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform, in Polynomes orthogonaux et applications, Lecture Notes in Math. 1171, Springer-Verlag, 1985, 174–183.
• Luke, Y.L., The Special Functions and Their Approximations, Academic Press, New York, Vol.II, 1969.
• Marriaga, M., P´erez, T.E., Pinar, M.A., Three term relations for a class of bivariate orthogonal polynomials, Mediterr. J. Math., 14(54)(2017).
• Masjed-Jamei, M., Marcellan, F., Huertas, E. J., A finite class of orthogonal functions generated by Routh-Romanovski polynomials, Complex Var. Elliptic Eqns., 59(2)(2014), 162–171.
• Masjed-Jamei, M., Koepf,W., Two classes of special functions using Fourier transforms of generalized ultraspherical and generalized Hermite polynomials, Proc. Am. Math. Soc., 140(2012), 2053–2063.
• Masjed-Jamei, M., Koepf, W., Two finite classes of orthogonal functions, Applicable Analysis, 92(11)(2013), 2392–2403.
• Mastroianni, G., Milovanovic, G.V., Interpolation Processes-Basic Theory and Applications, Springer-Verlag, Berlin-Heidelberg, 2008.
• Milovanovic´, G.V., Öztürk, G., Aktaş, R., Properties of some of two-variable orthogonal polynomials, Bull. Malays Math. Sci. Soc., 43(2)(2020), 1403–1431.
• Olver, S., Townsend, A.,Vasil, G.M., Recurrence relations for a family of orthogonal polynomials on a triangle, In: Spectral and High Order Methods for Partial Differential Equations, ICOSAHOM 2018, Lecture Notes in Computational Science and Engineering, Springer, Berlin, 134(2020) (arXiv:1801.09099).
• Rainville, E.D., Special Functions, The Macmillan Company, New York, 1960.
• Suetin, P.K., Orthogonal Polynomials in Two Variables, Gordon and Breach Science Publishers, Moscow, 1988.
• Szegö, G., Orthogonal Polynomials, 4th ed., American Mathematical Society Colloquium Publications, 23, (1975).