This paper explores the convergence in the W1,p\left((-1,1), (w(a,b),w(a+1,b+1))\right) norm of the Fourier expansion in terms of {qn(a,b)}n\geq 0 with 1< p
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:Let {qn(a,b)}n \geq 0 be the sequence of polynomials orthonormal with respect to the Sobolev inner product \langle f,g\rangleS:=\int-11f(x)g(x)w(a,b)(x)dx+\int-11f'(x)g'(x)w(a+1,b+1)(x)dx, where w(a,b)(x)=(1-x)a(1+x)b, x\in [-1,1] and a,b>-1.
📊
Temel Bulgular & Veri:Let {qn(a,b)}n \geq 0 be the sequence of polynomials orthonormal with respect to the Sobolev inner product \langle f,g\rangleS:=\int-11f(x)g(x)w(a,b)(x)dx+\int-11f'(x)g'(x)w(a+1,b+1)(x)dx, where w(a,b)(x)=(1-x)a(1+x)b, x\in [-1,1] and a,b>-1.
💡
Sonuç & Katkı:This paper explores the convergence in the W1,p\left((-1,1), (w(a,b),w(a+1,b+1))\right) norm of the Fourier expansion in terms of {qn(a,b)}n\geq 0 with 1< p
Let {qn(a,b)}n \geq 0 be the sequence of polynomials orthonormal with respect to the Sobolev inner product \langle f,g\rangleS:=\int-11f(x)g(x)w(a,b)(x)dx+\int-11f'(x)g'(x)w(a+1,b+1)(x)dx, where w(a,b)(x)=(1-x)a(1+x)b, x\in [-1,1] and a,b>-1. This paper explores the convergence in the W1,p\left((-1,1), (w(a,b),w(a+1,b+1))\right) norm of the Fourier expansion in terms of {qn(a,b)}n\geq 0 with 1< p
Abstract
Let {qn(a,b)}n \geq 0 be the sequence of polynomials orthonormal with respect to the Sobolev inner product \langle f,g\rangleS:=\int-11f(x)g(x)w(a,b)(x)dx+\int-11f'(x)g'(x)w(a+1,b+1)(x)dx, where w(a,b)(x)=(1-x)a(1+x)b, x\in [-1,1] and a,b>-1. This paper explores the convergence in the W1,p\left((-1,1), (w(a,b),w(a+1,b+1))\right) norm of the Fourier expansion in terms of {qn(a,b)}n\geq 0 with 1< p
MARCELLÁN, F., QUINTANA, Y., URIELES, A. (2013). On the Pollard decomposition method applied to some Jacobi--Sobolev expansions. Turkish Journal of Mathematics, Cilt: 37 - Sayı: 6, 934–948. https://ajindex.com/articles/723323
Askey, R.: Orthogonal Polynomials and Special Functions, Regional Conference Series in Applied Mathematics, SIAM. Bristol, UK. J. W. Arrowsmith Ltd. 1975.
Badkov, V.M.: The uniform convergence of Fourier series in orthogonal polynomials. Math. Notes 5, 174–179 (1969). Ciaurri, ´ O., Roncal, L.: The Bochner-Riesz means for Fourier-Bessel expansions. J. Funct. Anal. 228, 89–113 (2005). De Jes´ us, M.N., Petronilho, J.: Sobolev orthogonal polynomials and (M, N ) -coherent pairs of measures. J. Comput. Appl. Math. 237, 83–101, (2012).
Fejzullahu, B. X.: Divergent Legendre-Sobolev polynomial series. Novi. Sad J. Math. 38, 35–41 (2008).
Fejzullahu, B. X.: Asymptotic properties and Fourier expansions of orthogonal polynomials with a non-discrete Gegenbauer-Sobolev inner product. J. Approx. Theory 162, 397–406 (2010).
Fejzullahu, B. X., Marcell´ an, F.: A Cohen type inequality for Laguerre-Sobolev expansions. J. Math. Anal. Appl. 352, 880–889 (2009).
Iserles, A., Koch, P.E., Nİrsett, S.P., Sanz-Serna, J.M.: Orthogonality and approximation in a Sobolev space. In: Algorithms for Approximations (Eds.: J.C. Mason and M.G. Cox) 117–124, London. Chapman & Hall (1990).
Johansson, M.: The Hilbert transform. Master Thesis. Vaxjo University. Sweden, 1999. Marcell´ an, F., Osilenker, B.P.: Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product. Math. Zamet. 62, 871–880 (1997). Marcell´ an, F., Osilenker, B.P., Rocha, I.A.: Problemas abiertos en series de Fourier de Jacobi-Sobolev. In: Margarita Mathematica en memoria de Jos´ e Javier (Chicho) Guadalupe Hern´ andez. (Eds.: L. Espa˜ nol and J.L. Varona) 183– 189, Logro˜ no, Spain. Servicio de Publicaciones, Universidad de La Rioja (2001) (in Spanish). Marcell´ an, F., Osilenker, B.P., Rocha, I.A.: On Fourier series of Jacobi-Sobolev orthogonal polynomials. J. of Inequal. & Appl. 7, 673–699 (2002). Marcell´ an, F., Rocha, I.A., Salto, L.: Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product. J. Approx. Theory 121, 336–356 (2003). Marcell´ an, F., Quintana, Y.: Polinomios ortogonales no-est´ andar. Propiedades algebraicas y anal´ıticas. XXII Escuela Venezolana de Matem´ aticas. Ediciones IVIC, M´ erida, Venezuela (2009) (in Spanish). Marcell´ an, F., Quintana, Y., Urieles, A.: On W 1,p -convergence of Fourier-Sobolev expansions. J. Math. Anal. Appl. 398, 594–599 (2013).
Mastroianni, G., Notarangelo, I.: L p -convergence of Fourier sums with expone