Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The Lagrange reconstructing polynomial [C.W. Shu, High-order WENO schemes for convection-dominated problems, SIAM Rev. 51 (1) (2009) 82-126] of a function f(x) on a given set of equidistant (@Dx=const) points {x"i+@?@Dx;@?@?{-M"-,...,+M"+}} is defined as the polynomial whose sliding (with x) averages on [x-12@Dx,x+12@Dx] are equal to the Lagrange interpolating polynomial of f(x) on the same stencil [G.A. Gerolymos, Approximation error of the Lagrange reconstructing polynomial, J. Approx. Theory 163 (2) (2011) 267-305. doi:10.1016/j.jat.2010.09.007].  first study the fundamental functions of Lagrange reconstruction, then show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) x"i+n12@Dx (n@?Z), and obtain several identities. Using these identities,  show that there exists a unique representation of the Lagrange reconstructing polynomial on {i-M"-,...,i+M"+} as a combination of the Lagrange reconstructing polynomials on Neville substencils [E. Carlini, R. Ferretti, G. Russo, A WENO large time-step scheme for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 27 (3) (2005) 1071-1091], with ights which are rational functions of @x (x=x"i+@x@Dx) [Y.Y. Liu, C.W. Shu, M.P. Zhang, On the positivity of the linear ights in WENO approximations, Acta Math. Appl. Sin. 25 (3) (2009) 503-538], and give an analytical recursive expression of the ight-functions.  show that all of the poles of the rational ight-functions are real, and that there can be no poles at half-points.  then use the analytical expression of the ight-functions, combined with the factorization of the fundamental functions of Lagrange reconstruction, to obtain a formal proof of convexity (positivity of the ight-functions) in the neighborhood of @x=12, iff all of the substencils contain either point i or point i+1 (or both).
['G. A. Gerolymos']
I am writing a research paper on a idea Representation of the Lagrange reconstructing polynomial by combination of substencils
[{"from": "human", "value": "I am writing a research paper on a idea Representation of the Lagrange reconstructing polynomial by combination of substencils"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The Lagrange reconstructing polynomial [C.W. Shu, High-order WENO schemes for convection-dominated problems, SIAM Rev. 51 (1) (2009) 82-126] of a function f(x) on a given set of equidistant (@Dx=const) points {x\"i+@?@Dx;@?@?{-M\"-,...,+M\"+}} is defined as the polynomial whose sliding (with x) averages on [x-12@Dx,x+12@Dx] are equal to the Lagrange interpolating polynomial of f(x) on the same stencil [G.A. Gerolymos, Approximation error of the Lagrange reconstructing polynomial, J. Approx. Theory 163 (2) (2011) 267-305. doi:10.1016/j.jat.2010.09.007].  first study the fundamental functions of Lagrange reconstruction, then show that these polynomials have only real and distinct roots, which are never located at the cell-interfaces (half-points) x\"i+n12@Dx (n@?Z), and obtain several identities. Using these identities,  show that there exists a unique representation of the Lagrange reconstructing polynomial on {i-M\"-,...,i+M\"+} as a combination of the Lagrange reconstructing polynomials on Neville substencils [E. Carlini, R. Ferretti, G. Russo, A WENO large time-step scheme for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 27 (3) (2005) 1071-1091], with ights which are rational functions of @x (x=x\"i+@x@Dx) [Y.Y. Liu, C.W. Shu, M.P. Zhang, On the positivity of the linear ights in WENO approximations, Acta Math. Appl. Sin. 25 (3) (2009) 503-538], and give an analytical recursive expression of the ight-functions.  show that all of the poles of the rational ight-functions are real, and that there can be no poles at half-points.  then use the analytical expression of the ight-functions, combined with the factorization of the fundamental functions of Lagrange reconstruction, to obtain a formal proof of convexity (positivity of the ight-functions) in the neighborhood of @x=12, iff all of the substencils contain either point i or point i+1 (or both)."}]