In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn（f;x,y） and a smooth function g（x,...In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn（f;x,y） and a smooth function g（x,y） on D converges to that of f（x,y） and g（x,y） on D when n →∞ , provided f（x,y） belongs to C（D） and all first partial derivatives of g（x,y） belong to the space LipM^α（0 〈 α ≤1）. We further show that provided all second partial derivatives of g（x,y） also belong to the space LipM^α and f（x,y） belongs to C^1 （D）, the inner product between the partial derivative of Hakopian interpolation polynomial δ/δx Hn（f;z,y） and g（x,y） on D converges to that between δ/δxf（x,y） and g（x,y） on D when n →∞. oo.展开更多
Weighted Lp mean convergence of Extended Hermite-Fejer operators based on the zeros of orthogonal polynomials with respct to the general weight and Jacobi weight is investigated. Suf ficient conditions for such conve...Weighted Lp mean convergence of Extended Hermite-Fejer operators based on the zeros of orthogonal polynomials with respct to the general weight and Jacobi weight is investigated. Suf ficient conditions for such convergence for all continuous functions are given.展开更多
A weight w(x) is provided with the properties: w(x)>(1-x2) l/2 and Lagrange's interpolation based on the zeros of orthogonal polynomials with respect to w diverges in Lp (p>6) for some [-1, 1], This gives a ...A weight w(x) is provided with the properties: w(x)>(1-x2) l/2 and Lagrange's interpolation based on the zeros of orthogonal polynomials with respect to w diverges in Lp (p>6) for some [-1, 1], This gives a negative answer to Problem 10 of P. Turan.展开更多
文摘In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn（f;x,y） and a smooth function g（x,y） on D converges to that of f（x,y） and g（x,y） on D when n →∞ , provided f（x,y） belongs to C（D） and all first partial derivatives of g（x,y） belong to the space LipM^α（0 〈 α ≤1）. We further show that provided all second partial derivatives of g（x,y） also belong to the space LipM^α and f（x,y） belongs to C^1 （D）, the inner product between the partial derivative of Hakopian interpolation polynomial δ/δx Hn（f;z,y） and g（x,y） on D converges to that between δ/δxf（x,y） and g（x,y） on D when n →∞. oo.
文摘Weighted Lp mean convergence of Extended Hermite-Fejer operators based on the zeros of orthogonal polynomials with respct to the general weight and Jacobi weight is investigated. Suf ficient conditions for such convergence for all continuous functions are given.
基金Project supported by the National Natural Science Foundation of China.
文摘A weight w(x) is provided with the properties: w(x)>(1-x2) l/2 and Lagrange's interpolation based on the zeros of orthogonal polynomials with respect to w diverges in Lp (p>6) for some [-1, 1], This gives a negative answer to Problem 10 of P. Turan.
