Summability of Jacobi Series
Copyright policy )Summability of Jacobi Series
The positivity of some Cesàro mean is proven for Jacobi series Σ a n P n ( α , β ) ( x ) , α , β ≧ − 1 2 \Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac {1}{2} . This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the ( C , α + β + 2 ) (C,\alpha + \beta + 2) means is conjectured and proven for some ( α , β ) (\alpha ,\beta ) . One consequence of this conjecture would be the complete monotonicity of x − c ( x 2 + 1 ) − c , c ≧ 1 {x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1 .
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.), General integral transforms, Cesàro, Euler, Nörlund and Hausdorff methods
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.), General integral transforms, Cesàro, Euler, Nörlund and Hausdorff methods
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