The Divergence of Lagrange Interpolation for |x|α at Equidistant Nodes

The divergence of Lagrange interpolation for |x|α (2 < α < 4) at equidistant nodes

“…Now consider| sin(πnx)/π|, by Rever( [3],theorem1), | sin(πnx)/π| is bounded away from zero infinitely often.…”

On Lagrange interpolation with equally spaced nodes

“…We point out that the reader may, if he wishes, easily find comparable estimates to (10). However, it is not to hard to see that the upper summation index 3 in (9) may not be replaced by a smaller index to give a result which is comparable to (10).…”

“…For example, see [2,3,7,9,10,11,12,14]. An extension of Bernstein's result is given in [10]: [2] Theorem 1 informs us that the divergence behaviour is rather general and does not depend on the special characteristics of \x\. In 1990, Byrne, Mills and Smith, amplifying the classical result of Bernstein, showed that the rate of divergence of the sequence <L n (|x| ,x 0 ) \ depends on the location of xo in [-1,1].…”

“…We combine estimate (9) and formula (5). By a careful, but absolute elementary calculation, one shows that (10) S…”

Fractional Hermite interpolation for non-smooth functions

“…In [20], a new representation of Hermite osculatory interpolation was presented in order to construct weighted Hermite quadrature rules with arithmetic and geometric nodes. For f (x) = |x| α , x ∈ [−1, 1], Revers [26], Lu [19], and Su [27] showed that the sequence of Lagrange interpolating polynomials with equidistant nodes is divergent everywhere in the interval except at zero and the endpoints, for 0 < α ≤ 1, 1 < α ≤ 2, and 2 < α < 4, respectively. For fractional smooth functions, Wang et al [35] derived a general form for a local fractional Taylor expansion based on the local fractional derivative at the singular points and obtained the remainder expansions for linear and quadratic interpolants.…”