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S.N.
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My research interest in interpolation theory is mainly focused in the investigation of pointwise divergence properties of equidistant Lagrange interpolation to the continuous functions on [-1,1]. We can illustrate the equidistant divergence by a well known example, discorverd by S.N. Bernstein in 1918.
|
|
On
the left-hand side you can see an "animated" running subsequence of
equidistant
interpolation polynomials for the function Abs[x] on the interval
[-1,+1].
In 1918, S.N. Bernstein proved the surprising result that for each choice of x in [-1,+1], apart from the values -1, 0, +1 the sequence of Lagrange polynomials is unbounded and thus does not converge to the correct function value. For the points -1, +1 divergence cannot occur in the equidistant interpolation array. However, for the point 0 the situation is more complicated: It was
proved in 1939 by
D.L. Berman that the Lagrange polynomials at zero converge to its true
function value. |
| In
1999 Bernstein's classical result was extended to the following
expression:
|
 An extension of Bernstein's result (red polynomial).. |
| Regarding
to the above mentioned results of D.L. Berman and S.M. Lozinskii, I was
able to amplify the bounds for the approximation error for Abs[x] at
the
point zero to the following expression:
|
 Approximation error for the point zero (green line). For Abs[x] it was shown that the approximation constant at the point zero is exactly 2/Pi. |
 |
It
is further conjectured that the Bernstein result reflects a more
general
situation. Here is the conjecture:
|
I succeeded in proving the following (special) situation (which gives a slight extension of a result from 1990 by Byrne/Mills/Smith (see "On Lagrange interpolation with equidistant nodes", Bull. Austral. Math. Soc. Vol. 42 (1990), 81-89):
