An article titled "Approximation of polynomials by Hermite interpolation" by Robert Kantrowitz '82, the Marjorie and Robert W. McEwen Professor of Mathematics, and Michael M. Neumann of Mississippi State University, appears in the current issue of the journal Elemente der Mathematik.
A theorem from functional analysis guarantees that a given polynomial on the unit interval of the real line is uniformly approximable by a sequence of polynomials whose derivatives vanish at the origin. Combining elements from numerical analysis, the authors go on to provide simple and manageable formulas for such polynomials that also agree with the given one at equally-spaced points of the interval. It turns out that these are the Hermite interpolation polynomials for this setting, named after the 19th century French mathematician, Charles Hermite.
When the derivative stipulation is instead placed on the midpoint of the interval, the authors show that a switch from equally-spaced interpolation points to Chebyshev nodes leads to approximating polynomial interpolants of Chebyshev-type, after the 19th century mathematician Pafnuty Chebyshev, considered to be the founding father of Russian mathematics.
