= Hermite interpolating polynomial
{c}
{title2=$H(x_i)=y_i,\quad H'(x_i)=d_i$}
= Hermite interpolant
{c}
{synonym}
At distinct <interpolation nodes> $x_0,\ldots,x_n$, prescribed values $y_i$ and first <derivatives> $d_i$ determine exactly one <Hermite interpolating polynomial> of <polynomial degree> at most $2n+1$. The homogeneous data force a double <root of a polynomial> at every <interpolation node>. Hence the <kernel of a linear map> given by these $2n+2$ conditions is zero: a nonzero <polynomial> with those double <roots of a polynomial> would have <polynomial degree> at least $2n+2$. Equal finite <dimensions of a vector space> then give existence as well as uniqueness. This <interpolation polynomial> is distinct from the <Hermite polynomials> forming a classical <orthogonal polynomial> family.
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