Hermite Polynomial Interpolation

    The cubic Hermite polynomial  p(x)  has the interpolative properties          and    both the function values and their derivatives are known at the endpoints of the interval  .  Hermite polynomials were studied by the French Mathematician Charles Hermite (1822-1901), and are referred to as a "clamped cubic," where "clamped" refers to the slope at the endpoints being fixed.  This situation is illustrated in the figure below.  

Theorem (Cubic Hermite Polynomial).  If    is continuous on the interval  , there exists a unique cubic polynomial    such that
        ,
        ,
        ,  
        .

Remark.  The cubic Hermite polynomial is a generalization of both the Taylor polynomial and Lagrange polynomial, and it is referred to as an "osculating polynomial."  Hermite polynomials can be generalized to higher degrees by requiring that the use of more nodes   and the extension to agreement at higher derivatives    for    and    .  The details are found in advanced texts on numerical analysis    

More Background. The Clamped Cubic Spline

    A clamped cubic spline is obtained by forming a piecewise cubic function which passes through the given set of knots   with the condition the function values, their derivatives and second derivatives of adjacent cubics agree at the interior nodes.  The endpoint conditions are , where   are given.

More Background. The Natural Cubic Spline

    A natural cubic spline is obtained by forming a piecewise cubic function which passes through the given set of knots with the condition the function values, their derivatives and second derivatives of adjacent cubics agree at the interior nodes.  The endpoint conditions are  . The natural cubic spline is said to be "a relaxed curve."