Interpolation by algebraic and trigonometric polynomials

The problem to be examined here is that of determining an algebraic or trigonometric polynomial of minimal degree exactly satisfying given constraints. The constraints to be considered are of the following forms: (A) Ordinates are specified at given distinct abscissas, (B) Slopes are specified at some or all of the points where the ordinates are constrained, (C) pth derivatives are specified at some points for which all lower-order derivatives, including the zeroth (i.e. the ordinate), are also specified. The polynomials which we will construct to satisfy such constraints will be of the form P.(x) = E crx' or S.(x) = 1:' b, sin rx. First we consider the standard Lagrangian problem with constraints of type (A) to be satisfied by an algebraic polynomial. Let flk (X) = Ho ( xi), and let Lk(x) be a kth degree polynomial with specified ordinates fi at the k + 1 distinct abscissas xi, i = 0, 1, ... , k. A recursive procedure for generating the sequence of polynomials Lk(X) is defined by writing Lo = fo, Ho = x xo , and then, for k = 0, 1, * *