An Introduction to Sampling Analysis

A basic property of algebraic polynomials $${{P}_{n}}\left( t \right): = \sum\nolimits_{{k = 0}}^{n} {{{a}_{k}}{{t}^{k}}}$$ of degree ≤ n is that any such polynomial is fully and uniquely determined by its values P _n (t _v ) at an arbitrary given set of n + 1 distinct points t₀, t₁,…, t_v,…, t _n .The converse question, whether it is always possible to find a suitable polynomial P _n (x) which takes on the function (polynomial) values y₀, y₁,…, y _n associated with any n + 1 distinct abscissas (interpolation or nodal points) t₀, t₁, …, t _n , is answered by the Lagrangian interpolation formula: (1)$$ P_n \left( t \right): = \sum\limits_{k = 0}^n {y_k l_k \left( t \right) = \sum\limits_{k = 0}^n {P_n \left( {t_k } \right)l_k \left( t \right)} } $$ where, for k = 0, 1,…, n, (2)$$\begin{array}{*{20}{c}} {{{l}_{k}}\left( t \right): = {{l}_{{k,n}}}\left( t \right) = \frac{{{{\omega }_{{n + 1}}}\left( t \right)}}{{\left( {t - {{t}_{k}}} \right){{{\omega '}}_{{n + 1}}}\left( {{{t}_{k}}} \right)}},} & {{{\omega }_{{n + 1}}}\left( t \right): = \left( {t - {{t}_{0}}} \right) \ldots \left( {t - {{t}_{n}}} \right).} \\ \end{array}$$ Indeed, l _k (t) is the only algebraic polynomial of degree n which possesses the property (3)$${{l}_{k}}\left( {{{t}_{\nu }}} \right) = {{\delta }_{{k,\nu }}} = \left\{ {\begin{array}{*{20}{c}} 0 & {for{\mkern 1mu} \nu \ne \kappa } \\ 1 & {for{\mkern 1mu} \nu \ne \kappa .} \\ \end{array} } \right.$$ Hence the polynomial P _n (t) of degree n, thus determined, is unique.