Characterization of polynomial functions and application to time-frequency analysis

We propose necessary and sufficient conditions for a function φ(t) to be a polynomial and establish a formula that allows us to compute the value of the derivative φ´(t) of a polynomial at any point in terms of the values of this polynomial. For this purpose, we establish that any polynomial φ(t) of degree ⩽Q can be represented as a linear combination of φ(t-t₁), …, Q(t-t_Q), where t₁, …, t_Q denote arbitrary parameters. A similar property is established for the derivative φ´(t). Some signal processing applications of these results are discussed,