Functions & Functionals
Fixed point of a Function is defined as an input that is equal to its output. If f(x) =x for a particular value of x, then that value of x is a fixed point of the function f(x). Some functions have no fixed points, some have one and some have many fixed points. Fix(f) denotes the set of fixed points of a function f. Example- f(x) =x2 - 2 has fixed points at x=(-1,2). When the input x is not a number but a function, the output also becomes a function at the fixed points if any and such functions are called Functionals. Hence functionals are functions that take a function for its input. Functional is also defined as a function from a vector space into its underlying scalar field, or a set of functions to the real numbers. If it is a Linear function from a linear vector space into its underlying scalar field, it is called linear functional. In other words, it is a function that takes a vector as its input argument, and returns a scalar. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument, then it is sometimes considered a function of a function. Functionals are additive -> f(x+y) =f(x) + f(y). Functional derivatives are used in Lagrangian mechanics. Functional integrals were used by Feynman for path integral formulation of his quantum mechanics. Suppose in real coordinate space, the vector x is represented as column vector then the linear functional f(x) is where
Using the concept of functionals and fixed points, one can eliminate explicit recursion for a function through 2 steps--
(1) Find a functional whose fixed point is the recursion function we select.
(2) Find the fixed point of a function without recursion.