Locally Estimated Operators
These are versions of operators which take a function and an element in the domain of the function to give a locally weight estimation of those operators. Specifically, we'll be focusing on averages and variances.
Local average operators are a class of operators. An operator A: (I: T -> Y) * T -> Y, where T is a subset of Y which is a subset of the set of real numbers R, is a local average operator if the following hold for some b, such that b is less than or equal to all t in T, and some L: T * T -> Y :
I is the input function. It returns a value dependent upon time t which is used as the input for the local average operator. The operator is dependent upon all values of x from the beginning b until the current time t. L is a local weighting function that defines how heavily the operator weighs values close to time t over older values. Rule 1 defines the operator in terms of the weighting function and input. Rule 2 specifies that the weighting function is normalized. This is mostly for clarity. Rule 2 could be integrated directly into rules 1 and 3. Rule 3 specifies that the weight function has a derivative which is on average 0 or higher. This ensures that the weighting doesn't favor values closer to b.
To clarify, this is a subset of all possible local average operators. This subset covers the case for real numbers.