Indicator functionIn mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset of some set X, then if and otherwise, where is a common notation for the indicator function. Other common notations are and The indicator function of A is the Iverson bracket of the property of belonging to A; that is, For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers.
Locally integrable functionIn mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.
Kronecker deltaIn mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets: For example, because , whereas because . The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.
Mehler kernelThe Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator. defined a function and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials H(.) based on weight function exp(−x2) as This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis. In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel.
Simple functionIn the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are.
