integroidu

Integroidu is a theoretical construct in mathematics and theoretical computer science intended to unify discrete summation and continuous integration. It characterizes a family of generalized integral operators that map functions on a domain to a scalar, depending on a parameter family of measures. In common terms, an integroidu operator I assigns to a function f a value I(f) that reduces to a sum when the domain is finite or countable and to a traditional integral when the domain carries a continuous measure.

Definition and basic properties: Let X be a measure space with a measure mu that can be decomposed into a discrete part and a continuous part. An integroidu operator I acts on suitable function spaces and satisfies linearity, i.e., I(af + bg) = a I(f) + b I(g). It also respects a coupling with convolution: I(f * g) = I(f) * I(g) under appropriate normalization, and it has an inversion property under a duality transform. In practice, I is often represented as I(f) = ∫ f dmu + sum over discrete weights, where mu encodes both components.

Relations and examples: In the purely discrete case, I reduces to a weighted sum ∑ f(n) w_n. In the purely continuous case, I reduces to a Lebesgue integral ∫ f(x) d mu(x). The integroidu framework allows a single formalism to describe hybrid signals and data streams.

Applications: It is proposed in theoretical models of hybrid analysis, signal processing, and type theory for programming languages that manipulate both discrete and continuous data.

History and terminology: The term integroidu blends "integration" with a Latin-root implying a grid or network structure; it appears in speculative literature on unified integral theories.

See also: Integration, summation, Lebesgue integration, Stieltjes integral, convolution, measure theory.