derivatiivid
Derivatiivid is a hypothetical mathematical construct described as a generalized derivative that blends local differentiation with a short-range nonlocal averaging. It is not part of standard calculus or established literature, but appears in speculative or exploratory discussions about nonlocal operators and their effects in analysis and signal processing. The concept serves as a toy model to study how small-scale averaging can interact with pointwise differentiation.
Formally, fix a small scale ε > 0 and a real parameter λ. Let Aε be a local averaging
Dε,λ f(t) = f′(t) + λ Aε f(t).
Thus derivatiivid combines the ordinary derivative with a nonlocal contribution given by the average of f
Properties and interpretation: Dε,λ reduces to the ordinary derivative when λ = 0. For fixed ε, as λ increases, the
Applications are primarily theoretical and pedagogical, helping researchers explore how nonlocal averaging interacts with differentiation and