A Foundational Analysis of Local Kernel-Based Calculus

We introduce the local kernel-based calculus, a unifying framework for local differential and integral operators based on an arbitrary positive continuous kernel function. This framework encompasses conformable, non-conformable, and our newly introduced local Euler-kernel derivatives as special cases. The parameter of the kernel is unrestricted and may take negative values, reflecting its role as a genuine parameter rather than an order of fractional differentiation. Within this general setting, we rigorously prove a complete set of foundational theorems: linearity, the product rule, continuity, Rolle’s theorem, the mean value theorem, and the fundamental theorem of calculus via the associated integral operator. We also derive a new formulation of the chain rule that expresses the chain rule entirely in terms of the kernel-based derivatives. While algebraically equivalent to the classical form, this representation preserves the intuitive structure of the chain rule without reference to the classical derivative. We further establish the Fundamental Theorem of Local Euler Calculus and its generalization, the Fundamental Theorem of Local Kernel-Based Calculus, confirming that the derivative and integral operators are genuine inverses, with the classical fundamental theorem recovered as special cases when the kernel reduces to unity. As an important illustration, we develop the local Euler calculus with the exponential kernel in full detail, providing explicit derivative and integral formulas for elementary functions. This special case demonstrates the simplicity and power of the functional approach. Overall, the local kernel-based calculus provides a solid, self-contained foundation that unifies a wide class of local operators and extends far beyond the traditional setting.