In this paper, we introduce q-calculus into the theory of fractal and classical splines by constructing a \({\textbf{q}}\) -fractal cubic Hermite interpolant using q-differentiation. Under suitable conditions, the proposed interpolant provides a q-analogue of the classical cubic Hermite interpolant. It also offers greater flexibility than both the \(\mathcal {C}^1\) -fractal cubic Hermite interpolant (fractal cubic Hermite interpolant) and the classical cubic Hermite interpolant. We establish the existence of a \({\textbf{q}}\) -fractal cubic Hermite interpolant whose graph lies inside a prescribed rectangle. By treating the first-order q-derivatives at the knots as free parameters, we derive conditions under which the \({\textbf{q}}\) -fractal cubic Hermite interpolant yields a \({\textbf{q}}\) -fractal cubic spline. The classical \(\mathcal {C}^2\) -cubic spline arises as a special case of the proposed \({\textbf{q}}\) -fractal cubic spline. We also establish a minimization property for the \({\textbf{q}}\) -fractal cubic spline using q-integration. Furthermore, under suitable assumptions on the original function, we analyze the convergence of the \({\textbf{q}}\) -fractal cubic spline by deriving a priori error estimates in the \(L_p\) -norm, for \(p \ge 2.\) Numerical examples illustrate the advantages of the proposed \({\textbf{q}}\) -fractal cubic Hermite interpolant and \({\textbf{q}}\) -fractal cubic spline over their existing counterparts.
