Multivariate Fractal Interpolation Functions: Some Approximation Aspects and an Associated Fractal Interpolation Operator

The paper proves existence and uniqueness of a multivariate fractal interpolation function f̃ on D=∏ Ik under: (i) boundary/vertex consistency via the parity map τ, (ii) Lipschitz-in-y contractivity of Fi1…in, and (iii) a matching condition across shared faces. It does so by defining the Read–Bajraktarević operator T on a closed subspace C* of C(D) that fixes corner data, proving T is well defined and contractive, then identifying f̃ as its unique fixed point. It also shows the graph invariance G=⋃ Wi(G) directly from the fixed-point equation and verifies interpolation at all grid nodes via τ and the corner constraints (2.5)–(2.6) together with (2.8) (see the definition of τ in (2.4), the boundary condition (2.5), the Lipschitz condition (2.6), the matching condition (2.8), and Steps I–IV establishing T’s properties and G’s invariance ). The candidate solution follows the same core structure: constructs T with the same piecewise definition, proves well-definedness via the matching condition, shows contractivity via the Lipschitz-in-y bound, derives graph invariance from the fixed-point identity, and uses τ to establish interpolation. A small difference is that the paper restricts T to C* from the outset, whereas the candidate starts on C(D) and then uses an invariant closed subset S to enforce corner interpolation before propagating to all grid nodes via invariance; both routes are standard and correct. Thus, both arguments are correct and essentially the same in substance.