Ricerche mat. DOI 10.1007/s11587-017-0340-1 A class of universal approximators of real continuous functions revisited Constantinos Siettos 1 · Francesco Giannino 2 · Lucia Russo 3 · Salvatore Cuomo 4 Received: 23 June 2017 / Revised: 30 September 2017 © Università degli Studi di Napoli "Federico II" 2017 Abstract We revisit the theorem stating that it is possible to approximate with any accuracy any real continuous function with a class of relational maps. In other words, relational maps are universal approximators. We review the key works that have proved this property, highlighting their limitations and providing yet another proof that it is not restricted by certain assumptions considered in early proofs. We also show how one can go inversely to approximate these systems with a series of polynomials. This provides us with analytical expressions of these maps which can facilitate a series of important analysis tasks such as modeling and numerical analysis of ill-defined- uncertain complex systems. Keywords Approximation of continuous nonlinear functions · Nonlinear systems · Relational maps · Polynomial series approximation · Numerical analysis Mathematics Subject Classification 93C10 · 41A10 · 37Mxx · 94D05 B Constantinos Siettos ksiet@mail.ntua.gr B Francesco Giannino Giannino@unina.it 1 School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Athens, Greece 2 Dipartimento di Agraria, Università di Napoli Federico II, Naples, Italy 3 Consiglio Nazionale delle Ricerche, Istituto di Ricerche sulla Combustione, Naples, Italy 4 Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Naples, Italy 123