Complex Anal. Oper. Theory (2010) 4:953–974 DOI 10.1007/s11785-009-0033-1 Complex Analysis and Operator Theory Fractal Approximation M. A. Navascués Received: 8 June 2009 / Accepted: 8 July 2009 / Published online: 31 July 2009 © Birkhäuser Verlag Basel/Switzerland 2009 Abstract In the present article every complex square integrable function defined in a real bounded interval is approached by means of a complex fractal function. The approximation depends on a partition of the interval and a vectorial parameter of the iterated function system providing the fractal attractor. The original may be discon- tinuous or undefined in a set of zero measure. The fractal elements can modify the features of the originals, for instance their character of smooth or non-smooth. The properties of the operator mapping every function into its fractal analogue are studied in the context of the uniform and least square norms. In particular, the transformation provides a decomposition of the set of square integrable maps. An orthogonal system of fractal functions is constructed explicitly for this space. Sufficient conditions for the uniform convergence of the fractal series expansion corresponding to this basis are also deduced. The fractal approximation of real functions is obtained as a particular case. Keywords Fractal interpolation functions · Least squares approximation · Orthogonal systems Mathematics Subject Classification (2000) Primary 28A80; Secondary 41A10 · 58C05 · 65D05 · 26A27 Communicated by Palle Jorgensen. M. A. Navascués (B ) Centro Politécnico Superior de Ingenieros, Universidad de Zaragoza, C/ María de Luna 3, 50018 Zaragoza, Spain e-mail: manavas@unizar.es