Investigation of summability of neural network operators

In this Ph.D. thesis, the approximation properties of neural network operators are examined in all aspects. Artificial neural networks, which have a wide range of applications from engineering to medical and finance to computer technologies, were first addressed in the approximation theory in 1989 by Cybenko and then developed by Cardaliaguet, Euvrard and Anastassiou in 1997. It is known that uniform continuous functions on the whole real axis can be pointwise approximated by means of neural network operators. However, since it is very difficult to calculate the values of neural network operators in test functions analytically, the classical Korovkin Theorem is often not useful. In our thesis, we have firstly modified the neural network operators having linear structure defined with the help of bell-type activation functions in order to obtain uniform approximation to uniform continuous functions. Later, we have used non-negative regular summability methods for situations where the classical approximation fails. As it is known, a summability method is an alternative formulation of convergence of a sequence (or a series) which is divergent in the conventional sense. We have also obtained graphical illustrations by using various mathematical software programs to support our results. We have also expanded our work on the theory of multivariable functions. In the second part of the thesis, we discussed the nonlinear neural network operators with the help of maximum-product operations. We have showed that similar approximation results are also valid for the nonlinear case. As far as we know, the approximation by neural network operators has not been conducted in this direction although the techniques of summability theory have been frequently used in the approximation theory, which gains the originality to our thesis.