Half-operatorisation of a function and physical applications

We define an half-operatorisation of a function by using its Fourier series such that the Fourier series basis changes into exponential phase operators with the Fourier coefficients remaining intact. We define and discuss the half-operator version of various Fourier Series properties. One application is the master field describing the large N limits of matrix models. We also define an half-operatorisation using gauge equivalent exponential phase operators which have a shifted vacuum.