Wavelet Calculus and Connection Coefficients

As we have seen in the previous chapters, one can use a wavelet series to represent a function of one or more variables which can be either an infinite series at the theoretical level or an approximating finite series in an applications context. The role of calculus, computing derivatives and solving differential equations, is an extremely important part of using mathematics to model the natural world. The question naturally arises: If one has a series representation of a given function, how can one efficiently compute derivatives of the function in terms of the wavelet expansion coefficients? More generally, suppose one considers algebraic operations of a nonlinear nature, e.g., the square of a function, can one efficiently represent the function that is the result of the nonlinear operation in terms of the wavelet coefficients of the original function? The answer to these and similar questions is given by the concept of connection coefficients. Before we give a general definition of this new concept, we will illustrate it with a special and informative case, which will lead naturally to the general case.