Abstract:   

   A frame, and more specifically, a tight frame is a generalization of the concept of an orthonormal basis in a separable Hilbert space. Frames have been used quite extensively in recent years in the study of wavelet and Gabor systems. Of course, in most situations, the interest lies in finding frames of a special type, such as those generated by the action of one or several unitary operators on a particular element of the Hilbert space. In these problems, it could happen that no orthonormal basis of the type we are interested in can be constructed while frames or tight frames of that same type do exist. The main purpose of this talk is to discuss the construction of such tight frames in concrete Hilbert spaces. The examples discussed will include certain Hilbert spaces of functions or distributions naturally associated with a positive-definite distribution on a symmetric interval (-R,R), R>0. The concept of representing measure arises here when the extension problem is considered. We will also discuss recent results concerning irregular Gabor systems for L^2(R^n) in which the problem of finding associated tight frames leads again to the construction of certain representing measures in the time-frequency space.