Frames in finite-dimensional inner product spaces

Abstract

In the study of vector spaces, one of the most important concepts is that of a basis. A basis provides us with an expansion of all vectors in terms of “elementary building blocks” and hereby helps us by reducing many questions concerning general vectors to similar questions concerning only the basis elements. However, the conditions to a basis are very restrictive – no linear dependence between the elements is possible, and sometimes we even want the elements to be orthogonal with respect to an inner product. This makes it hard or even impossible to find bases satisfying extra conditions, and this is the reason that one might look for a more flexible tool.