In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing, a frame provides a redundant, stable way of representing a signal. Frames are used in error detection and correction and the design and analysis of filter banks and more generally in applied mathematics, computer science, and engineering. Definition and motivation Motivating example: computing a basis from a linearly dependent set Suppose we have a set of vectors in the vector space V and we want to express an arbitrary element as a linear combination of the vectors , that is, we want to find coefficients such that If the set does not span , then such coefficients do not exist for every such . If spans and also is linearly independent, this set forms a basis of , and the coefficients are uniquely determined by . If, however, spans but is not linearly independent, the question of how to determine the coefficients becomes less apparent, in particular if is of infinite dimension. Given that spans and is linearly dependent, one strategy is to remove vectors from the set until it becomes linearly independent and forms a basis. There are some problems with this plan: Removing arbitrary vectors from the set may cause it to be unable to span before it becomes linearly independent. Even if it is possible to devise a specific way to remove vectors from the set until it becomes a basis, this approach may become unfeasible in practice if the set is large or infinite. In some applications, it may be an advantage to use more vectors than necessary to represent . This means that we want to find the coefficients without removing elements in . The coefficients will no longer be uniquely determined by . Therefore, the vector can be represented as a linear combination of in more than one way. Formal definition Let V be an inner product space and be a set of vectors in . These vectors satisfy the frame condition if there are positive real numbers A and B such that and for each in V, A set of vectors that satisfies the frame condition is a frame for the vector space. The numbers A and B are called the lower and upper frame bounds, respectively. The frame bounds