Highly Nonlinear Approximations for Sparse Signal Representation

Frames and bases for finite a dimension vector space

If are some elements of a vector space , by a linear combination of we mean an element in of the form , with .

Let be a subset of element of . The set of all linear combinations of elements of is called the span of and is denoted by .

A subset of is said to be linearly independent if and only if

A subset is said to be linearly dependent if it is not linearly independent.

is said to be a basis of if it is linearly independent and . The dimension of a finite dimensional vector space is the number of elements in a basis for . The number of elements in a set is termed the cardinality of the set.

If the number of elements spanning a finite dimensional vector space is larger than the number of elements of a basis for the same space, the set is called a redundant frame. In other words a redundant frame (henceforth called simply a frame) for a finite dimensional vector space is a linearly dependent set of vectors spanning .

Let be a spanning set for . Then every can be expressed as

with

If the spanning set is a basis for the numbers in the above decomposition are unique. In the case of a redundant frame, however, these numbers are not longer unique. For further discussion about redundant frames in finite dimension see [1].