Highly Nonlinear Approximations for Sparse Signal Representation
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
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