# Highly Nonlinear Approximations for Sparse Signal Representation

## Linear operators and linear functionals

Let and be vectors spaces. A mapping is a linear operator if

for all and . is called the domain of and its codomain or image. If the codomain of a linear operator is a scalar field, the operator is called a linear functional on . The set of all linear functionals on is called the dual space of .

The adjoint of an operator is the unique operator satisfying that

If the operator is self-adjoint or Hermitian

An operator has an inverse if there exists such that

and

where and denote the identity operators in and , respectively. By a generalised inverse we shall mean an operator satisfying the following conditions

The unique generalized inverse satisfying

is known as the Moore Penrose pseudoinverse.