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Definition of operators

Mathematically speaking an operator is a function of a function, i.e. a rule (or mapping) according to which a function $ f$ is transformed into another function $ g$ . We use the notation $ g=R[f]$ or simply $ g=Rf$ , where $ R$ denotes the operator. Examples of operators are the derivative, the integral, convolution (with a specific function), multiplication by a scalar and others. Note that in general the domains of $ f$ and $ g$ are not necessarily the same. For example, in the case of the derivative, the domain of $ g=Rf$ is the subset of the domain of $ f$ , in which $ f$ is smooth. In particular if $ f=\vert x\vert$ , $ x\in[-1,1]$ , then the domain of $ g$ is $ (-1,0)\cup(0,1)$ .

An important class of operators are the linear operators. An operator $ L$ is linear if for any two functions $ f_1$ , $ f_2$ and any two scalars $ a_1$ , $ a_2$ , $ L[a_1f_1+a_1f_2]=a_1Lf_1 + a_2Lf_2$ . The derivative, integral, convolution and multiplication by scalar are all linear operators.

In the discrete world, operators act on vectors and linear operators are in fact matrices, with which the vectors are multiplied. (Multiplication by a matrix is a linear operation, since $ \mathbf{M}(a_1\mathbf{x}_1+a_2\mathbf{x}_2) = a_1\mathbf{M}\mathbf{x}_1+a_2\mathbf{M}\mathbf{x}_2$ ). In fact many of the calculations performed routinely in science and engineering are essentially matrix multiplications in disguise. For example assume a vector $ \mathbf{x}=[x_1\;x_2\;\ldots\;x_n]^T$ with length $ n$ (superscript $ {}^T$ denotes transpose). Padding this vector with $ m$ zeros, produces another vector $ \mathbf{y}$ with

where $ \mathbf{0}$ is the zero vector of length $ m$ . One can readily verify that zero padding is a linear operation with operator matrix $ \mathbf{L}=\begin{bmatrix}\mathbf{I} \mathbf{O}\end{bmatrix}$ , where $ \mathbf{I}$ is the $ n\times n$ identity matrix and $ \mathbf{O}$ is the $ m\times n$ zero matrix, since

$\displaystyle \mathbf{y} = \begin{bmatrix}x_1 x_2 \vdots  x_n 0 \vdot...
...}\end{bmatrix}\mathbf{x} = \begin{bmatrix}\mathbf{x} \mathbf{0}\end{bmatrix}.$    

Note that as in the case of functions, the domains of $ \mathbf{x}$ and $ \mathbf{y}$ are different: $ \mathbf{x}\in\mathbb{R}^n$ (or more generally $ \mathbf{x}\in\mathbb{C}^n$ ), while $ \mathbf{y}\in\mathbb{R}^{n+m}$ (or $ \mathbb{C}^{n+m}$ ).

Similarly, one can define convolution of $ \mathbf{x}$ with $ \mathbf{a}=[a_1\;a_2\;\dots\;a_m]^T$ as the multiplication of $ \mathbf{x}$ with

$\displaystyle \mathbf{A} = \begin{bmatrix}a_1 & 0 & 0 & \cdots & 0 & 0 a_2 & ...
...& 0 & 0 & \cdots & a_m & a_{m-1}  0 & 0 & 0 & \cdots & 0 & a_m \end{bmatrix}.$    

and many other operations as matrix multiplications. Other operators are the identity operator is the identity matrix $ \mathbf{I}$ and is implemented by [sec:copy]copy.c and [sec:ccopy]ccopy.c and the null operator (or zero matrix $ \mathbf{O}$ ), which is implemented by [sec:adjnull]adjnull.c. For the rest of this introduction, the boldface notation will imply specifically discrete operators, while the normal fonts will imply operators on either continuous or discrete mathematical entities.

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