# Mathematical Notation and Definitions

The following notation is necessary to explain the underlying mathematical definitions used in the text:

`R = (-∞, +∞)` | The set of real numbers.
`Z = {0, ±1, ±2, ...}` | The set of integer numbers.
`ZN = Z× ... ×Z` | The set of N-dimensional series of integer numbers.
`p = (p1, ..., pN) ∈ ZN` | N-dimensional series of integers.
u:ZN→R | Function u with arguments from ZN and values from R.
u(p) = u(p1, ..., pN) | The value of the function u for the argument (p1, ..., pN).
`w = u*v` | Function w is the convolution of the functions u, v.
`w = u•v` | Function w is the correlation of the functions u, v.

Given series p, q ∈ ZN:

- series `r = p + q` is defined as `rn = pn + qn` for every `n=1,...,N`

- series `r = p - q` is defined as `rn = pn - qn` for every `n=1,...,N`

- series `r = sup{p, q}` is defines as `rn = max{pn, qn}` for every `n=1,...,N`

- series `r = inf{p, q}` is defined as `rn = min{pn, qn}` for every `n=1,...,N`

- inequality `p ≤ q` means that pn ≤ qn for every `n=1,...,N`.

A function `u(p)` is called a finite function if there exist series Pmin, Pmax ∈ ZN such that:

`u(p) ≠ 0` implies
`Pmin ≤ p ≤ Pmax.`

Operations of convolution and correlation are only defined for finite functions.

Consider functions u, v and series Pmin, PmaxQmin, Qmax ∈ ZN such that:

`u(p) ≠ 0` implies `Pmin ≤ p ≤ Pmax`.
`v(q) ≠ 0` implies `Qmin ≤ q ≤ Qmax.`

Definitions of linear correlation and linear convolution for functions u and v are given below.

## Linear Convolution

If function `w = u*v` is the convolution of u and v, then:

`w(r) ≠ 0` implies `Rmin ≤ r ≤ Rmax`,
where `Rmin = Pmin + Qmin` and `Rmax = Pmax + Qmax.`

If `Rmin ≤ r ≤ Rmax`, then:

w(r) = `∑u(t)·v(r−t)` is the sum for all t ∈ ZN such that `Tmin ≤ t ≤ Tmax`,
where `Tmin = sup{Pmin, r − Qmax}` and `Tmax = inf{Pmax, r − Qmin}.`

## Linear Correlation

If function w = u • v is the correlation of u and v, then:

`w(r) ≠ 0` implies `Rmin ≤ r ≤ Rmax`,
where `Rmin = Qmin - Pmax` and `Rmax = Qmax - Pmin`.

If `Rmin ≤ r ≤ Rmax`, then:

`w(r) = ∑u(t)·v(r+t)` is the sum for all `t ∈ ZN` such that `Tmin ≤ t ≤ Tmax`,
where `Tmin = sup{Pmin, Qmin − r}` and `Tmax = inf{Pmax, Qmax − r}`.

Representation of the functions u, v, w as the input/output data for the Intel MKL convolution and correlation functions is described in the Data Allocation.

Parent topic: Convolution and Correlation