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. 
Z^{N} = Z× ... ×Z 
The set of Ndimensional series of integer numbers. 
p = (p_{1}, ..., p_{N}) ∈ Z^{N} 
Ndimensional series of integers. 
u:Z^{N}→R  Function u with arguments from Z^{N} and values from R. 
u(p) = u(p_{1}, ..., p_{N})  The value of the function u for the argument (p_{1}, ..., p_{N}). 
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 ∈ Z^{N}:

series
r = p + q
is defined asr^{n} = p^{n} + q^{n}
for everyn=1,...,N

series
r = p  q
is defined asr^{n} = p^{n}  q^{n}
for everyn=1,...,N

series
r = sup{p, q}
is defines asr^{n} = max{p^{n}, q^{n}}
for everyn=1,...,N

series
r = inf{p, q}
is defined asr^{n} = min{p^{n}, q^{n}}
for everyn=1,...,N

inequality
p ≤ q
means that p^{n}≤ q^{n} for everyn=1,...,N
.
A function u(p)
is called a finite function if there exist series P^{min}, P^{max}∈ Z^{N} such that:
u(p) ≠ 0
implies
P^{min}≤ p ≤ P^{max}.
Operations of convolution and correlation are only defined for finite functions.
Consider functions u, v and series P^{min}, P^{max}Q^{min}, Q^{max}∈ Z^{N} such that:
u(p) ≠ 0
implies P^{min}≤ p ≤ P^{max}
.v(q) ≠ 0
implies Q^{min}≤ q ≤ Q^{max}.
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 R^{min}≤ r ≤ R^{max}
,
where R^{min} = P^{min} + Q^{min}
and R^{max} = P^{max} + Q^{max}.
If R^{min}≤ r ≤ R^{max}
, then:
w(r) = ∑u(t)·v(r−t)
is the sum for all t ∈ Z^{N} such that T^{min}≤ t ≤ T^{max}
,
where T^{min} = sup{P^{min}, r − Q^{max}}
and T^{max} = inf{P^{max}, r − Q^{min}}.
Linear Correlation
If function w = u • v is the correlation of u and v, then:
w(r) ≠ 0
implies R^{min}≤ r ≤ R^{max}
,
where R^{min} = Q^{min}  P^{max}
and R^{max} = Q^{max}  P^{min}
.
If R^{min}≤ r ≤ R^{max}
, then:
w(r) = ∑u(t)·v(r+t)
is the sum for all t ∈ Z^{N}
such that T^{min}≤ t ≤ T^{max}
,
where T^{min} = sup{P^{min}, Q^{min}− r}
and T^{max} = inf{P^{max}, Q^{max}− 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.