In[]:=
A={3,7,0,0}​​B={-1,2,1,0}​​A1=2FourierMatrix[4].A​​B1=2FourierMatrix[4].B
Out[]=
{3,7,0,0}
Out[]=
{-1,2,1,0}
Out[]=
{10,3+7,-4,3-7}
Out[]=
{2,-2+2,-2,-2-2}
In[]:=
.5InverseFourier[A1*B1]
Out[]=
{-3.,-1.,17.,7.}
In[]:=
A2={3,7,0,1,0,0,0,0}​​B2={-1,2,1,0,0,0,0,0}​​A3=sqrt[8]FourierMatrix[8].A2​​B3=sqrt[8]FourierMatrix[8].B2
Out[]=
{3,7,0,1,0,0,0,0}
Out[]=
{-1,2,1,0,0,0,0,0}
Out[]=

11sqrt[8]
2
2
,
3
2
2
+
7
π
4

2
2
+
3π
4

2
2
sqrt[8],

3
2
+3sqrt[8]
2
,
3
2
2
+
π
4

2
2
+
7
3π
4

2
2
sqrt[8],-
5sqrt[8]
2
2
,
3
2
2
+
-
π
4

2
2
+
7
-
3π
4

2
2
sqrt[8],

3
2
-3sqrt[8]
2
,
3
2
2
+
7
-
π
4

2
2
+
-
3π
4

2
2
sqrt[8]
Out[]=

sqrt[8]
2
,-
1
2
-

2
2
+
π
4

2
sqrt[8],-
(1-)sqrt[8]
2
,-
1
2
+

2
2
+
3π
4

2
sqrt[8],-
sqrt[8]
2
,-
1
2
-

2
2
+
-
3π
4

2
sqrt[8],-
(1+)sqrt[8]
2
,-
1
2
+

2
2
+
-
π
4

2
sqrt[8]
In[]:=
1/8InverseFourier[A3*B3]
InverseFourier
:Argument 
11
2
sqrt[8]
4
,-
1
2
-

2
2
+
π
4

2
3
2
2
+
7
π
4

2
2
+
3π
4

2
2
2
sqrt[8]
,-
9
4
-
3
4
2
sqrt[8]
,-
1
2
+

2
2
+
3π
4

2
3
2
2
+
1
4

2
2
+
7
1
4

2
2
2
sqrt[8]
,
5
2
1
4
,-
1
2
-

2
2
+
-
3π
4

2
3
2
2
+
-
π
4

2
2
+
7
-
3π
4

2
2
2
sqrt[8]
,-
9
4
+
3
4
2
sqrt[8]
,-
1
2
+

2
2
+
-
π
4

2
3
2
2
+
7
-
π
4

2
2
+
-
3π
4

2
2
2
sqrt[8]
 is not a non-empty list or rectangular array of numeric quantities.
Out[]=
1
8
InverseFourier
11
2
sqrt[8]
4
,-
1
2
-

2
2
+
π
4

2
3
2
2
+
7
π
4

2
2
+
3π
4

2
2
2
sqrt[8]
,-
9
4
-
3
4
2
sqrt[8]
,-
1
2
+

2
2
+
3π
4

2
3
2
2
+
π
4

2
2
+
7
3π
4

2
2
2
sqrt[8]
,
5
2
sqrt[8]
4
,-
1
2
-

2
2
+
-
3π
4

2
3
2
2
+
-
π
4

2
2
+
7
-
3π
4

2
2
2
sqrt[8]
,-
9
4
+
3
4
2
sqrt[8]
,-
1
2
+

2
2
+
-
π
4

2
3
2
2
+
7
-
π
4

2
2
+
-
3π
4

2
2
2
sqrt[8]
