Orthonormal Sets in C[-Pi,Pi]
Orthonormal Sets in C[-Pi,Pi]
In[3]:=
Clear[u,v]
In[4]:=
u[x_]:=Table[Cos[k x],{k,20}]
In[5]:=
ColumnForm[u[x]]
Out[5]=
Cos[x] Cos[2 x] Cos[3 x] Cos[4 x] Cos[5 x] Cos[6 x] Cos[7 x] Cos[8 x] Cos[9 x] Cos[10 x] Cos[11 x] Cos[12 x] Cos[13 x] Cos[14 x] Cos[15 x] Cos[16 x] Cos[17 x] Cos[18 x] Cos[19 x] Cos[20 x]
In[6]:=
v[x_]:=Table[Sin[k x],{k,20}]
In[7]:=
ColumnForm[v[x]]
Out[7]=
Sin[x] Sin[2 x] Sin[3 x] Sin[4 x] Sin[5 x] Sin[6 x] Sin[7 x] Sin[8 x] Sin[9 x] Sin[10 x] Sin[11 x] Sin[12 x] Sin[13 x] Sin[14 x] Sin[15 x] Sin[16 x] Sin[17 x] Sin[18 x] Sin[19 x] Sin[20 x]
To get a particular function, use u[x][[i]]
In[8]:=
u[x][[4]]
v[x][[6]]
Out[8]=
Cos[4 x]
Out[9]=
Sin[6 x]
The norms of these functions can be computed. Remember, the norm is defined by the integral. For example, norm[u[[1]]] is really the square root of the Integral from -Pi to Pi of u[[1]] u[[1]].
In[10]:=
Sqrt[Integrate[Cos[x]Cos[x],{x,-Pi,Pi}]]
Out[10]=
Sqrt[Pi]
In[11]:=
nu:=Table[norm[u[x][[i]]],{i,5}]
In[12]:=
ColumnForm[nu]
Out[12]=
Sqrt[Pi] Sqrt[Pi] Sqrt[Pi] Sqrt[Pi] Sqrt[Pi]
In[13]:=
nv:=Table[norm[v[x][[i]]],{i,5}]
In[14]:=
ColumnForm[nv]
Out[14]=
Sqrt[Pi] Sqrt[Pi] Sqrt[Pi] Sqrt[Pi] Sqrt[Pi]
Well. . .
In[15]:=
Clear[d,e]
In[16]:=
e[x_]:=u[x]/Sqrt[Pi]
In[17]:=
d[x_]:=v[x]/Sqrt[Pi]
Not only are these new sequences of functions normal, but they are orthogonal as well, (hence orthonormal). The following computations should convince you of this. Remember, we are computing the integrals from -Pi to Pi of (1/Sqrt[Pi])cos[kx](1/Sqrt[Pi])cos[mx] or (1/Sqrt[Pi])cos[kx](1/Sqrt[Pi])sin[mx] or (1/Sqrt[Pi])sin[kx](1/Sqrt[Pi])sin[mx].
In[18]:=
Integrate[(1/Pi)Cos[x]Cos[2x],{x,-Pi,Pi}]
Out[18]=
0
In[19]:=
Table[innerproduct[e[x][[i]],e[x][[j]],x],{i,5},{j,5}];
MatrixForm[%]
Out[19]=
1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
In[20]:=
Table[innerproduct[d[x][[i]],d[x][[j]],x],{i,5},{j,5}];
MatrixForm[%]
Out[20]=
1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
In[21]:=
Table[innerproduct[e[x][[i]],d[x][[j]],x],{i,5},{j,5}];
MatrixForm[%]
Out[21]=
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Up to Fourier Series