Orthogonal functions

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, orthogonal functions belong to a <a href="/facts/Function_space/lLUE2R1t">function space</a> that is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> equipped with a <a href="/facts/Bilinear_form/gyvzn9ym">bilinear form</a>. When the function space has an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> as the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>, the bilinear form may be the <a href="/facts/Integral/V7lyV997">integral</a> of the product of functions over the interval:

⟨
        f
        ,
        g
        ⟩
        =
        ∫
        
          
            
              f
              (
              x
              )
            
            ¯
          
        
        g
        (
        x
        )
        
        d
        x
        .
      
    
    {\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.}

The functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
 are <a href="/facts/Orthogonality_(mathematics)/eKmrl6oQ">orthogonal</a> when this integral is zero, i.e. 
 
 
 
 ⟨
 f
 ,
 
 g
 ⟩
 =
 0
 
 
 {\displaystyle \langle f,\,g\rangle =0}
 
 whenever 
 
 
 
 f
 ≠
 g
 
 
 {\displaystyle f\neq g}
 
. As with a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector <a href="/facts/Dot_product/tNz8MLjT">dot product</a>; two vectors are mutually independent (orthogonal) if their dot-product is zero.
Suppose 
 
 
 
 {
 
 f
 
 0
 
 
 ,
 
 f
 
 1
 
 
 ,
 …
 }
 
 
 {\displaystyle \{f_{0},f_{1},\ldots \}}
 
 is a sequence of orthogonal functions of nonzero <a href="/facts/L2-norm/xIbR4uE1">L2-norms</a> 
 
 
 
 
 
 ‖
 
 f
 
 n
 
 
 ‖
 
 
 2
 
 
 =
 
 
 ⟨
 
 f
 
 n
 
 
 ,
 
 f
 
 n
 
 
 ⟩
 
 
 =
 
 
 (
 
 ∫
 
 f
 
 n
 
 
 2
 
 
  
 d
 x
 
 )
 
 
 
 1
 2
 
 
 
 
 
 {\textstyle \left\|f_{n}\right\|_{2}={\sqrt {\langle f_{n},f_{n}\rangle }}=\left(\int f_{n}^{2}\ dx\right)^{\frac {1}{2}}}
 
. It follows that the sequence 
 
 
 
 
 {
 
 
 f
 
 n
 
 
 
 /
 
 
 
 ‖
 
 f
 
 n
 
 
 ‖
 
 
 2
 
 
 
 }
 
 
 
 {\displaystyle \left\{f_{n}/\left\|f_{n}\right\|_{2}\right\}}
 
 is of functions of L2-norm one, forming an <a href="/facts/Orthonormal_sequence/JfgL1nAh">orthonormal sequence</a>. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being <a href="/facts/Square-integrable_function/IASblcJJ">square-integrable</a>.

Orthogonal functions open-in-new

Orthogonal functions