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The usual (although not unique) mathematical definition of the
continuous dot product is
 
(38) 
where the bar over f_{1} stands for complex conjugate (in the case of
complexvalued functions). Applying definition () to the
dot product in equation () and approximating the integral
by a finite sum on the regular grid N, we arrive at the approximate
equality
 
(39) 
We can consider equation () not only as a useful
approximation, but also as an implicit definition of the
regular grid. Grid regularity means that approximation ()
is possible. According to this definition, the more regular the grid
is, the more accurate is the approximation.
Substituting equality () into equations ()
and () yields a solution to the interpolation problem.
The solution takes the form of equation () with
 
(40) 
We have found a constructive way of creating the linear
interpolation operator from a specified set of basis functions.
It is important to note that the adjoint of the linear operator in
formula () is the continuous dot product of the
functions W (x, n) and f (x). This simple observation follows from
the definition of the adjoint operator and the simple equality
 

 (41) 
In the final equality, we have assumed that the discrete dot product
is defined by the sum
 
(42) 
Applying the adjoint interpolation operator to the function f,
defined with the help of formula (), and employing
formulas () and (), we discover that
 

 (43) 
This remarkable result shows that although the forward linear
interpolation is based on approximation (), the adjoint
interpolation produces an exact value of f (n)! The approximate
nature of equation () reflects the fundamental
difference between adjoint and inverse linear operators
Claerbout (1992).
When adjoint interpolation is applied to a constant function , it is natural to require the constant output f (n) = 1.
This requirement leads to yet another general property of the
interpolation functions W (x,n):
Property 9380
 
(44) 
The functional basis approach to interpolation is well developed in
the sampling theory Garcia (2000). Some classic examples are discussed
in the next section.
Next: Interpolation with Fourier basis
Up: Forward interpolation
Previous: Function basis
Stanford Exploration Project
12/28/2000