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The usual (although not unique) mathematical definition of the
continuous dot product is
 
(11) 
where the bar over f_{1} stands for complex conjugate (in the case of
complexvalued functions.) Applying definition (11) to the
dot product in formula (10) and approximating the integral by
a finite sum on the regular grid N, we arrive at the approximate
equality
 
(12) 
We can consider equation (12) not only as a useful
approximation, but also as an implicit definition of the
regular grid. The grid regularity means that approximation
(12) is possible. According to this definition, the more
regular the grid is, the more accurate is the approximation.
Substituting equality (12) into formulas (10) and
(7) gives us a solution to the interpolation problem. The
solution takes the form of equation (1) with
 
(13) 
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 (1) is the continuous dot product of functions
W (x, n) and f (x). This simple observation follows from the
definition of the adjoint operator and the simple equality
 

 (14) 
where we have assumed that the discrete dot product is defined by the
sum
 
(15) 
Applying the adjoint interpolation operator to the function f,
defined with the help of formula (13), and employing
formulas (7) and (10), we discover that
 

 (16) 
This remarkable result shows that although the forward linear
interpolation is based on approximation (12), the adjoint
interpolation produces an exact value of f (n)! The approximate
nature of equation (13) 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 1233
 
(17) 
Next: Interpolation with Fourier basis
Up: Fomel: Interpolation
Previous: Function basis
Stanford Exploration Project
11/11/1997