Solution

The usual (although not unique) mathematical definition of the continuous dot product is  

$$(f_1, f_2) = \int \bar{f}_1 (x) f_2 (x) dx \;,$$ (9)

where the bar over f₁ stands for complex conjugate (in the case of complex-valued 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  

$$(\psi_j (x), f (x)) = \int \bar{\psi}_j (x) f (x) dx \approx \sum_{n \in N} \bar{\psi}_j (n) f (n)\;.$$ (10)

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  

$$W (x, n) = \sum_{k \in K} \sum_{j \in K} \Psi^{-1}_{kj} \psi_k (x) \bar{\psi}_j (n)\;.$$ (11)

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

$$\begin{eqnarray} \left(f_1 (x), \sum_{n \in N} W (x, n) f_2 (n)\right) = \sum_{... ...mber \ \left(\left(W (x, n), f_1 (x)\right), f_2 (n) \right) \;,\end{eqnarray}$$ (12)

where we have assumed that the discrete dot product is defined by the sum  

$$(f_1 (n), f_2 (n)) = \sum_{n \in N} \bar{f}_1 (n) f_2 (n) \;.$$ (13)

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

$$\begin{eqnarray} \left(W (x, n), f (x)\right) = \sum_{k \in K} \sum_{j \in K} ... ...si_k (x), f (x)\right) = \sum_{j \in K} c_j \psi_j (n) = f (n)\;.\end{eqnarray}$$ (14)

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 $f (x) \equiv 1$, 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):