We can not quantize the coordinates of approximation nodes (post-quantization approach), because in such a case we cannot guarantee that a final

approximation error is below the given error bound [[epsilon].sub.j] (Kolesnikov and Akimov, 2007).

For a large class of planar structures, the

approximation error for [A.sub.d] tends to 0 as fast as [([[DELTA].sub.1][[DELTA].sub.2]).sup.3/4].

The Euler Equation error is also important because we know that under certain conditions, the

approximation error of the policy function is of the same order of magnitude as the size of the Euler equation residual and correspondingly the change in welfare is of the square order of the Euler equation residual (Santos, 2000).

with the Fourier

approximation error bound based on standard results (e.g., see Atkinson [1989, p.

The difference between this direct estimate and the second-order approximation reveals the magnitude of the

approximation error. The methods used and our results are reported in the next section of the paper.

Because of the increase in the total number of degrees of freedom, the h-method ultimately reduces the

approximation error to zero.

They are also consequences of Weyl's theorem and yield upper bounds for the

approximation error of similar order as those stated above.

The importance of this observation is that the proposed scheme gives us an intuition on which approximation of all the approximations results in large

approximation error. The approach presented in [14] does not show any intuition on which approximation results in large

approximation error, since in [14] the authors only explicitly consider Taylor series expansion for general problem.

The

approximation error of the Nystrom method is O([m.sup.-1/2]) [11].

Though the larger the number of neural network nodes, the smaller the

approximation error, neural network with a large hidden node number tends to complicate the control and the computation of the control system.

where [W.sup.*] is the ideal weight vector of W and [eta](Z) is any small

approximation error which satisfies [absolute value of ([eta](Z))] [less than or equal to] [[eta].sup.*].