Functional Analysis and Applications by Abul Hasan Siddiqi

Author:Abul Hasan Siddiqi
Language: eng
Format: epub, pdf
Publisher: Springer Singapore, Singapore

Thus

where we have a smaller constant as .

Remark 8.1

Inequality (8.10) indicates that the problem of estimating the error is equivalent to a problem in approximation theory, namely, the determination of the distance between a function and a subspace of H. Under appropriate condition on u, one can show that for , where C(u) is independent of h and dependent on u, and so . In this case, we say that the order of convergence is or, equivalently, we have an order , and we write .

We prove below a theorem for the abstract error estimation, called as the first Strang lemma due to Gilbert Strang, [180]. Here, the forms and are not defined on the space H, since the point values are not defined in general for functions in the space .

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