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 Oct10 comment second-order divided differenc Thanks a lot, doraemonpaul. This is similar to @nikita2's suggestion, isn't it? By the same token $\sum\limits_{i = 0}^{i = 4} {\left(5 sin \left( i \frac{\pi}{2} \right) - 10 \right) \left(5 cos \left( i \frac{\pi}{2} \right) \right)} < 0$ which shows that treating $f'(t)$ in terms of $\delta$-functions is incorrect since it yields 0. Do you agree with that? Oct8 comment second-order divided differenc This seems to me to be the case as well but I'd like to hear objections, if a all, from other mathematicians. If there are none, I'll accept the answer. Oct8 comment second-order divided differenc Under that assumprion, said sum (with $i$ running from 0 to 4) will be negative for a function having the values -5, 0, -5, -10, -5, right? Is the assumption acceptable, though? Sep23 comment Is the following substitution legitimate? Emmad, I quess your reply also answers my question about the seeming similarity with boundary value problems in differential equations. In the boundary value problems the equalities given as boundary conditions are absolute, unlike the case at hand, right? Sep23 comment Is the following substitution legitimate? Thanks, Emmad to you too. Sep23 comment Is the following substitution legitimate? Ted, but the first equation shows that $b_1$ depends on $x$, doesn't it? Therefore, Deven's reply holds. Sep23 comment Is the following substitution legitimate? Thanks a bunch, Deven. This appeared to be something akin to the boundary value problems in differential equations. How would you comment the seeming similarity of the above incorrect substitution to the boundary value problems?