Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be bounded by the second derivative in $C$ norm of the function multiplied by $(x_2-x_1)^2$, i.e. $\sup_{x∈[x1,x2]}f^{′′}(x)(x_2-x_1)^2$ and some multiplying constant. If $f^{''}$ doesn't exist, can I replace the error estimate with $L^{\infty}$ norm for $f^{''}$, provided second weak derivative exist? How can I treat such an estimate? I sort of bounded the error and have it as a second order, but the constant is measured a.s. because of the weak derivative, so I should say that the error estimate holds almost everywhere?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.