Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given $f\in C^{w}[0,1]$ with periodic conditions $f(0)^{(j)}=f(1)^{(j)},\ j=0,\dots, w-1$ and its Fourier series are $f(x)=\sum_{l}f_{i}\exp(2\pi ix)$.

I need to find the asymptotic order of errors $$\sum_{\left|l\right|>K/2}f_{l}\exp(2\pi i lx)\mbox{ and }K^{-4q}\sum_{l=0}^{K/2}f_{l-K}l{}^{4q}\exp\left(2\pi i lt\right)\mbox{ as }K\rightarrow \infty.$$

Is it enough to say that $\left|f_{i}\right|=O(K^{-w})$ (from integration by parts) and replace the summation by integration to show that this order is $O(K^{1-w})$?

Or when the orders of corresponding sum and integral are the same?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.