FFT with modified fourier coeffitients

Dear MKL forum,
I solve such a problem. Can you help me please?
Lets have a function Y=∑ _k=−∞^∞ iY_ne^ikπy and then I have a function which is defined as X=∑_k=−∞^∞ ik²Y_ne^ikπy.

I know the Y. The i is imaginary unit.

How can I compute the X? I think I do the FFT on Y and obtain thus the Y_n, right? And then I think I will do the backward FFT of function defined as f=ik²Y_n. But what have I do with the summation index k here in the function f?

It is right that FFT(ik²Y_n)=X?

I'm not sure absolutely what to do with k when the FFT sum is summated per k. Or can I change something in MKL FFT directly?