# Thread: L1-norm approximation

1. ## L1-norm approximation

Dear friends,

Do you think it is possible to approximate the L1-norm of a finite discrete sequence {x[n]} (i.e., ||x|| = sum(|x[n]|) ) given the values of its discrete Fourier transform? If not, is there a theorem asserting that, in this case, it is impossible to arrive at a "Parseval-like" (approximative) relation?

Thank you very much in advance for your responses.

2. If I remember right, the fourier transform of an L^1 sequence, need not belong in L^1. So the norm would be meaningless

3. Originally Posted by Rebesques
If I remember right, the fourier transform of an L^1 sequence, need not belong in L^1. So the norm would be meaningless
The sequence is finite, so we are dealing with the Discrete FT here.
As the DFT of a bounded finite sequence is bounded it is of necessity
L^1.

RonL

4. Yes, this is the case. For simplicity (and in order to avoid the "definiteness issue") I assumed the sequence to be finite. In this case, its Fourier transform is bounded and hence well-defined. Moreover, if the norm in question were the L2 norm, the Parseval theorem guarantees that the L2 norm of the sequence is equal to that of its (discrete) Fourier coefficients. However, it is possible to figure out (even approximately) what is the L1 norm of the sequence given only the Fourier coefficients of the latter?

Thank you very much for your interest and responses.

5. Originally Posted by Neighbor
Yes, this is the case. For simplicity (and in order to avoid the "definiteness issue") I assumed the sequence to be finite. In this case, its Fourier transform is bounded and hence well-defined. Moreover, if the norm in question were the L2 norm, the Parseval theorem guarantees that the L2 norm of the sequence is equal to that of its (discrete) Fourier coefficients. However, it is possible to figure out (even approximately) what is the L1 norm of the sequence given only the Fourier coefficients of the latter?
In principle it is. This is because the DFT of a finite sequence contains all the information
embodied in the original sequence, and infact the inverse DFT will reconstruct the
orginal exactly.

However I'm not aware of any simple elegant equivalent of Parseval's
theorem for the L1 norm, but that does not mean that there isn't one

RonL