Convolution with a sinc gives uniform approximation to a function

[Fernsanz]

Hi everybody.
Some students have asked me about problem 2.13 in Mallat's book "A wavelet Tour of Signal Proccessing". After some work on it, I think is not completely correct. I think some hypostesis on modulus of continuity are needed.
I attach the statement.

Esentially, what it says, is that the convolution of a continuous and bounded variation function with a sinc converges UNIFORMLY to the function. I would need a kind of uniform Riemann-Lebesgue lemma to achieve that conclussion.
It is a relevant problem cause it could justify why we can truncate the spectrum of a signal and recover something not too diferent from that truncate spectra for L2 signals.
Let me know what do you think.
Thanks

 

[ericmachorro]

You've probably had this addressed by now - but the bounded variation assumption that Mallat should provide you with the modulus of continuity that you need, because it says the difference in two values of the function are bounded.

The result is interesting for a deeper reason too ... it says that the sinc function converges to the delta function.