Fernsanz
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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.
![Text of problem 2.13: Let f_ξ = f * h_ξ with ĥ_ξ = 1_[-ξ,ξ]. Suppose f has bounded variation, ||f||∞ < +∞, and is continuous in a neighborhood of t0. Prove f_ξ(t) converges uniformly to f(t) as ξ goes to +∞.](https://www.physicsforums.com/attachments/mallat-jpg.6005/ "mallat.jpg")
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
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