John Pryce wrote:
Nate
Checking your quadratic convolution kernel example as best I can, I agree
that with xx=[1/5,1] you get h(xx) = ([-1/4,7/8],D3). The given h(x) has
continuous first derivative on the whole line, so D3 is telling the
truth.
But suppose thanks to a typo, my h(x) is discontinuous: say, I mistakenly
write 9/4 for 9/8 in the formula. I STILL get decoration D3 on the answer
don't I? What is your D3 telling us? Or is the onus entirely on the
programmer to get the code right?
I've pondered that as well. These are my thoughts:
Two facts to keep in mind: a) the function h is piecewise-defined and b)
C(f,X) only tells us the truth (by definition) about the restriction of
f to
X over those pieces. So careful scrutiny to the definitions I believe shows
that decoration D3 is still the truth. In other words, no claim is being
made by the decoration about the global continuity of h.