@robjohn: I distinctly remember the peak of the Perseides in the early nineties. We were traveling with an old VW bus through the mountains and arranged it so that we were on mountain passes during the night.
@J.M. : in some neat real world applications, information is present generally as a physical quantity varying with some independent parameter, say for example time (it could be anything).......and people genreally tend to represent it with a function ....i would like know some things regarding this representation
i meant to mention a particular case of real valued functions of a real parameter....say an audio signal as a variation of acoustic pressure with time.......all i am trying is to give present context.....i am not done with it yet
Anyway, I was going to tell a little bit more about my project. The Hardy Space H^1 turns out to be quite useful in the Euclidean theory for example in PDEs. Now many of those results can also be useful in SPDEs if we have a different measure, in this case that is the Gaussian measure.
The Gaussian measure is non-doubling so that messes everything up.
So, I'm trying to figure out what an appropriate Hardy space would be in this case. Certainly we have to replace the Laplacian by something else since that is not symmetric. It turns out that the Ornstein-Uhlenbeck operator works.
@robjohn Yes I know :).
And then we of course want to know if the dual is BMO (or what is BMO in this case?) and if the Riesz transform is bounded and so on.
@t.B : it is because of the perception.....human ears tend to distinguish sounds as generally high frequency or low frequencies....there are more intricacies with them...i am not intending to get into them
@Rajesh: a very rough model would be that we can detect signals depending on their frequencies. So if we have a pure 440 Hz signal there is exactly one detector stimulated, so it makes sense to decompose an audio signal according to its frequencies.
The philosophy with the method of using a mean square error (or the likes of Lp norms) is essentially under the assumption that the information of the signal is present in the shape of the function representing the signal...this is what happening in almost all of our real world applications....barring some which i am not aware of
@Rajesh: if we have a complicated signal and we want to determine the contribution of the pure A sound to that signal, that corresponds to projecting the function representing the signal orthogonally (with respect to the L^2-product) onto the 1-dimensional space generated by the function representing the pure A sound
@t.b. , @J.M. : I'd like to consider a case where the information is present in the signal not just in the form of the shape of the function representing it but also present in the manner in which the function is behaving............it can apply to any signal or any function used in any application........the info the function is carrying is not just in its shape but also in its behaviour.....stating precisely..........shape is not all that matters...the behaviour also matters.........contd..
it also means that. taking smooth approximation to such functions (Stone-Weierstrass approximation) is not going to be good enough..........do you think its a cool idea....especially if i am able to influence real world applications (could be anything including Physics)
ses this class of function........mathoverflow.net/questions/61797/a-class-of-functions-closed : note in the comments by Pietro Majer (in his answer) that the set of rationals can be generalized with any countable dense subset. If we take a smooth approximation to this (stone-Weierstrass) we still loose information about the behaviour of the function which is quite important...........
if a function were to carry any real world information embedded in its behavoiur also, then these class of functions seem to be the best cadidates for me....intuitively (not math in it) i this what i feel
what J.M. said. + I'm sorry but I fail to see why differentiability of a function depending on the denominator of a rational and smoothness at the irrationals should in any way be tied to the real world
Above all, our theory explains why simplicity is so highly desirable. To understand this there is no need for us to assume a ‘principle of economy of thought’ or anything of the kind. Simple statements, if knowledge is our object, are to be prized more highly than less simple ones because they tell us more; because their empirical content is greater; and because they are better testable. Karl Popper, The Logic of Scientific Discovery (1959), 43. Simplicity and degree of falsifiability. P. 142
In addition, you're depriving yourself of all mathematical tools by focusing on your class of functions. You don't even have a vector space of functions, because the zero function does not belong to your class, you can't add, you cant subtract, etc
if the existing things aren't good .......there might be new things hidden somewhere.........anyway...i know there's no point in arguing when I don't really have anything solid in place....but i have some hope somewhere on these lines