
2:58 PM (0 minutes ago)



Dearest Soulmate,
I’m about to prepare a program of study of Mainardi’s papers in order to contact him rather than rely only on David Donoho whose patience with me may have been been injured in the recent months of trying to get his interest in my projects. His papers are quite insightful and brilliant. He understands signal processing of signals of the type of sample path regularity one might expect from these superdiffusions and subdiffusions filled with fractional derivative norms and scaling. A function of class C^gamma with 0<gamma<1 will result from a C^1 function with a fractional derivative of RiemannLiouville type of order 1gamma because the semigroup property of fractional differentiation. It is fascinating to me the subtleties involved in the relationship between semigroup theory which tells us that there is a semigroup that results from say a stable law in some canonical manner and that semigroup is associated to a Bochnerian subordination of a Brownian diffusion, and at the same time this can be described using a functional analytic Besov norm which was developed completely within general ‘deterministic’ setup of calculating the differentiability properties of individual functions in terms of certain convolution operators. In analysis generally, convolution operators are extremely useful because they often appear as the solution operators to certain partial differential equations. For example, the heat equation has a unique solution once boundary conditions and initial conditions are given and that solution can be expressed as a convolution with a socalled fundamental solution of the heat equation which happens to be the Gaussian Bell curve. The kernel in one dimension is expressed as:
k(x,t) = 1/sqrt(2 *pi* t) exp( – x^2/ 2t )
So if u(x,t) satisfies the heat equation with initial condition u(x,0)=f(x) as the initial condition and Kg(x,) = k(,t)*g(x) will solve the heat equation. These are niceties that are lost when one looks at the NavierStokes equations instead of the heat equation because in that case, one cannot expect a linear operator on any function space to solve the nonlinear nonhomogeneous equation — the offending term is the grad(u) u convection term which is responsible for via the Reynold’s number to switch between laminar and turbulent flow.
Now David Donoho’s signal processing expertise includes the study of Besov signal estimation in WHITE noise. This is fabulous work by the way because this gives us a model with a cutoff of determinism and stochasticity at the level of finer detail than what these fractional differential equation models are doing which can be solved by stochastic models as we saw in a recent note I sent of BaeumerMeerchaert. What the David Donoho optimal denoising model is assuming is that DETERMINISTIC signals that are obscured by WHITE noise can be optimally recovered using the wavelet thresholding technology he and his coworkers have developed early in 1990s.
2 Attachments
Preview attachment BaeumerMeerschaertStochasticSolFDE.pdf
BaeumerMeerschaertStochasticSolFDE.pdf
Preview attachment unknownsmoothnessdonohojohnstone1995.pdf
unknownsmoothnessdonohojohnstone1995.pdf
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