README.md

aimath

The project is designed to automatically estimate functions in symbolic form

Function Spaces

We define function spaces $L^p$, $W^{k,p}$, $\dot H^s$, $H^s$, $C^{r,\alpha}$ by using the sympy, each of then taking a function symbol and parameter as input and output their corresponding norms.

For instance, we call

Lp(p,f)

to define the standard $L^p$ space for function $f$.

Sobolev Embedding

We build the [Sobolev inequality] (https://en.wikipedia.org/wiki/Sobolev_inequality) for the. Here we need to input our function, which will output all possible embedding spaces as a common rule. We know that they are in a linear relation. So we pick 10 possible pairs where each pair are rational numbers.

For $f$ in space $W^{k,p}(\mathbb{R}^n)$, we call

Sobolev_Embed_auto(k,p,n,f)

to get a list of embedded spaces.

Example: For $f \in W^{2,2}(\mathbb{R})$, we call

Sobolev_Embed_auto(2,2,1,f**2)

And we have

$$ \left[ \left \| f^{2} \right \|{C^{0,\frac{3}{2}}}, \ \left \| f^{2} \right \|{C^{\frac{1}{6},\frac{4}{3}}}, \ \left \| f^{2} \right \|{C^{\frac{1}{3},\frac{7}{6}}}, \ \left \| f^{2} \right \|{C^{\frac{1}{2},1}}, \ \left \| f^{2} \right \|{C^{\frac{2}{3},\frac{5}{6}}}, \ \left \| f^{2} \right \|{C^{\frac{5}{6},\frac{2}{3}}}, \ \left \| f^{2} \right \|{C^{1,\frac{1}{2}}}, \ \left \| f^{2} \right \|{C^{\frac{7}{6},\frac{1}{3}}}, \ \left \| f^{2} \right \|{C^{\frac{4}{3},\frac{1}{6}}}, \ \left \| f^{2} \right \|{C^{\frac{3}{2},0}}\right] $$

For $f \in W^{2,2}(\mathbb(\R)^4)

Sobolev_Embed_auto(2,2,4,f)

We have

$$ \left[ \left \| f \right \|{W^{\frac{1}{25},100}}, \ \left \| f \right \|{W^{\frac{25}{97},\frac{450}{29}}}, \ \left \| f \right \|{W^{\frac{39}{82},\frac{757}{90}}}, \ \left \| f \right \|{W^{\frac{52}{75},\frac{75}{13}}}, \ \left \| f \right \|{W^{\frac{41}{45},\frac{180}{41}}}, \ \left \| f \right \|{W^{\frac{35}{31},\frac{287}{81}}}, \ \left \| f \right \|{W^{\frac{101}{75},\frac{199}{67}}}, \ \left \| f \right \|{W^{\frac{97}{62},\frac{225}{88}}}, \ \left \| f \right \|{W^{\frac{139}{78},\frac{101}{45}}}, \ \left \| f \right \|{W^{2,2}}\right] $$

Para Product

We build the Bony's Paraproduct decomposition. We include the high-high, low-high, and low-high paraproduct. Moreover, we also have the helpful operator $\nabla$ and $\triangle$ build-in. The paraproduct can be used with the Sobolev Embedding and Holder inequality to estimate functions.

For instance, if we want to calculate the paraproduct of

$$ \nabla f * \triangle g= \pi_{hh}(\nabla f * \triangle g)+ \pi_{hl}(\nabla f * \triangle g)+\pi_{lh}(\nabla f * \triangle g), $$

we can simply call:

Paraproduct(nabla*f,lap*g)

The output will be in latex as the following:

$$ \sum_{k=-\infty}^{\infty} 2^{3 k} P_{k}f P_{k}g + \sum_{\substack{-\infty \leq j \leq k - 10\\-\infty \leq k \leq \infty}} 2^{j} 2^{2 k} P_{j}f P_{k}g + \sum_{\substack{-\infty \leq j \leq k - 10\\-\infty \leq k \leq \infty}} 2^{2 j} 2^{k} P_{j}g P_{k}f $$