# Padua transform

This demonstrates the Padua transform and inverse transform, explaining precisely the normalization and points

`using FastTransforms`

We define the Padua points and extract Cartesian components:

```
N = 15
pts = paduapoints(N)
x = pts[:,1]
y = pts[:,2];
nothing #hide
```

We take the Padua transform of the function:

```
f = (x,y) -> exp(x + cos(y))
f̌ = paduatransform(f.(x , y));
nothing #hide
```

and use the coefficients to create an approximation to the function $f$:

```
f̃ = (x,y) -> begin
j = 1
ret = 0.0
for n in 0:N, k in 0:n
ret += f̌[j]*cos((n-k)*acos(x)) * cos(k*acos(y))
j += 1
end
ret
end
```

`#3 (generic function with 1 method)`

At a particular point, is the function well-approximated?

`f̃(0.1,0.2) ≈ f(0.1,0.2)`

`true`

Does the inverse transform bring us back to the grid?

`ipaduatransform(f̌) ≈ f̃.(x,y)`

`true`

*This page was generated using Literate.jl.*