Add RKHS document with kernel definition and reproducing kernel explanation

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Alexis BAYLET 2026-03-27 14:37:36 +01:00
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# RKHS
## Kernel definition
Let `X` be a non empty set. Let $k: \mathbb{X} \times \mathbb{X} \to \mathbb{R}$ symetric and positive definite.
$$
\forall (x_{1}, ..., x_{n}) \in \mathbb{X}^{n},
\forall c \in \mathbb{R}^{n},
\sum_{i=1}^{n} \sum_{j=1}^{n} c_{i} c_{j} k(x_{i}, x_{j}) \geq 0
$$
$ K_{i, j} = k(x_{i}, x_{j}) $
### 1.2 Reproducing Kernel
Let `H` be a Hilbert space of functions of real value functions $ f: \mathbb{X} \to \mathbb{R} $ endowed with the inner product $ \langle ., . \rangle_{H} $ k is a reproducing kernel if :
- $ \forall x \in \mathbb{X}, k(., x) \in H $
- $ \forall f \in H, \forall x \in \mathbb{X}, f(x) = \langle f, k(., x) \rangle_{H} $
`H` is called the Reproducing Kernel Hilbert Space (RKHS) associated to `k`.
**remark:** $ f = K(., x) then k $