From 265401d85c8414a13cedc8c08ccc4205014efd35 Mon Sep 17 00:00:00 2001
From: Alexis BAYLET <alexisbaylet6@gmail.com>
Date: Fri, 27 Mar 2026 14:37:36 +0100
Subject: [PATCH] Add RKHS document with kernel definition and reproducing
 kernel explanation

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 RKHS.md | 27 +++++++++++++++++++++++++++
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 create mode 100644 RKHS.md

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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 $
\ No newline at end of file