www-ai.cs.tu-dortmund.de/en/LEHRE/FACHPROJEKT/SS14/Papers/kernelmethods.pdf
Kernel methods in machine learning
instance, if µ is a normal distribution of the form (2π/σ2)−d/2e−σ2‖ω‖2/2 dω,
then the corresponding positive definite function is the Gaussian e−‖x‖2/(2σ2); see (20).
Bochner’s theorem allows us to interpret [...] Cauchy– Schwarz inequality: If k is a positive definite kernel, and x1, x2 ∈X , then
k(x1, x2) 2 ≤ k(x1, x1) · k(x2, x2).(9)
2.2.1. Construction of the reproducing kernel Hilbert space. We now de- fine [...] given by
F [f ](ω) := (2π)−d/2 ∫
f(x)e−i〈x,ω〉 dx,(44)
with the inverse F−1[f ](x) = (2π)−d/2 ∫
f(ω)ei〈x,ω〉 dω.(45)
Next, compute the Fourier transform of k as
F [k(x, ·)](ω) = (2π)−d/2 ∫ ∫
(υ(ω′)e−i〈x,ω′〉)ei〈x …