𝑓 2.

𝑓

𝒇

𝑃

𝑄

𝐷𝑓 (𝑃 ∥ 𝑄)

𝑄

𝑃

𝑓

𝑓

𝑄

𝑃

𝛺

𝑓(1) = 0

𝑓

𝑃

𝑄

𝑃

𝑄

𝑓

𝑑𝑃 𝐷𝑓 (𝑃 ∥ 𝑄) = ∫ 𝑓 ( ) 𝑑𝑄 𝑑𝑄 Ω 𝛺 𝑝

𝑞

𝜇

𝑃

𝑑𝑃 = 𝑝𝑑𝜇, 𝑑𝑄 = 𝑞𝑑𝜇

𝑄 𝑓

𝑝(𝑥) 𝐷𝑓 (𝑃 ∥ 𝑄) = ∫ 𝑓 ( ) 𝑞(𝑥)𝑑𝜇(𝑥) 𝑞(𝑥) Ω 𝑓

𝜒

𝑓 𝑓 𝑓 𝑓 𝑓(𝑡) 𝑡 ln 𝑡 , − ln 𝑡 2

(√𝑡 − 1) , 2(1 − √𝑡)

(𝜒 2 )

|𝑡 − 1| 𝜒

(𝑡 − 1)2 , 𝑡 2 − 1

2

1+𝛼 4 (1 − 𝑡 2 ) , if 𝛼 ≠ ±1 2 {1 − 𝛼 𝑡 ln 𝑡 , if 𝛼 = 1 − ln 𝑡 , if 𝛼 = −1

𝛼



𝑓

𝑃

𝑄

𝑑𝑃 𝑑𝑃 𝐷𝑓 (𝑃 ∥ 𝑄) = ∫ 𝑓 ( ) 𝑑𝑄 ≥ 𝑓 (∫ 𝑑𝑄) = 𝑓(1) = 0 𝑑𝑄 𝑑𝑄 

𝑃 𝑄 𝐷𝑓 (𝑃 ∥ 𝑄) ≥ 𝐷𝑓 (𝑃𝜅 ∥ 𝑄𝜅 )

𝑃𝜅

𝑄𝜅

𝜅

*𝑃, 𝑄+ 

0≤𝜆≤1 𝐷𝑓 (𝜆𝑃1 + (1 − 𝜆)𝑃2 ∥ 𝜆𝑄1 + (1 − 𝜆)𝑄2 ) ≤ 𝜆𝐷𝑓 (𝑃1 ∥ 𝑄1 ) + (1 − 𝜆)𝐷𝑓 (𝑃2 ∥ 𝑄2 ) ℝ2+

𝑝

(𝑝, 𝑞) ↦ 𝑞𝑓 ( ) 𝑞