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--- function_integral_on_ℝⁿ [2016/02/06 15:06]
nikolaj
+++ function_integral_on_ℝⁿ [2017/01/05 00:00]
nikolaj
@@ Line 14: / Line 14: @@
 ^ $ {\mathrm d}\left(\int_{v(y)}^{w(y)}\,f(x)\,{\mathrm d}x\right) = f(v(y))\,{\mathrm d}v(y)-f(w(y))\,{\mathrm d}w(y) $ ^
-For $f$ convex and $\langle f\rangle_{[a,b]}:=\dfrac{1}{b - a}\int_a^b f(x)\,{\mathrm d}x$
+For $f$ convex and
+$\langle f\rangle_{[a,b]}:=\dfrac{1}{b - a}\int_a^b f(x)\,{\mathrm d}x$
 ^ $\dfrac{f(a) + f(b)}{2} \ge \langle f\rangle_{[a,b]} \ge f \left(\dfrac{a+b}{2}\right) $ ^
-==== Parents ====
+See references.
+== Kernel of he integral ==
+A linear combination of functions that are zero under an integral are again zero.
+Special case
+$$\int_{-a}^a E(x) \left( \dfrac{1}{2} + \sum_{k=0}^\infty c_k U_k(x)^{2k+1} \right) \,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$
+e.g. all $U_k$ the same and $c_k$ so that you get $\frac{1}{1\pm e^{y}}$:
+$$\int_{-a}^a E(x) \dfrac{1}{1\pm {\mathrm e}^{U(x)}}\,{\mathrm d}x = \int_0^a E(x) \,{\mathrm d}x$$
+$$\int_{-a}^a f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,{\mathrm d}x = \int_0^a f(x^2) \,{\mathrm d}x$$
+=== References ===
+Wikipedia:
+[[https://en.wikipedia.org/wiki/Hermite%E2%80%93Hadamard_inequality|Hermite–Hadamard inequality]]
+-----
 === Subset of ===
 [[Function integral]]