Q1: What is correct for the distribution $T = f \cdot \delta$ with $f(x) = x^2$?

A1: $T(\varphi) = \varphi(0)$

A2: $T(\varphi) = 2^2 \varphi(0)$

A3: $T(\varphi) = 0$

A4: $T(\varphi) = x^2 \cdot \varphi(0)$

Q2: What is correct for the distribution $T = f \cdot \delta$ with a smooth function $f \in C^\infty(\mathbb{R}^n)$?

A1: $T(\varphi) = 2 \cdot \varphi(0)$

A2: $T(\varphi) = f(x) \varphi(0)$

A3: $T(\varphi) = f(0) \varphi(0)$

Q3: Consider two smooth functions $f,g \in C^\infty(\mathbb{R}^n)$. What is correct for the product $T_f \cdot T_g$?

A1: $ \langle T_f \cdot T_g , \varphi \rangle = 0$

A2: $ \langle T_f \cdot T_g , \varphi \rangle = \langle T_{f \cdot g} , \varphi \rangle$

A3: $ \langle T_f \cdot T_g , \varphi \rangle = \int_{\mathbb{R}^n} f(x) g(x) dx$

A4: $ \langle T_f \cdot T_g , \varphi \rangle = \int_{\mathbb{R}^n} \varphi(x)dx$