The following equation may be meaningful, but how can we make it well-defined $$\delta(x-a)\cdot\delta(x-b)=0$$ Question How do we defined this equation? Or more broadly define product between generalized functions with certain restrictions. Whether this definition satisfies the product rule [$D(fg)=Df\cdot g+f\cdot Dg$].

The following equation may be meaningful, but how can we make it well-defined $$\delta(x-a)\cdot\delta(x-b)=0$$ Question: How do we defined this equation? Or more broadly define product between generalized functions with certain restrictions. Whether this definition satisfies the product rule [$D(fg)=Df\cdot g+f\cdot Dg$].

Added: Maybe theory of hyperfunction can explain this, which I am not familiar with. Should its product satisfy the product rule?