Note that O(g) depends on g only through its differential dg.
Then T tends to zero only through positive values.
Most distribution spaces can be characterized through the action of appropriate convolution operators.
Exercises 2 to 5 [= Exercises 2--5; amer. Exercises 2 through 5]
We now go through the clauses of Definition 3.
The rest of the proof goes through as for Corollary 2, with hardly any changes.
On dividing through by f, we see that......
Multiplying through by f and integrating shows that......
As t runs from 0 to 1, the point f(t) runs through the interval [a,b].
rotation through π/3 <through an angle θ>
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