I'm reading some lecture notes about differential geometry. I reached to a point where the author prove that a given integral is well defined. I will not give the precise integral I'm talking about, since this will require to introduce a lot of stuff. I think he used an argument similar to the following:
Let $V$ be a vector space of dimension $n$, with basis $(E^1,...,E^n)$. Let $\phi: V \rightarrow \mathbb{R}$ be a smooth map with compact support. Then the condition $$ \sum_i|a^i(X)| \geq c, \quad c >0 ,$$ implies that the integral $$ \int_V P(X) \hat{\phi} (- \sum_{i=1}^n a^i(X)E^i),$$ Converges, where $P$ is polynomial function on $V$, $a^i(X) \in \mathbb{R}$ and $\hat{ \phi}$ denotes the Fourier transform of $\phi$.
Is the argument above true ? Namely does the condition $ \sum_i|a^i(X)| \geq c, \quad c >0 ,$ implies that the integral $ \int_V P(X) \hat{\phi} (- \sum_{i=1}^n a^i(X)E^i)$ converges ?