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On the \(B\) transform of van der Corput. (Sur la transformation \(B\) de van der Corput.) (French) Zbl 0969.11028
The \(B\)-transform of van der Corput is a discrete analogue of the method of stationary phase often used in the estimation of exponential sums of the shape \(S=\sum_{a\leq m\leq b}e(f(m))\) (with \(e(x)=e^{2\pi ix})\). Under the standard assumptions \(|f''(x)|\geq\lambda_2>0\) and \(|f^{(3)}(x) |\leq \lambda_3\) for \(a\leq x\leq b\), \(f'\) is injective and we can define \(J=f' ([a,b])\), \(z=f^{\prime-1}\) and \(f^*(y)= f(z(y))- yz(y)\) for \(y\) in \(J\). The result of the \(B\)-transform is \[ S=e^{\varepsilon i\pi/4} \sum_{\nu\in J}{e(f^* (\nu))\over \sqrt{|f''(z(\nu)) |}}+E \] where \(\varepsilon=1\) if \(f''> 0\), \(\varepsilon= -1\) if \(f''<0\) and \(E\) is an error term. In the standard applications, \(|f^{(j)}(x) |\asymp T/ \mu^j\). In such circumstances, the method of stationary phase leads to estimates of the shape \[ I(b)=\int^b_a e \bigl(f(x)) dx\sim e^{\varepsilon i\pi/4} {e\bigl(f^* (0)\bigr) \over\sqrt {\biggl |f''\bigl(z(0) \bigr)\biggr |}} \] based on the expansion \(f(x)= f(c)+ {1\over 2}f''(c) (x-c)^2+ O(T|x-c|^3/ \mu^3)\) about the critical point \(c=z(0)\) and imitating the behaviour of the Cornu spiral \(I(x)\). The machinery applies more generally to sums \(\sum_{a\leq x\leq b}\varphi(m) e(f(m))\) and integrals \(\int^b_a \varphi(x) e(f(x))dx\) and can be made more delicate to take account of finer arithmetic properties.
The calculus calls on formulae for expanding \(f^*(y)\) and its derivatives and perturbations. All of this is classical machinery for treating exponential sums and the paper provides a useful compendium to relieve the tedium which is often a burden in the applications.

11L07 Estimates on exponential sums