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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.

##### MSC:
 11L07 Estimates on exponential sums
##### Keywords:
$$B$$-transform of van der Corput