ATS Theorem - Van Der Corput Lemma

Van Der Corput Lemma

Let be a real differentiable function in the interval moreover, inside of this interval, its derivative is a monotonic and a sign-preserving function, and for the constant such that satisfies the inequality Then

\sum_{a<k\le b} e^{2\pi i f(k)} = \int_a^be^{2\pi i f(x)}dx + \theta\left(3 + \frac{2\delta}{1-\delta}\right),

where

Read more about this topic:  ATS Theorem

Famous quotes containing the words van and/or der:

    “My position is a naturalistic one; I see philosophy not as an a priori propaedeutic or groundwork for science, but as continuous with science. I see philosophy and science as in the same boat—a boat which, to revert to Neurath’s figure as I so often do, we can rebuild only at sea while staying afloat in it. There is no external vantage point, no first philosophy.”
    —Willard Van Orman Quine (b. 1908)

    “Under the lindens on the heather,
    There was our double resting-place.”
    —Walther Von Der Vogelweide (1170?–1230?)