Van der Corput's method
In mathematics, van der Corput's method generates estimates for exponential sums. The method applies two processes, the van der Corput processes A and B which relate the sums into simpler sums which are easier to estimate.
The processes apply to exponential sums of the form
where f is a sufficiently smooth function and e(x) denotes exp(2πix).
To apply process A, write the first difference fh(x) for f(x+h)−f(x).
Assume there is H ≤ b−a such that
Process B transforms the sum involving f into one involving a function g defined in terms of the derivative of f. Suppose that f' is monotone increasing with f'(a) = α, f'(b) = β. Then f' is invertible on [α,β] with inverse u say. Further suppose f'' ≥ λ > 0. Write
Applying Process B again to the sum involving g returns to the sum over f and so yields no further information.
The method of exponent pairs gives a class of estimates for functions with a particular smoothness property. Fix parameters N,R,T,s,δ. We consider functions f defined on an interval [N,2N] which are R times continuously differentiable, satisfying
uniformly on [a,b] for 0 ≤ r < R.
We say that a pair of real numbers (k,l) with 0 ≤ k ≤ 1/2 ≤ l ≤ 1 is an exponent pair if for each σ > 0 there exists δ and R depending on k,l,σ such that
uniformly in f.
By Process A we find that if (k,l) is an exponent pair then so is . By Process B we find that so is .
A trivial bound shows that (0,1) is an exponent pair.
The set of exponents pairs is convex.
The exponent pair conjecture states that for all ε > 0, the pair (ε,1/2+ε) is an exponent pair. This conjecture implies the Lindelöf hypothesis.
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