364 research outputs found
Exponential Sums Related to Maass Forms
We estimate short exponential sums weighted by the Fourier coefficients of a
Maass form. This requires working out a certain transformation formula for
non-linear exponential sums, which is of independent interest. We also discuss
how the results depend on the growth of the Fourier coefficients in question.
As a byproduct of these considerations, we can slightly extend the range of
validity of a short exponential sum estimate for holomorphic cusp forms.
The short estimates allow us to reduce smoothing errors. In particular, we
prove an analogue of an approximate functional equation previously proven for
holomorphic cusp form coefficients.
As an application of these, we remove the logarithm from the classical upper
bound for long linear sums weighted by Fourier coefficients of Maass forms, the
resulting estimate being the best possible. This also involves improving the
upper bounds for long linear sums with rational additive twists, the gains
again allowed by the estimates for the short sums. Finally, we shall use the
approximate functional equation to bound somewhat longer short exponential
sums.Comment: 58 p
Shape identification in inverse medium scattering problems with a single far-field pattern
Consider time-harmonic acoustic scattering from a bounded penetrable obstacle
embedded in a homogeneous background medium. The index
of refraction characterizing the material inside is supposed to be H\"older
continuous near the corners. If is a convex polygon, we
prove that its shape and location can be uniquely determined by the far-field
pattern incited by a single incident wave at a fixed frequency. In dimensions
, the uniqueness applies to penetrable scatterers of rectangular type
with additional assumptions on the smoothness of the contrast. Our arguments
are motivated by recent studies on the absence of non-scattering wavenumbers in
domains with corners. As a byproduct, we show that the smoothness conditions in
previous corner scattering results are only required near the corners
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