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We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^{p}$ norm of the discrepancy with respect to the translation variable as the dilation parameter goes to infinity. If there is a single flat point with normal in a rational direction we obtain an asymptotic expansion for this norm. Anomalies may appear when two flat points have opposite normals. When all the flat points have normals in generic irrational directions, we obtain a smaller discrepancy. Our proofs depend on careful estimates for the Fourier transform of the characteristic function of the convex body.

We estimate the $L^{p}$ norms of the discrepancy between the volume and the number of integer points in $r\Omega-x$, a dilated by a factor $r$ and translated by a vector $x$ of a convex body $\Omega$ in $\mathbb{R}^{d}$ with smooth boundary with strictly positive curvature, \[ \left\{ {\displaystyle\int_{\mathbb R}}{\displaystyle\int_{\mathbb{T}^{d}}}\left\vert \sum_{k\in\mathbb{Z}^{d}}\chi _{r\Omega-x}(k)-r^{d}\left\vert \Omega\right\vert \right\vert ^{p}dxd\mu(r-R) \right\} ^{1/p}, \] where $\mu$ is a Borel measure compactly supported on the positive real axis and $R\to+\infty$., Comment: 37 pages, 6 figures. arXiv admin note: text overlap with arXiv:1706.04419

The classical Riemann localization principle states that if an integrable function of one variable vanishes in an open set, then its Fourier expansion converges to zero in this set. This principle does not immediately extend to several dimensions, and here we study the Hausdorff dimension of the sets of points where localization for Riesz means of Fourier expansions may fail.

We estimate some mixed $L^{p}\left( L^{2}\right) $ norms of the discrepancy between the volume and the number of integer points in $r\Omega-x$, a dilated by a factor $r$ and translated by a vector $x$ of a convex body $\Omega$ in $\mathbb{R}^{d}$, $ \left\{ {\int_{\mathbb{T}^{d}}}\left( \frac{1}{H} {\int_{R}^{R+H}}\left\vert \sum_{k\in\mathbb{Z}^{d}}\chi _{r\Omega-x}(k)-r^{d}\left\vert \Omega\right\vert \right\vert^{2}dr\right)^{p/2}dx\right\} ^{1/p}. $ We obtain estimates for fixed values of $H$ and $R\to\infty$, and also asymptotic estimates when $H\to\infty$.

Revisiting and extending a recent result of M. Huxley, we estimate the Lp($\mathbb{T}$d) and Weak–Lp($\mathbb{T}$d) norms of the discrepancy between the volume and the number of integer points in translated domains.

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R^2. The proof depends on simultaneous diophantine approximation and a general version of the Erdos-Turan inequality., Comment: 14 pages, 2 figures

The classical Koksma-Hlawka inequality does not apply to functions with simple discontinuities. Here we state a Koksma-Hlawka type inequality which applies to piecewise smooth functions [email protected]"@W, with f smooth and @W a Borel subset of [0,1]^d: |N^-^[email protected]?j=1N([email protected]"@W)(x"j)[email protected]!"@Wf(x)dx|@?D(@W,{x"j}"j"="1^N)V(f), where D(@W,{x"j}"j"="1^N) is the discrepancy D(@W,{x"j}"j"="1^N)=2^[email protected]?[0,1]^d{|N^-^[email protected][email protected]"@W"@?"I(x"j)-|@[email protected]?I||}, the supremum is over all d-dimensional intervals, and V(f) is the total variation V(f)[email protected][email protected]@?{0,1}^d2^d^-^|^@a^|@!"["0","1"]"^"d|(@[email protected]?x)^@af(x)|dx. We state similar results with variation and discrepancy measured by L^p and L^q norms, 1/p+1/q=1, and we also give extensions to compact manifolds.

We state some equiconvergence results between Bochner Riesz means of expansions in eigenfunctions of suitable Sturm Liouville operators. Then we determine the Hausdorff dimension of the divergence set of Bochner Riesz means of radial functions in Sobolev classes on Euclidean and non Euclidean spaces. © 2013 Springer Science+Business Media New York.

This paper contains an $L^{p}$ improving result for convolution operators defined by singular measures associated to hypersurfaces on the motion group. This needs only mild geometric properties of the surfaces, and it extends earlier results on Radon type transforms on $\mathbb{R}^{n}$. The proof relies on the harmonic analysis on the motion group., Comment: 12 pages, 1 figure

There exists a positive function ψ(t) on t ≥ 0, with fast decay at infinity, such that for every measurable set O in the Euclidean space and R > 0, there exist entire functions A(x) and B (x) of exponential type R, satisfying A(x) ≤ xω(x) ≤ B(x) and |B(x) - A(x)| ≤ψ (Rdist (x,θomega;)). This leads to Erd?os Turán estimates for discrepancy of point set distributions in the multi-dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds. © 2010 American Mathematical Society.