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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 study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz–Zygmund inequality and ad hoc definitions of function spaces on metric measure spaces. The same techniques are used to prove the existence of point distributions in metric measure spaces with small $$L^p$$ discrepancy with respect to certain classes of subsets, for example, metric balls.

We produce low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a smooth convex domain with positive curvature in \(\mathbb {R}^{d}\). The proof depends on simultaneous Diophantine approximation and on appropriate estimates of the decay of the Fourier transform of characteristic functions.

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 $H\to\infty$.

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

We study the error in quadrature rules on a compact manifold. Our estimates are in the same spirit of the Koksma-Hlawka inequality and they depend on a sort of discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

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 estimate the error in the approximation of the integral of a smooth function over a parallelepiped Ω or a simplex S by Riemann sums with deterministic ℤ d -periodic nodes. These estimates are in the spirit of the Koksma–Hlawka inequality, and depend on a quantitative evaluation of the uniform distribution of the sampling points, as well as on the total variation of the function. The sets used to compute the discrepancy of the nodes are parallelepipeds with edges parallel to the edges of Ω or S. Similarly, the total variation depends only on the derivatives of the function along directions parallel to the edges of Ω or S.

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.