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We show how to build a kernel \[ K_X(x,y)=\sum_{m=0}^Xh(\lambda_m/{\lambda_X})\varphi_m(x)\overline{\varphi_m(y)} \] on a compact Riemannian manifold $M$, which is positive up to a negligible error and such that $K_X(x,x)\approx X$. Here $0=\lambda_0^2\le\lambda_1^2\le\ldots$ are the eigenvalues of the Laplace-Beltrami operator on $M$, listed with repetitions, and $\varphi_0,\,\varphi_1,\ldots$ an associated system of eigenfunctions, forming an orthonormal basis of $L^2(M)$. The function $h$ is smooth up to a certain minimal degree, even, compactly supported in $[-1,1]$ with $h(0)=1$, and $K_X(x,y)$ turns out to be an approximation to the identity., Comment: 17 pages

We consider a loosely coupled algorithm for fluid-structure interaction based on a Robin interface condition for the fluid problem (explicit Robin-Neumann scheme). We study the dependence of the stability of this method on the interface parameter in the Robin condition. In particular, for a model problem we find sufficient conditions for instability and stability of the method. In the latter case, we found a stability condition relating the time discretization parameter, the interface parameter, and the added mass effect. Numerical experiments confirm the theoretical findings and highlight optimal choices of the interface parameter that guarantee an accurate solution with respect to an implicit one.

We give asymptotic estimates of the variance of the number of integer points in translated thin annuli in any dimension.

We show asymptotic expansions of the eigenfunctions of certain perturbations of the Jacobi operator in a bounded interval, deducing equiconvergence results between expansions with respect to the associated orthonormal basis and expansions with respect to the cosine basis. Several results for pointwise convergence then follow., 38 pages

In this work, we focus on the Optimized Schwarz Method for circular flat interfaces and geometric heterogeneous coupling arising when cylindrical geometries are coupled along the axial direction. In the first case, we provide a convergence analysis for the diffusion-reaction problem and jumping coefficients and we apply the general optimization procedure developed in Gigante and Vergara (Numer. Math. 131 (2015) 369–404). In the numerical simulations, we discuss how to choose the range of frequencies in the optimization and the influence of the Finite Element and projection errors on the convergence. In the second case, we consider the coupling between a three-dimensional and a one-dimensional diffusion-reaction problem and we develop a new optimization procedure. The numerical results highlight the suitability of the theoretical findings.

Let$$( \mathcal {M},g ) $$(M,g)be ad-dimensional compact connected Riemannian manifold and let$$\{ \varphi _{m}\} _{m=0}^{+\infty }$${φm}m=0+∞be a complete sequence of orthonormal eigenfunctions of the Laplace–Beltrami operator on$$\mathcal {M}$$M. We show that there exists a positive constantCsuch that for all integersNandXand for all finite sequences ofNpoints in$$\mathcal {M}$$M,$$\{x( j)\} _{j=1}^{N}$${x(j)}j=1N, and positive weights$$\{ a_{j}\} _{j=1}^{N}$${aj}j=1Nwe have$$\begin{aligned} \sum _{m=0}^{X}\left| \sum _{j=1}^{N}a_{j}\varphi _{m}( x( j)) \right| ^{2}\ge \max \left\{ CX\sum _{j=1}^{N}a_{j} ^{2},\left( \sum _{j=1}^{N}a_{j}\right) ^{2}\right\} . \end{aligned}$$∑m=0X∑j=1Najφm(x(j))2≥maxCX∑j=1Naj2,∑j=1Naj2.

In this work we consider the optimized Schwarz method designed for computational domains that feature spherical or almost spherical interfaces. In the first part, we consider the diffusion-reaction...

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.

The original version of the article unfortunately contained an error in the acknowledgments section. The corrected Acknowledgements is given below.

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 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

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.

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 $H\to\infty$.

Let ${A_{k}}_{k=0}^{+infty}$ be a sequence of arbitrary complex numbers, let $alpha,, eta>-1$, let ${P_{n}^{alpha,, eta}}_{n=0}^{+infty}$ be the Jacobi polynomials and define the functions% egin{align*} H_{n}left( alpha,,z ight) & =sum_{m=n}^{+infty}frac{A_{m}z^{m}% }{Gammaleft( alpha+n+m+1 ight) left( m-n ight) !}\ Gleft( alpha,, eta,,x,,y ight) & =sum_{r,s=0}^{+infty}% frac{A_{r+s}x^{r}y^{s}}{Gammaleft( alpha+r+1 ight) Gammaleft( eta+s+1 ight) r!s!}. end{align*} Then, for any non negative integer $n$,% egin{multline*} int_{0}^{frac{pi}{2}}Gleft( alpha,, eta,,x^{2}sin^{2}phi ,,y^{2}cos^{2}phi ight) P_{n}^{alpha,, eta}left( cos2phi ight) ,sin^{2alpha+1}phicos^{2 eta+1}phi,dphi\ =frac{1}{2}H_{n}left( alpha+ eta+1,,x^{2}+y^{2} ight) P_{n}% ^{alpha,, eta}left( frac{y^{2}-x^{2}}{y^{2}+x^{2}} ight) . end{multline*} When $A_{k}=left( -1/4 ight) ^{k}$, this formula reduces to Bateman's expansion for Bessel functions. For particular values of $y$ and $n$ one obtains generalizations of several formulas already known for Bessel functions, like Sonine's first and second finite integrals and certain Neumann series expansions. Particular choices of ${A_{k}}_{k=0}^{+infty}$ allow to write all these type of formulas for specific special functions, like Gegenbauer, Jacobi and Laguerre polynomials, Jacobi functions, or hypergeometric functions.

The Optimized Schwarz Method (OSM) is a domain decomposition method based on the introduction of generalized Robin interface conditions obtained by linearly combining the two physical interface conditions through the introduction of suitable symbols, and then on the optimization of such symbols within a proper subset, see [10, 13]. This method has been considered so far for many problems in the case of flat interfaces, see, e.g., [3, 5–7, 11, 16, 17]. Recently, OSM has been considered and analyzed for the case of cylindrical interfaces in [8, 9], and for the case of circular interfaces in [2]. In particular, in [8] we developed a general convergence analysis of the Schwarz method for elliptic problems and an optimization procedure within the constants, with application to the fluid-structure interaction (FSI) problem.

A long standing conjecture for the linear Schroedinger equation states that 1/4 of derivative in $L^2$, in the sense of Sobolev spaces, suffices in any dimension for the solution to that equation to converge almost everywhere to the initial datum as the time goes to 0. This is only known to be true in dimension 1 by work of Carleson. In this paper we show that the conjecture is true on spherical averages. To be more precise, we prove the $L^2$ boundedness of the associated maximal square function on the Sobolev class $H^{1/4}(R^n)$ in any dimension n., Comment: 20 pages, 3 figures

We propose a unified convergence analysis of the generalized Schwarz method applied to a linear elliptic problem for a general interface (flat, cylindrical or spherical) in any dimension. In particular, we provide the exact convergence set of the interface symbols related to the operators involved in the transmission conditions. We also provide a general procedure to obtain estimates of the optimized interface symbols within the constants. We apply such general results to a simple fluid---structure interaction model problem given by the interaction between an incompressible, inviscid fluid and the wave equation. Finally, we assess the effectiveness of the theoretical findings through three-dimensional numerical experiments in the haemodynamic context, obtained by solving the coupling between the Navier---Stokes equations and the linear infinitesimal elasticity.

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

In Geometric Discrepancy we usually test a distribution of N points against a suitable family of sets. If this family consists of dilated, translated and rotated copies of a given d-dimensional convex body \(D \subset \left [0,1\right )^{d}\), then a result proved by W. Schmidt, J. Beck and H. Montgomery shows that the corresponding L 2 discrepancy cannot be smaller than \(c_{d}N^{\left (d-1\right )/2d}\). Moreover, this estimate is sharp, thanks to results of D. Kendall, J. Beck and W. Chen. Both lower and upper bounds are consequences of estimates of the decay of \(\left \Vert \hat{\chi }_{D}\left (\rho \cdot \right )\right \Vert _{L^{2}\left (\varSigma _{d-1}\right )}\) for large ρ, where \(\hat{\chi }_{D}\) is the Fourier transform (expressed in polar coordinates) of the characteristic function of the convex body D, while Σ d−1 is the unit sphere in \(\mathbb{R}^{d}\). In this chapter we provide the Fourier analytic background and we carefully investigate the relation between the L 2 discrepancy and the estimates of \(\left \Vert \hat{\chi }_{D}\left (\rho \cdot \right )\right \Vert _{L^{2}\left (\varSigma _{d-1}\right )}\).

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.

In this work we consider the Optimized Schwarz Method for the three-dimensional (3D) diffusion-reaction problem. In particular, we treat the case of cylindrical interfaces between the subdomains, and we provide a convergence analysis of the Schwarz method, both in the case of Dirichlet interface conditions and in that of general transmission conditions. This allows us to recover, for the latter case, optimal symbols for the interface conditions, which are supposed to work well for geometries which feature cylindrical interfaces. Moreover, starting from these optimal symbols, we propose effective and easily computable constant interface parameters, derived both from Taylor expansions and from an optimization procedure. We finally present several 3D numerical results aiming at validating the theoretical findings.

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.

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

We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates., We have learned about an important reference on the subject matter, and revised the paper accordingly

We define a continuous Gabor transform for strong hypergroups and prove a Plancherel formula, an L2 inversion formula and an uncertainty principle for it. As an example, we show how these techniques apply to the Bessel–Kingman hypergroups and to the dual Jacobi polynomial hypergroups. These examples have an interpretation in the setting of radial functions on Rd and zonal functions on compact two-point homogeneous spaces, where they provide a new transform which possesses many properties of the classical Gabor transform.

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.

The problem solved here rises in a natural way from certain estimates associated with the almost everywhere convergence to the initial datum of the solution to the Schrodinger equation, assuming minimum smoothness. The result gives precise information about the uniform integrability properties of Bessel functions against the oscillatory term e^{itx2}.

We consider a Sturm-Liouville operator of the kind d 2 dt2 + A (t) A(t) d dt on (0;+1) and the related eigenfunction expansion. We prove that, under suitable assumptions on A (t), the partial sums of the Fourier integral associated to such expansion behave like the partial sums of the classical Fourier-Bessel transform. This implies an almost everywhere convergence result for L (A (t) dt) functions. Our methods rely on asymptotic expansions for the eigenfunctions and the Harish-Chandra function that we prove under very weak hypotheses. MSC2000: Primary 43A62. Secondary 43A32, 34L10 Di¤erential operators of the kind L = d 2 dt2 + A0 (t) A (t) d dt (0.1) and the associated spectral decompositions arise naturally in harmonic analysis. For example when A (t) = t , the operator L is the radial part of the Laplacian in R and in this case the associated spectral decomposition is the so called Fourier-Bessel expansion that corresponds to the harmonic analysis of radial functions in R. This transform is de…ned for any > 1 2 by F f ( ) = Z +1 0 f (t) 2 ( + 1) J ( t) ( t) t 2 dt but of course it can be interpreted as a Fourier transform of a radial function only when = (n 2)=2. The inversion formula associated to this transform is given by