Leonardo Colzani

Università degli Studi di Milano-Bicocca | UNIMIB · Department of Mathematics and its Applications

We provide a multidimensional weighted Euler–MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if $$\chi _{\tau {\mathcal {P}}}$$ χ τ P denotes the characteristic...

Given an irrational vector $\alpha$ in $\mathbb{R}^{d}$, a continuous function $f(x)$ on the torus $\mathbb{T}^{d}$ and suitable weights $\Phi(N,n)$ such that $\sum_{n=-\infty}^{+\infty}\Phi(N,n)=1$, we estimate the speed of convergence to the integral $\int_{\mathbb{T}^{d}}f(y)dy$ of the weighted sum $\sum_{n=-\infty}^{+\infty}\Phi(N,n) f(x+n\alph...

We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half‐spaces.

We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.

We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences of a more general series expansion; namely, if $\chi _{\tau\mathcal{P}}$ denotes the characteristic function of...

We add another brick to the large building comprising proofs of Pick’s theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick’s theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.

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

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

We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, u...

We study the asymptotic decay as t→+∞ of integral transforms∫0+∞g(x)Φ(tx)dx. Examples are the cosine and sine Fourier transforms, the Hankel transforms, and the Laplace transforms. Under appropriate assumptions on the kernels and on the functions involved, we prove that the integral transforms can be controlled by the support, or by the first oscil...

We add another brick to the large building comprising proofs of Pick's theorem. Although our proof is not the most elementary, it is short and reveals a connection between Pick's theorem and the pointwise convergence of multiple Fourier series of piecewise smooth functions.

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, $$\begin{aligned} \left\{ {\displaystyle \int _{{\mathbb {R}}}}{\d...

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

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...

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 characteristi...

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...

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 defin...

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...

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.

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

The classical Riemann localization principle states that if an in- tegrable 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 o...

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.

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 fχΩ, with f smooth and Ω a Borel subset of [0,1]d: |N-1∑j=1N(f χΩ)(xj)-∫Ωf(x) dx|≤D(Ω,(xj)j=1N)V(f), where D(Ω,(xj)j=1N) is the discrepancy D(Ω,(xj)j=1N)=2dsupI E[0...

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 ℤd -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...

The Bochner–Riesz means are defined by the Fourier multiplier operators \((S_{R}^{\alpha}\ast f)\hat{\ }(\xi)=( 1-|R^{-1} \xi|^{2})^{\alpha}_{+}\hat{f}(\xi)\). Here we prove that if f has β derivatives in L p (R d ), then \(S_{R}^{\alpha}\ast f(x)\) converges pointwise to f(x) as R→+∞ with a possible exception of a set of points with Hausdorff dime...

We study the error in quadrature rules on a compact manifold. As in the Koksma-Hlawka inequality, we consider a 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.

Although Fourier series or integrals of piecewise smooth functions may be slowly convergent, sometimes it is possible to accelerate their speed of convergence by adding and subtracting suitable combination of known functions. Mathematics Subject Classification (2010)42A10–42A20

There exists a positive function $\psi(t)$ on $t\geq 0$, with fast decay at infinity, such that for every measurable set $\Omega$ in the Euclidean space and $R>0$, there exist entire functions $A(x) $ and $B(x)$ of exponential type $R$, satisfying $A(x)\leq \chi_{\Omega}(x)\leq B(x)$ and $|B(x)-A(x)| \leq \psi(R\operatorname*{dist}(x,\partial\Omega...

We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and H\"older functions on proper subintervals of $\mathbb{R}$ are $\operatorna...

We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure |E| of such a set. We exploit a rational change of variable of the type used by George Boole in his paper "On the comparison of transcendents, with certain appl...

We prove that when a function on the real line is symmetrically rearranged, the distribution function of its uncentered Hardy–Littlewood maximal function increases pointwise, while it remains unchanged only when the function is already symmetric. Equivalently, if ℳ is the maximal operator and 𝒮 the symmetrization, then 𝒮ℳf(x)≤ℳ𝒮f(x) for every x, an...

Let X be a quasi-Banach rearrangement invariant space and let T be a (", )-atomic operator for which a restricted type estimate of the form kT EkX D(|E|) for some positive function D and every measurable set E is known. Then, we show that this estimate can be extended to the set of all positive functions f 2 L1 such that kfk1 1, in the sense that k...

In the first part of the paper we establish the pointwise convergence as t → +∞ for convolution operators $\int _{ℝ^{d}}\ t^{d}K\ (ty)\ \varphi (x-y)dy$ under the assumptions that φ(y) has integrable derivatives up to an order α and that $|K(y)|\leq c(1+|y|)^{-\beta}$ with α+β &gt d. We also estimate the Hausdorff dimension of the set where diverge...

We show that if log(2 − ∆)f ∈ L 2 (R d) then the inverse Fourier transform of f converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0. Our proof is based on a simple application of the Rademacher-Menshov Theore...

Given a positive constant �, there exists a constant c such that for every measurable set in the Euclidean space and R > 0, there exist entire functions of exponential type R with A(x) � �(x) � B(x) and jB(x) A(x)j 6 c(1 + Rdist(x;@)) �. Analogous results hold

We show that if log ⁡ ( 2 − Δ ) f ∈ L 2 ( R d ) \log (2-\Delta )f\in L^2({\mathbb R}^d) , then the inverse Fourier transform of f f converges almost everywhere. Here the partial integrals in the Fourier inversion formula come from dilates of a closed bounded neighbourhood of the origin which is star shaped with respect to 0 0 . Our proof is based o...

We give bounds for the mean square deviation with respect to arbitrary probability measures of the number of integer points in translated or dilated convex bodies. The proofs are based on Fourier analytic methods.

Over the course of the last century, the systematic exploration of the relationship between Fourier analysis and other branches of mathematics has lead to important advances in geometry, number theory, and analysis, stimulated in part by Hurwitz’s proof of the isoperimetric inequality using Fourier series. This unified, self-contained volume is ded...

Eigenfunctions of elliptic boundary value problems can be well approximated by entire functions of exponential type and, as a consequence, it is possible to transfer approximation results with entire functions to eigenfunction expansions. Here, in particular, we consider Jackson and Bernstein type theorems.

We estimate the time decay and regularity of solutions to the wave equation and study the localization and convergence of Fourier integrals. While for generic square integrable functions the results hold almost everywhere, for radial functions the only exception is at the origin.

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂t 2-▵x )u(t,x)=0 and meanwhile we give a new proof of the solution formula.

Let A be an appropriate planar domain and let f be a piecewise smooth function on . We discuss the rate of convergence of in terms of the interaction between the geometry of A and the geometry of the singularities of f. The most subtle case is when x belongs to the singular set of f.

Let Ω be a convex planar domain, with no curvature or regularity assumption on the boundary. Let Nθ(R) = card{RΩθ2}, where Ωθ denotes the rotation of Ω by θ. It is proved that, up to a small logarithmic transgression, Nθ(R) = |Ω|R2 + O(R2/3), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of...

We study the asymptotic behavior of the quadratic means on spheres of increasing radius of the Fourier transforms of characteristic functions of sets with smooth boundary. Beside to give different proofs of known Euclidean results, we also consider the non-Euclidean Fourier transform on the two dimensional sphere and the hyperbolic disk.

. Let A be an appropriate planar domain and let f be a piecewise smooth function on R 2 . We discuss the rate of convergence of S f(x) = Z A b f() exp(2i x)d in terms of the interaction between the geometry of A and the geometry of the singularities of f . The most subtle case is when x belongs to the singular set of f and here Hilbert transform te...

Fourier coefficients ∫ 0 1 f(x)exp(-2πinx)dx of piecewise smooth functions are of the order of |n| -1 and Fourier series ∑ n=-∞ +∞ f ^(n)exp(2πinx) converge everywhere. Here we consider analogs of these results for eigenfunction expansions f(x)=∑ λ ℱf(λ)φ λ (x), where {λ 2 } and {φ λ (x)} are eigenvalues and an orthonormal complete system of eigenf...

We state a localization principle for expansions in eigenfunctions of a self-adjoint second order elliptic operator and we prove an equiconvergence result between eigenfunction expansions and trigonometric expansions. We then study the Gibbs phenomenon for eigenfunction expansions of piecewise smooth functions on two-dimensional manifolds.

We study convolution operators bounded on the non-normable Lorentz spaces L1,q of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure...

Sets thrown at random in space contain, on average, a number of integer points equal to the measure of these sets. We determine the mean square error in the estimate of this number when the sets are homothetic to a domain with fractal boundary. This is related to the problem of approximating Lebesgue integrals by random Riemann sums.

Restrict a smooth function to a domain bounded by a smooth surface. We study the summability of the Fourier integral of this function at points near the boundary of the domain.

We study the maximal function Mf(x) = sup \f(x + y, t)\ when omega is a region in the [GRAPHICS] upper half space R+N+1 and f(x,t) is the harmonic extension to R+N+1 of a distribution in the Besov space B(p,q)alpha(R(N)) or in the Triebel-Lizorkin space F(p,q)alpha(R(N)). In particular, we prove that when OMEGA = {\y\N/(N-alphap) < t < 1} the opera...

Le medie di Bochner-Riesz di opportune funzioni test sono definite per mezzo della trasformata di Fourier da (S R δ)^Γ(δ+1)−1\( \left( {1 - \frac{{\left| \xi \right|^2 }} {{R^2 }}} \right)_ + ^\delta \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi ) \). In questo lavoro studiamo la limitatezza da\(L^p (R^N )\) in\(L^q (R^N )\...

We shall prove an equiconvergence theorem between Fourier-Bessel expansions of functions in certain weighted Lebesgue spaces and the classical cosine Fourier expansions of suitable related functions. These weighted Lebesgue spaces arise naturally in the harmonic analysis of radial functions on euclidean spaces and we shall use the equiconvergence r...

The Bochner-Riesz means of order delta greater-than-or-equal-to 0 for suitable test functions on R(N) are defined via the Fourier transform by (S(R)(delta)f)(xi) = (1 - \xi\2/R2)+(delta)f(xi). We show that the means of the critical index delta = N/P - N + 1/2, 1 < p < 2N/N + 1, do not map L(p,infinity)(R(N)) into L(p,infinity) (R(N)), but they map...

We study the operator H f(x) = 2-x ∫+∞ 0 2y f(y)/x - y dy on Lorentz spaces on R+ with respect to the measure 4x dx. This is related to the harmonic analysis of radial functions on hyperbolic spaces. We prove that this operator is bounded on the Lorentz spaces $L^{2, 9} (\mathbb{R}_+, 4^x dx), 1

We study the operator \[ H f ( x ) = 2 − x ∫ 0 + ∞ 2 y f ( y ) x − y d y \mathcal {H}f(x) = {2^{ - x}}\int _0^{ + \infty } {\frac {{{2^y}f(y)}}{{x - y}}dy} \] on Lorentz spaces on R + {\mathbb {R}_ + } with respect to the measure 4 x d x {4^x}dx . This is related to the harmonic analysis of radial functions on hyperbolic spaces. We prove that this...

Let {λ2} and {θ{symbol}λ} be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator on a compact N-dimensional manifold M. The Riesz means of order δ of an integrable function on M are defined by RΛδf{hook}(x) = ∝M∑λ < Λ ( 1 - λ2 Λ2)δθ{symbol}λ(x)θ{symbol}λ(y)f{hook}(y)dμ(y). In this paper we stud...

Let ƒ be a function in Lp(T), 1 ⩽ p < +∞, or le ƒ be a continuous function on the torus T if p = + ∞, and let Kn be the nth Fejér kernel. We prove that, although the sequence {} is not monotone in general, it still has a monotonicity property. Namely, if .

Let {λ 2} and {ϕ λ } be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator Δ on a compact manifoldM of dimensionN. We prove that the Riesz means of order δ, defined by\(R_\Lambda ^\delta f = \sum\limits_{\lambda< \Lambda } {\left( {1 - \frac{{\lambda ^2 }}{{\Lambda ^2 }}} \right)^\delta \hat f...

An analog of the theorem of D. Jackson on the approximation of periodic functions by means of trigonometric polynomials is established for some Hardy spaces of several variables.

Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g G, t A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spheri...

We prove direct and converse theorems of approximation for distributions in the Hardy spaces Hp, 0< +="">.

It is proved that for every orthonormal complete system in L[1](0, 1) there exists a set A, of measure arbitrarily close to 1, which carries no nonzero function with Fourier transform in lp> for every p <2.

In this paper we study a class of kernels $F_R$ which generalize the Bochner-Riesz kernels on the $N$-dimensional torus. Our main result consists in upper estimates for the $L^p$ norms of $F_R$ as $R$ tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces.

In this paper we study a class of kernels F R {F_R} which generalize the Bochner-Riesz kernels on the N N -dimensional torus. Our main result consists in upper estimates for the L p {L^p} norms of F R {F_R} as R R tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces.

A class of convolution operators is studied, which transform the Hardy space H p (ℝ n ) into H q (ℝ n ).

Typescript. Thesis (Ph. D.)--Washington University, 1982. Dept. of Mathematics. Vita. Includes bibliographical references (leaves 108-109).