Spherical Averages Research Articles

The discrete spherical averages over a family of sparse sequences

We initiate the study of the ℓp(ℤd)-boundedness of the arithmetic spherical maximal function over sparse sequences. We state a folklore conjecture for lacunary sequences, a key example of Zienkiewicz and prove new bounds for a family of sparse sequences that achieves the endpoint of the Magyar-Stein-Wainger theorem for the full discrete spherical maximal function in [MSW02]. Perhaps our most interesting result is the boundedness of a discrete spherical maximal function in ℤ4 over an infinite, albeit sparse, set of radii. Our methods include the Kloosterman refinement for the Fourier transform of the spherical measure (introduced in [Mag07]) and Weil bounds for Kloosterman sums which are utilized by a new further decomposition of spherical measure.

Exploring the origin of clumpy dust clouds around cool giants

Context. Dust grains forming in the extended atmospheres of AGB stars are critical for the heavy mass loss of these cool luminous giants, as they provide radiative acceleration for the stellar winds. Characteristic mid-IR spectral features indicate that the grains consist mainly of silicates and corundum. The latter species seems to form in a narrow zone within about 2 stellar radii, preceding the condensation of silicate dust, which triggers the outflow. Recent high-angular-resolution observations show clumpy, variable dust clouds at these distances. Aims. We explore possible causes for the formation of inhomogeneous dust layers, using 3D dynamical simulations. Methods. We modeled the outer convective envelope and the dust-forming atmosphere of an M-type AGB star with the CO5BOLD radiation-hydrodynamics code. The simulations account for frequency-dependent gas opacities, and include a time-dependent description of grain growth and evaporation for corundum (Al2O3) and olivine-type silicates (Mg2SiO4). Results. In the inner, gravitationally bound, and corundum-dominated layers of the circumstellar envelope, a patchy distribution of the dust emerges naturally, due to atmospheric shock waves that are generated by large-scale convective flows and pulsations. The formation of silicate dust at somewhat larger distances probably indicates the outer limit of the gravitationally bound layers. The current models do not describe wind acceleration, but the cloud formation mechanism should also work for stars with outflows. Timescales of atmospheric dynamics and grain growth are similar to observed values. In spherical averages of dust densities, more easily comparable to unresolved observations and 1D models, the variable 3D morphology manifests itself as cycle-to-cycle variations. Conclusions. Grain growth in the wake of large-scale non-spherical shock waves, generated by convection and pulsations, is a likely mechanism for producing the observed clumpy dust clouds, and for explaining their physical and dynamical properties.

An L2-identity and pinned distance problem

Let $\mu$ be a Frostman measure on $E\subset\mathbb{R}^d$. The spherical average decay $$\int_{S^{d-1}}|\widehat{\mu}(r\omega)|^2\,d\omega\lesssim r^{-\beta} $$ was originally used to attack Falconer distance conjecture, via Mattila's integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag's result on pinned distance problem significantly. The idea is to reduce the pinned distance problem to an integral where spherical averages apply. The key ingredient is the following identity. Using a group action argument, we show that for any Schwartz function $f$ on $\mathbb{R}^d$ and any $x\in\mathbb{R}^d$, $$\int_0^\infty |\omega_t*f(x)|^2\,t^{d-1}dt\,=\int_0^\infty|\widehat{\omega_r}*f(x)|^2\,r^{d-1}dr,$$ where $\omega_r$ is the normalized surface measure on $r S^{d-1}$. An interesting remark is that the right hand side can be easily seen equal to $$c_d\int\left|D_x^{-\frac{d-1}{2}}e^{-2\pi i t\sqrt{-\Delta}}f(x)\right|^2\,dt=c_d'\int\left|D_x^{-\frac{d-2}{2}}e^{2\pi i t\Delta}f(x)\right|^2\,dt.$$ An alternative derivation of Mattila's integral via group actions is also given in the Appendix.

Discrete Multilinear Spherical Averages

AbstractIn this note we give a characterization of $\ell ^{p}\times \cdots \times \ell ^{p}\rightarrow \ell ^{q}$ boundedness of maximal operators associated with multilinear convolution averages over spheres in $\mathbb{Z}^{n}$ .

Hypercontractivity of Spherical Averages in Hamming Space

Consider the linear space of functions on the binary hypercube and the linear operator $S_\delta$ acting by averaging a function over a Hamming sphere of radius $\delta n$ around every point. It is...

Average decay of the Fourier transform of measures with applications

We consider spherical averages of the Fourier transform of fractal measures and improve the lower bound on the rate of decay by taking advantage of multilinear estimates. Maximal estimates with respect to fractal measures are deduced for the Schrödinger and wave equations. This refines the almost everywhere convergence of the solution to its initial datum as time tends to zero. A consequence is that the solution to the wave equation cannot diverge on a (d-1) -dimensional manifold if the data belongs to the energy space \dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d) .

Sharp Strichartz estimates in spherical coordinates

We prove almost Strichartz estimates found after adding regularity in the spherical coordinates for Schr\"odinger-like equations. The estimates are sharp up to endpoints. The proof relies on estimates involving spherical averages. Sharpness is discussed making use of a modified Knapp-type example.

Spherical averages in the space of marked lattices

A marked lattice is a d-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on {mathbb {Z}}^d. We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit.

Mean convergence of Markovian spherical averages for measure-preserving actions of the free group

Mean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space. We endow the set of generators with a generalized Markov chain and establish the mean convergence of resulting spherical averages in this case under mild nondegeneracy assumptions on the stochastic matrix $\Pi$ defining our Markov chain. Equivalently, we establish the triviality of the tail sigma-algebra of the corresponding Markov operator. This convergence was previously known only for symmetric Markov chains, while the conditions ensuring convergence in our paper are inequalities rather than equalities, so mean convergence of spherical averages is established for a much larger class of Markov chains.

Spiraling of approximations and spherical averages of Siegel transforms

J.S.A. partially supported by NSF grant DMS 1069153, and NSF grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). A.G. partially supported by the Royal Society. J.T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147.

Variability of seismic source spectra, estimated stress drop, and radiated energy, derived from cohesive‐zone models of symmetrical and asymmetrical circular and elliptical ruptures

AbstractLarge variability of earthquake stress drops and scaled energy has been commonly reported in the literature, but it is difficult to assess how much of this variability is caused by underlying physical source processes rather than simply observational uncertainties. Here we examine a variety of dynamically realistic rupture scenarios for circular and elliptical faults and investigate to what extent the variability in seismically estimated stress drops and scaled energy comes from differences in source geometry, rupture directivity, and rupture speeds. We numerically simulate earthquake source scenarios using a cohesive‐zone model with the small‐scale yielding limit, where the solution approaches a singular crack model with spontaneous healing of slip. Compared to symmetrical circular source models, asymmetrical models result in larger variability of estimated corner frequencies and scaled energy over the focal sphere. The general behavior of the spherical averages of corner frequencies and scaled energy in the subshear regime extends to the supershear regime, although shear Mach waves generated by the propagation of supershear rupture lead to much higher corner frequency and scaled energy estimates locally. Our results suggest that at least a factor of 2 difference in the spherical average of corner frequencies is expected in observational studies simply from variability in source characteristics almost independent of the actual stress drops, translating into a factor of 8 difference in estimated stress drops. Furthermore, radiation efficiency estimates derived from observed seismic spectra should not be directly interpreted as describing rupture properties unless there are independent constraints on rupture speed and geometry.

Von Neumann and Birkhoff ergodic theorems for negatively curved groups

We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT(-1) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.

Exact reconstruction formula for the spherical mean Radon transform on ellipsoids

Many modern imaging and remote sensing applications require reconstructing a function from spherical averages (mean values). Examples include photoacoustic tomography, ultrasound imaging or SONAR. Several formulas of the back-projection type for recovering a function in n spatial dimensions from mean values over spheres centered on a sphere have been derived by D Finch, S K Patch and Rakesh (2004 SIAM J. Math. Anal. 35 1213–1240) for odd spatial dimension and by D Finch, M Haltmeier and Rakesh (2007 SIAM J. Appl. Math. 68 392–412) for even spatial dimension. In this paper we generalize some of these formulas to the case where the centers of integration lie on the boundary of an arbitrary ellipsoid. For the special cases n = 2 and n = 3 our results have recently been established by Y Salman (2014 J. Math. Anal. Appl. 420 612–20). For the higher dimensional case we establish proof techniques extending the ones in the above references. Back-projection type inversion formulas for recovering a function from spherical means with centers on an ellipsoid have first been derived by F Natterer (2012 Inverse Problems Imaging 6 315–20) for n = 3 and by V Palamodov (2012 Inverse Problems 28 065014) for arbitrary dimension. The results of Natterer have later been generalized to arbitrary dimension by M Haltmeier (2014 SIAM J. Math. Anal. 46 214–32). Note that these formulas are different from the ones derived in the present paper.

Maximal inequality and ergodic theorems for Markov groups

The paper shows that for actions of Markov semigroups, in particular, of finitely generated word hyperbolic groups, the Cesaro means of spherical averages converge almost everywhere for any function from the class Lp, p > 1.

Standard Tensorial Analysis of Local Ordering in Proteins from Residual Dipolar Couplings

Residual dipolar couplings (RDCs) in proteins arise from independent external medium-related and internal protein-related ordering of the spin-bearing probe. Griesinger et al. developed a method for treating RDCs in proteins. The global ordering is given in the standard manner by a rank 2 tensor specified in a known molecular frame, MF. The local ordering is described by the spherical harmonic ensemble averages, <Y(2m)(θ, φ)>, m = 0, ±1, ±2, also given in MF. From these quantities, a method we call mf-RDC derives the squared generalized order parameter (S(rdc)(2)), the amplitude (direction) of the anisotropic disorder, η (Φ′), and an approximation, (N−H)(eff), to the average probe orientation, i.e., to the local director. (N−H)(eff) is determined through a frame transformation where <Y(20)> is maximized. Φ′ is associated with a subsequent frame transformation where <Y(22) + Y(2−2)> is maximized. The mf-RDC method was applied previously to N−H and C−C(methyl) sites in ubiquitin. In this study, we convert the respective <Y(2m)(θ, φ)>'s into a Saupe tensor, which is diagonalized. This is the standard procedure. It yields the eigenvalues, S(xx), S(yy), and S(zz), and the Principal Axis System (PAS) of the rank 2 local ordering tensor, S(l). S(rdc)(2), η, and Φ′ can be recast as S(xx), S(yy), and S(zz). The mf-RDC frame transformations are not the same as the conventional Wigner rotation. The standard tensorial analysis provides new information. The contribution of local ordering rhombicity to S(rdc)(2) is evaluated. For the α-helix of ubiquitin, the main local ordering axis is assigned as C(i−1)(α) − C(i)(α); for the methyl sites, it is associated with the C−C(methyl) axis, as in mf-RDC. Ordering strength correlates with methyl type. The strength (rhombicity) of S(l) associated with picosecond−nanosecond local motions is reduced moderately (substantially) by nanosecond−millisecond local motions. A scheme for analyzing experimental RDCs based on the standard tensorial perspective, which allows for arbitrary orientation of the local director in the protein and of the PAS of S(l) in the probe, is formulated.

Cesàro Convergence of Spherical Averages for Measure-Preserving Actions of Markov Semigroups and Groups

Ces aro convergence of spherical averages is proven for measure-preserving actions of Markov semigroups and groups. Convergence in the mean is established for functions in L p , 1 ≤ p < ∞, and pointwise convergence for functions in L ∞. In particular, for measure-preserving actions of word hyperbolic groups (in the sense of Gromov) we obtain Ces aro convergence of spherical averages with respect to any symmetric set of generators.

A pointwise ergodic theorem for Fuchsian groups

AbstractWe use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesàro averages of spherical averages in a Fuchsian group.

Invariant measures on the space of horofunctions of a word hyperbolic group

AbstractWe introduce a natural equivalence relation on the space ℋ0 of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen–Marcus theorem. Furthermore, if η is such a measure and G acts on a probability space (X,μ) by measure-preserving transformations then η×μ is virtually ergodic with respect to a natural equivalence relation on ℋ0×X. This is comparable to a special case of the Howe–Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.

On Beckenbach–Radó type integral inequalities

Beckenbach and Rado characterized logarithmically subharmonic functions in the plane in terms of integral inequalities involving spherical averages. In this work we generalize this result to higher dimensions and thus answer to a question raised by Beckenbach and Rado. We also consider generalizations of integral inequalities suggested by Beckenbach and Rado and discuss connections to reverse Holder inequalities and Muckenhoupt weights.

On Spherical Averages of Radial Basis Functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration.