On the dimensional weak-type (1,1) bound for Riesz transforms

Let [Formula: see text] denote the [Formula: see text] Riesz transform on [Formula: see text]. We prove that there exists an absolute constant [Formula: see text] such that [Formula: see text] for any [Formula: see text] and [Formula: see text], where the above supremum is taken over measures of the form [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text] with [Formula: see text]. This shows that to establish dimensional estimates for the weak-type [Formula: see text] inequality for the Riesz transforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderón–Zygmund operators.

A consistent estimator for spectral density matrix of a discrete time periodically correlated process

In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed periodogram and exhibit its asymptotic consistency using simulated data.

Weak and Strong Constraints Variational Data Assimilation with the NCOM-4DVAR in the Agulhas Region Using the Representer Method

The difference between the strong and weak constraints four-dimensional variational (4DVAR) analyses is examined using the representer method formulation, which expresses the analysis as the sum of a first guess and a finite linear combination of representer functions. The latter are computed analytically for a single observation under both strong and weak constraints assumptions. Even though the strong constraints representer coefficients are different from their weak constraints counterparts, that difference is unable to help the strong constraints compensate for the loss of information that the weak constraints includes. Numerical experiments carried out in the Agulhas retroflection for single and multiobservation assimilations clearly show that the weak constraint 4DVAR produces analyses that fit the observations with significantly higher accuracy than the strong constraints.

THE INVARIANT MEASURE OF RANDOM WALKS IN THE QUARTER-PLANE: REPRESENTATION IN GEOMETRIC TERMS

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these linear combinations such that the resulting measure may yield an invariant measure of a random walk. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space and further show that the geometric terms in an invariant measure must have a pairwise-coupled structure. Finally, we show that at least one of the coefficients in the linear combination must be negative.

On the number of representations of a positive integer as a sum of two binary quadratic forms

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text]

Examples of linear multi-box splines

Let S1=S1(v0,…,vr+1) be the space of compactly supported C0 piecewise linear functions on a mesh M of lines through ℤ2 in directions v0,…,vr+1, possibly satisfying some restrictions on the jumps of the first order derivative. A sequence ϕ=(ϕ1,…,ϕr) of elements of S1 is called a multi-box spline if every element of S1 is a finite linear combination of shifts of (the components of) ϕ. We give some examples for multi-box splines and show that they are stable. It is further shown that any multi-box spline is not always symmetric

Some remarks on reconstruction from local weighted averages

We solve the convolution equation of the type $f\star\mu=g,$ where $f\star \mu$ is the convolution of $f$ and $\mu$ defined by $(f\star \mu)(x)=\int_{{\mathbb{R}}}f(x-y)d\mu(y),$ $g$ is a given function and $\mu$ is a finite linear combination of translates of an indicator function on an interval.

ON PERTURBATION OF BANACH FRAMES

A necessary and sufficient condition for the perturbation of a Banach frame by a non-zero functional to be a Banach frame has been obtained. Also a sufficient condition for the perturbation of a Banach frame by a sequence in E* to be a Banach frame has been given. Finally, a necessary condition for the perturbation of a Banach frame by a finite linear combination of linearly independent functionals in E* to be a Banach frame has been given.

Another way to say subsolution: the maximum principle and sums of Green functions

Consider an elliptic second order differential operator $L$ with no zeroth order term (for example the Laplacian $L=-\Delta$). If $Lu \leq 0$ in a domain $U$, then of course $u$ satisfies the maximum principle on every subdomain $V \subset U$. We prove a converse, namely that $Lu \leq 0$ on $U$ if on every subdomain $V$, the maximum principle is satisfied by $u+v$ whenever $v$ is a finite linear combination (with positive coefficients) of Green functions with poles outside $\overline{V}$. This extends a result of Crandall and Zhang for the Laplacian. We also treat the heat equation, improving Crandall and Wang's recent result. The general parabolic case remains open.

Characters of Depth-Zero, Supercuspidal Representations of the Rank-2 Symplectic Group

This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of p-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.