Divergent fourier series: Numerical experiments

An open problem in the theory of Fourier series is whether there are functions f L ₁ such that the partial sums S _n(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly bad functions Kahane proved that the rate of divergence is faster than o(log log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums S _n(f, x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense optimal, and conclude that, under this assumption, ...(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for L ₁ functions.     

Dyadic harmonic analysis beyond doubling measures

We characterize the Borel measures μ   on R$\mathbb{R}$ for which the associated dyadic Hilbert transform, or its adjoint, is of weak-type (1,1)$(1,1)$ and/or strong-type (p,p)$(p,p)$ with respect to μ  . Surprisingly, the class of such measures is strictly bigger than the traditional class of dyadically doubling measures and strictly smaller than the whole Borel class. In higher dimensions, we provide a complete characterization of the weak-type (1,1)$(1,1)$ for arbitrary Haar shift operators, cancellative or not, written in terms of two generalized Haar systems and these include the dyadic paraproducts. Our main tool is a new Calderón–Zygmund decomposition valid for arbitrary Borel measures which is of independent interest.     

Two-weightedL1 p -inequalities for singular integral operators on Heisenberg groups

Some sufficient conditions are found for a pair of weight functions, providing the validity of two-weighted inequalities for singular integrals defined on Heisenberg groups.     

Sampling multipliers and the Poisson Summation Formula

Periodization and sampling operators are defined, and the Fourier transform of periodization is uniform sampling in a well-defined sense. Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces. These Poisson Summation Formulas can be used to prove corresponding sampling theorems. The sampling operators used to understand and prove the aforementioned Poisson Summation Formulas lead to the introduction of spaces of continuous linear operators which commute with integer translations. Operators L of this type are appropriately called sampling multipliers. For a given function f, they give rise to new sampling formulas, whose sampling coefficients are of the form Lf. In practice, Lf can be used to model noisy data or data where point values are not available. By representation theorems of the second named author, some of these operator spaces are proved to be mixed norm spaces. The approach and results of this paper were developed in the context of Duffin and Schaeffer’s theory of frames. In particular, sampling multipliers L are related to the Bessel map used by Duffin and Schaeffer in their definition of the frame operator. The first named author was supported in part by AFOSR contract F49620-96-1-0193. The second named author was supported by the Cusanuswerk.     

On dualizing a multivariable Poisson summation formula

It is known [7] that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function ø of one variable into a function and its cosine transform in a generalized sense. The present work presents conditions on ø for which the transform relation holds in the classical sense, and extends this result to a class of generalizations of the Poisson formula in any number of dimensions.     

Heat-diffusion maximal operators for Laguerre semigroups with negative parameters

We prove L^p boundedness for the maximal operator of the heat semigroup associated to the Laguerre functions, , when the parameter α is greater than -1. Namely, the maximal operator is of strong type (p,p) if p>1 and , when -1<α<0. If α?0 there is strong type for 1<p?∞. The behavior at the end points is studied in detail.     

On the zeroth stable A1-homotopy group of a smooth curve

We provide a cohomological interpretation of the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor–Witt K-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable ${\mathbb{A}}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.     

Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces

Under the appropriate definition of sampling density D_ϕ, a function f that belongs to a shift invariant space can be reconstructed in a stable way from its non-uniform samples only if D_ϕ≥1. This result is similar to Landau's result for the Paley-Wiener space B _1/2 . If the shift invariant space consists of polynomial splines, then we show that D_ϕ<1 is sufficient for the stable reconstruction of a function f from its samples, a result similar to Beurling's special case B _1/2 .     

A necessary condition for Calderón-Zygmund singular integral operators

Calderón-Zygmund singular integral operators have been extensively studied for almost half a century. This paper provides a context for and proof of the following result: If a Calderón-Zygmund convolution singular integral operator is bounded on the Hardy space H¹ (Rⁿ), then the homogeneous of degree zero kernel is in the Hardy space H¹(S^n–1) on the sphere.     

Calderón's formula associated with a differential operator on (0, ∞) and inversion of the generalized abel transform

We prove a Calderón reproducing formula for a continuous wavelet transform associated with a class of singular differential operators on the half line. We apply this result to derive a new inversion formula for the generalized Abel transform.     

On a monotonic trigonometric sum

By establishing a cosine analogue of a result of Askey and Steinig on a monotonic sine sum, this paper sharpens and unifies several results associated with Young's inequality for the partial sums of k ^–1 cosk.     

New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-λ inequality with two parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all Lp spaces for 1<p<∞. Pointwise estimates are then replaced by appropriate localized Lp-Lq estimates. We obtain weighted Lp estimates for a range of p that is different from (1,∞) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.     

Bilinear oscillatory integrals and boundedness for new bilinear multipliers

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.     

The banach algebra A and its properties

Beurling’s algebra is considered. A* arises quite naturally in problems of summability of the Fourier series at Lebesgue points, whereas Wiener’s algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with[−π, π], and its dual space is indicated. Analogs of Herz’s and Wiener-Ditkin’s theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.     

On uncertainty principles in the finite dimensional setting

The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known “qualitative” uncertainty principles into more quantitative estimates. We then show how to transfer this result to the discrete version of the short time Fourier transform.     

A weak estimate of Calderón-Zygmund operators

We obtain a weak type theorem of Calderón-Zygmund operators in the Hardy space.     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

Divergence of the greedy algorithm in the Faber-Schauder system on a continuum cardinality set

The paper establishes that there exist a continuum cardinality set E ₀ ? [0, 1] and a function f ₀(x) ∈ C _[0,1], such that the greedy algorithm of f ₀(x) with respect to the Faber-Schauder system converges to +∞ at all points of E ₀.