Smooth local sinusoidal bases on two-dimensional L-shaped regions

In this article, we construct two-dimensional continuous/smooth local sinusoidal bases (also called Malvar wavelets) defined onL-shaped regions. With this construction, one is able to construct local sinusoidal bases and lapped orthogonal transforms (LOT) on arbitrarily shaped regions. This work is motivated from and useful in object-based video coding, where a segmented moving object may have arbitrary shape and block transform coding of this object is needed. Acknowledgements and Notes. His work was partially supported by an initiative grant from the Department of Electrical and Computer Engineering, University of Delaware, the Air Force Office of Scientific Research (AFOSR) under Grant No. F49620-97-1-0253, and the National Science Foundation CAREER Program under Grant MIP-9703377.     

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.     

Proof of a conjecture of José L. Rubio de Francia

Given a compact connected abelian group G, its dual group Γ can be ordered (in a non-canonical way) so that it becomes an ordered group. It is known that, for any such ordering on Γ and p in the range 1<p<∞, the characteristic function χ_I of an interval I in Γ is a p—multiplier with a uniform bound (independent of I) on the corresponding operator S_I on L^p(G). In this note it is shown that, for 1<p,q<∞, there is a constant C_p,q, independent of G and the particular ordering on Γ, such that for all sequences {I_j} of intervals in Γ and all sequences {f_j} in L^p(G). Such a result was conjectured by J.L. Rubio de Francia, who noted its validity when The present proof uses a transference argument, an approach which shows that any constant C_p,q for which the inequality holds when G = will serve for every G and every ordering on Γ. An added advantage of this approach is that it adapts to give an extension of the result for functions taking values in a UMD space.The work of the first author was partially supported by a grant from the National Science Foundation (U.S.A.). The second and third authors were partially supported by the HARP network HPRN-CT-2001-00273 of the European Commission and by grant BFM2001-0188 of Ministerio de Ciencia y Tecnologia.     

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 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.     

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.     

Gibbs' phenomenon and arclength

The arclengths of the graphs Γ(s_N(f)) of the partial sums s_N(f) of the Fourier series of a piecewise smooth function f with a jump discontinuity grow at the rate O(logN). This problem does not arise if f is continuous, and can be removed by using the standard summability methods.     

Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations

The paper revisits the concept of S^p$S^p$-pseudo-almost periodicity recently introduced by the author. In particular, we study the existence of pseudo-almost periodic solutions to some nonautonomous differential equations in the case when the semilinear forcing term is both continuous and S^p$S^p$-pseudo-almost periodic for p>1$p>1$. Applications include the existence of pseudo-almost periodic solutions to the heat equation with a forcing term, which is S^p$S^p$-pseudo-almost periodic and jointly continuous.     

Fourier Series with Linear Splines on an Interval

We construct orthonormal bases of linear splines on a finite interval [a, b] and then we study the Fourier series associated to these orthonormal bases. For continuous functions defined on [a, b], we prove that the associated Fourier series converges pointwisely on (a, b) and also uniformly on [a, b], if it convergences pointwisely at a and b.     

On basicity of exponentials inL P (dμ) and general prediction problems

Let be a nonnegative measure on the unit circle in the complex plane and 1<p<. It is of interest to find conditions on so that the set of exponentialse ⁱⁿ form a strongM-basis forL ^p (d). Some partial results are proved which can shed some light on this important open question. These results are of fundamental importance in the prediction theory of stochastic processes and other fields of applications. These results is then used to obtain a theorem which reduces some prediction problems to easier ones.To 80^th birthday of Paul ErdsThis research is supported by Office of Naval Research Grant No N00014-89-J-1824.     

Stepanov-like pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations

We obtain new existence and uniqueness theorems of pseudo almost periodic mild solutions to nonautonomous neutral partial evolution equations

     

Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.     

OMC for scalar multiples of unitaries

We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137-1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153-154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a₁, … , a_n and b₁, … , b_n, respectively. Then for any scalars t and s

     

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 .     

Walks and the spectral radius of graphs

Given a graph G, write μ(G) for the largest eigenvalue of its adjacency matrix, ω(G) for its clique number, and w_k(G) for the number of its k-walks. We prove that the inequalities

     

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.