Completeness of Integer Translates in Function Spaces on

We show that each of the Banach spacesC₀( ) andL^p( ), 2<p<∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system inL²( ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weightedL²spaces, and in the multivariate setting. We also prove some results in the opposite direction.     

Remarks Concerning Completeness of Translates in Function Spaces

This note explains how to translate the author's old result on cyclic vectors of the multiple shift operator into the language of completeness theorems for integer translates. This translation, together with those results, turns out to be a source for many completeness theorems. In particular, there follows the existence of functions f whose positive integer translates f(x−k), where k₊ are complete in the spaces C^l₀(), L^p(), W^l_p(), 2<p<∞, l=0, 1, …, as well as in their weighted and/or vector-valued analogues.     

On Series of Translates of Positive Functions

For , a discrete infinite set of nonnegative real numbers, and a nonnegative measurable function f: R R ⁺, consider . The sets naturally break into two types. Type 1 consists of such that either C = R almost everywhere or else C = Ø a.e., for every f. Type 2 consists of all the other . We introduce a notion of asymptotic density for and the complementary notion of asymptotic lacunarity. We demonstrate that is of type 2 if it is asymptotically lacunary or else is asymptotically dense and exhibits asymptotically large Q-independent sets. We also give some examples of sets of both types.     

On the One Sided Translates of Powers of Continuous Functions on Locally Compact Groups

We prove that the one sided translates of powers of a real valued continuous (respectively compactly supported) function with a unique maximum (or minimum) span a dense subspace in C(G) and C(X) (respectively in L ^P (G) and L ^P (X)) for any locally compact group G and for any Riemannian symmetric space X.     

Limit Properties of Systems of Integer Translates and Functions Generating Tight Gabor Frames

Mathematical Notes - Tangent-valued forms, tangent and cotangent vectors of the first and the second order are considered. For an affine connection, second-order tangent-valued (vertical and...     

On Completeness of a System of Functions in an Angular Domain

We show that the system {e ^–z /(1 + z ²) : > 0} is complete in a class of functions analytic in an angle.     

Covering the Plane with Translates of a Triangle

The minimum density of a covering of the plane with translates of a triangle is frac32frac{3}{2} .     

Representability of de Branges Matrices as Blaschke--Potapov Products and Completeness of Some Families of Functions

Criteria for the representability of meromorphic second-order matrix functions J-expanding in the upper half-plane (de Branges matrices) as left, right, and two-sided Blaschke--Potapov products are stated. Results on the spectral structure of operators whose characteristic matrix functions are de Branges matrices are obtained.     

Discretely Compact Embeddings of Spaces of Cell-Centered Grid Functions

Compactness of embeddings in discrete counterparts of Sobolev spaces is considered. We study the embeddings in spaces of cell-centered grid functions, in one- and two-dimensional domains. No restrictions are made on the mesh-ratios of the underlying meshes.     

On the Maximum Number of Translates in a Point Set

Given a finite set P⊆ℝ ^d , called a pattern, t _P (n) denotes the maximum number of translated copies of P determined by n points in ℝ ^d . We give the exact value of t _P (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that t _P (n)=n−m _r (n), where r is the rational affine dimension of P, and m _r (n) is the r -Kruskal–Macaulay function. We note that almost all patterns in ℝ ^d are rational simplices. The function t _P (n) is also determined exactly when | P |≤3 or when P has rational affine dimension one and n is large enough. We establish the equivalence of finding t _P (n) and the maximum number s _R (n) of scaled copies of a suitable pattern R⊆ℝ⁺ determined by n positive reals. As a consequence, we show that s_Ak(n)=n-\varTheta (n^1-1/p(k))s_{A_{k}}(n)=n-\varTheta (n^{1-1/\pi(k)}) , where A _k ={1,2,…,k} is an arithmetic progression of size k, and π(k) is the number of primes less than or equal to k.     

Line Transversals to Translates of Unit Discs

Let be a family of convex figures in the plane. We say that has property T if there exists a line intersecting every member of . Also, the family has property T(k) if every k-membered subfamily of has property T. Let B be the unit disc centered at the origin. In this paper we prove that if a finite family of translates of B has property T(4) then the family , where , has property T. We also give some results concerning families of translates of the unit disc which has either property T(3) or property T(5).     

On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ? Rⁿ is complete in L₂(D), where D is a bounded domain in Rⁿ, n ≥ 3, such that \(\bar D\) ∩ L = ?. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R³.     

Line Transversals to Translates of a Convex Body

Let K be a convex body in the plane. Define λ(K,t) as the smallest number satisfying the following: if F\mathcal{F} is any family of translates of K such that every t members of F\mathcal{F} have a common transversal, then all the members of l(K,t)F\lambda(K,t)\mathcal{F} have a common transversal. We give bounds for λ(K,3) and λ(K,4) for a general convex figure K. In particular, we obtain that λ(K,3)≤1.79 when K is the Euclidean disc.     

On the Determination of Convex Bodies by Translates of Their Projections

It is a well-known fact that a three-dimensional convex body is, up to translations, uniquely determined by the translates of its orthogonal projections onto all planes. Simple examples show that this is no longer true if only lateral projections are permitted, that is orthogonal projections onto all planes that contain a given line. In this article large classes of convex bodies are specified that are essentially determined by translates or homothetic images of their lateral projections. The problem is considered for all dimensions , and corresponding stability results are proved. Finally, it is investigated to which degree of precision a convex body can be determined by a finite number of translates of its projections. Various corollaries concern characterizations and corresponding stability statements for convex bodies of constant width and spheres.