Boris Sergeevich Kashin

A remark on Compressed Sensing

Recently, a new direction in signal processing — “Compressed Sensing” is being actively developed. A number of authors have pointed out a connection between the Compressed Sensing problem and the problem of estimating the Kolmogorov widths, studied in the seventies and eighties of the last century. In this paper we make the above mentioned connection more precise.

Improved Lower Bounds on the Rigidity of Hadamard Matrices

We writef=Ω(g) iff(x)≥cg(x) with some positive constantc for allx from the domain of functionsf andg. We show that at least Ω(n2/r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank belowr. This improves the previously known bound Ω(n2/r2). If we additionally know that the changes are bounded above in absolute value by some numberθ≥n/r, then the number of these entries is bounded below by Ω(n3/(rθ2)), which improves upon the previously known bound Ω(n2/θ2).

A Note on the Description of Frames of General Form

where 0 < A ≤ B < ∞ are absolute constants and ‖ · ‖ and ( · , · ) are the norm and the inner product in H . The constants B and A are respectively called the upper and lower bounds of the frame Φ, and their ratio κ = B/A is called the condition number and is denoted by κ(Φ) . For the case in which κ(Φ) = 1, i.e., A = B , the frame Φ is said to be tight. Frames were introduced in 1952 by Duffin and Schaeffer [1] (see also [2]), but some implicit results about frames were obtained earlier. In particular, in quantum information theory and in several areas of functional analysis, the following result has been known as Naimark’s theorem since 1940.

О равномерном приближении частной суммы ряда Дирихле более короткой суммой@@@On the Uniform Approximation of the Partial Sum of the Dirichlet Series by a Shorter Sum

1. Введение и формулировка основного результата. Установленная в этой заметке теорема по формулировке и методу доказательства может быть отнесена к группе результатов об оценках поперечников, полученных с использованием техники функционального анализа и теории вероятностей. При этом основной причиной для ее рассмотрения была некоторая аналогия с утверждениями, находящими важные приложения в теории чисел (см., например, [1; гл. 3]). Отметим также, что интерес к результатам общего характера о равномерном приближении частной суммы ряда Дирихле более короткой суммой высказывал несколько лет назад Карацуба. Ниже мы используем обычные обозначения: R, Z – множества действительных и целых чисел соответственно, C , N = 1, 2, . . . , – N -мерное комплексное пространство. Для x = {xi}i=1 ∈ C , 1 6 p < ∞

On n-term approximations with respect to frames bounded in L p (0, 1), 2 < p < ∞

In this paper, best canonical n-term approximations in the norm of the spaces L2(0, 1) of the family I of characteristic functions of intervals are studied.

A Note on the Approximation Properties of Frames of General Form

The constants A and B are called, respectively, the lower and upper bounds of the frame Φ and their ratio κ = κ(Φ) = B A is called the condition of the frame. Finally, a frame is said to be tight if κ(Φ) = 1, i.e., A = B . Here let us note the paper of Kozlov [2] in which tight frames were studied (in other terms). Obviously, any complete orthonormal system (o.n.s.) in H , as well as an arbitrary Riesz basis, is a frame (see the definition, for example, in [3, p. 17]). Frame systems are, in general, not minimal; however, some properties of orthogonal expansions remain valid for them. In particular, we use the inequality (see [1]) ∥∥∥∥ ∞ ∑

Erratum to: “On n-term approximations with respect to frames bounded in L p (0, 1), 2 < p < ∞”

“and Uβk is a Walsh orthogonal matrix of order βk whose entries satisfy the equality |(Uβk)rs| = β −1/2 k , 1 ≤ r, s ≤ βk. For brevity, we introduce the notation uk = Uβk , k = 2, 3, . . . . Let us define the sequence of natural numbers sk, k = 2, 3, . . . , i = 1, 2, . . . , 2 k − 1, as follows: s12 = 1, s i k + βk − 1 = s i+1 k , i = 1, 2, . . . , 2 k − 2, s k −1 k + βk − 1 = s 1 k+1. Let us now construct the required complete (in L(0, 1)) orthonormal system Ψ. To each of the χ̃ i k, except the last of the bundles, we add βk − 1 functions wj and transform them using the Walsh matrix. The elements of Ψ are numbered by three indices k, i, ν, setting    ψ i,1 k = χ i k for k = 0, 1, 1 ≤ i ≤ 2 k,

Remark on estimates of orthomassivity

Xt(x) = { 1 if 0 ≤ x ≤ t, 0 if t < x ≤ 1, the order relation OMn(K) log n, n → ∞, (4) was noted in [1], while (2) yields only the trivial estimate OMn(K) ≥ 1. The estimate of the quantities (1) is related to other problems in the theory of orthogonal series and combinatorics. So relation (4) actually is equivalent to the combination of the classical Men’shov– Rademacher and Men’shov theorems on convergence and divergences almost everywhere of orthogonal series (see [2, Chap. 9]). Thus, these classical theorems can be regarded as assertions on the complexity of the family of characteristic functions of the intervals (3). In the statement given below which generalizes, in effect, the proof of the Men’shov–Rademacher theorem, we establish an upper bound for orthomassivity; this bound leads to sharp results in a number of cases. In what follows, by#A we denote the number of elements in a finite set A. E-mail: kashin@mi.ras.ru

Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

AbstractIn this paper, we establish lower bounds for n-term approximations in the metric of L2(I2) of characteristic functions of plane convex subsets of the square I2 with respect to arbitrary orthogonal systems. It is shown that, as n→∞, these bounds cannot decrease more rapidly than