Boris Kashin

Boris Kashin
Russian Academy of Sciences | RAS · function theory

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160
Publications
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Publications

Publications (160)
Article
This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending w...
Preprint
This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending with very recent achievements. We also show how sampling discretization connects to spectral properties an...
Preprint
Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any $N$-dimensional subspace of the spa...
Article
Для конечной ортогональной системы равномерно ограниченных функций установлено существование достаточно плотных подсистем со свойством лакунарности и с хорошей оценкой нормы оператора мажоранты частных сумм.
Article
We consider conditions on a matrix A with unit operator (2,1)-norm ensuring the existence of a partition of this matrix into two submatrices with (2,1)-norms close to 1/2.
Article
Рассматриваются условия на матрицу $A$ с единичной операторной $(2,1)$-нормой, которые обеспечивают существование разбиения этой матрицы на две подматрицы с $(2,1)$-нормами, близкими к $1/2$. Библиография: 8 названий.
Article
Closely related notions of the Kolmogorov width and the approximate rank of a matrix are considered. New estimates are established in approximation problems related to the width of the set of characteristic functions of intervals; the multidimensional case (characteristic functions of parallelepipeds) is also considered.
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Full-text available
Using an approach proposed by Lunin in 1989, upper bounds are found for the norms of large submatrices of a fixed -matrix which defines an operator from into with unit norm. Bibliography: 15 titles.
Article
In this paper, best canonical n-term approximations in the norm of the spaces L (2)(0, 1) of the family I of characteristic functions of intervals are studied.
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It is shown that each Dirichlet polynomial P of degree N which is bounded in a certain natural Euclidean norm, admits a nontrivial uniform approximation on the corresponding interval on the real axis by a Dirichlet polynomial with spectrum containing significantly fewer than N elements. Moreover, this spectrum is independent of P. Bibliography: 19...
Article
We present a survey of the scientific results obtained by E. A. Storozhenko and related results of her disciples and give brief information about the seminar on the theory of functions held under her guidance.
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We introduce a space of quasicontinuous functions and study its approximate characteristics, i.e., ε-entropy and widths. We establish inequalities for norms of trigonometric polynomials in this space. In addition, we obtain exponents of the ε-entropy and widths of some classes of functions with low smoothness.
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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 connecti...
Article
This article gives a generalization of a theorem of Ryll and Wojtaszczyk on the existence of a sequence of homogeneous polynomials [$ P_N$], [$ N=1,\,2,\,\dots$], in [$ d$] variables with degree [$ P_N=N$] for which [$\displaystyle \Vert P_N\Vert _{L^2(S^d)}\geq c_d\Vert P_N\Vert _{C(S^d)}>0,$] where [$ S^d$] is the sphere in [$ d$]-dimensional com...
Article
The paper contains a proof of the following Theorem. Suppose [$ \sum_1^\infty f_k(x)$] converges unconditionally in [$ L_1\lbrack 0,\,1\rbrack $]. Then for any [$ \epsilon>0$] there exists a set [$ E_\epsilon\subset\lbrack 0,\,1\rbrack $], [$ \mu E_\epsilon>1-\epsilon$], such that [$ \sum_1^\infty f_k(x)$] converges unconditionally in [$ L_q(E_\eps...
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Compressed Sensing). В заметке уточ-няются отмеченные ранее рядом авторов связи этой темы с задачами об оценке поперечников по Колмогорову, рассмотренными в 70–80 гг. прошлого века. Библиография: 11 названий. 1. Введение. В последнее время, "сжатое измерение" (Compressed Sensing или Compressive Sampling) привлекает все большее внимание как математи...
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In this paper, a novel approach to the proof of inequalities of Lieb-Thirring type based on the standard apparatus of the theory of orthogonal series is proposed.
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In this paper, we establish lower bounds for canonical n -term approximations with respect to frames of general form of a family of functions which is a natural one from the point of view applications. The results proved in [1] for orthogonal bases in the spaces L 2 (0, 1) are generalized. The expediency of such a step is connected mainly with the...
Article
We study questions of the following type: Given positive semi-definite matrix G, does there exist a sequence of vectors in ℝn whose Grammian equals to G and which has some specified additional properties (typically related to the sup norm)? In particular, we show that the answer to the 1947 Knaster problem about real functions on spheres is negativ...
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Let A N , N=1,2,⋯, be the set of the Gram matrices of systems {e j } j=1 N formed by vectors e j of a Hilbert space H with norms ∥e j ∥ H ≤1, j=1,⋯,N. Let B N (K) be the set of the Gram matrices of systems {f j } j=1 N formed by functions f j ∈L ∞ (0,1) with ∥f j ∥ L ∞ (0,1) ≤K, j=1,⋯,N. It is shown that, for any K, the set B N (K) is narrower than...
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We prove new estimates for the entropy numbers of classes of multi­ variate functions with bounded mixed derivative. It is known that the investigation of these classes requires development of new techniques comparing to the univariate classes. In this paper we continue to de­ velop the technique based on estimates of volumes of sets of the Fourier...
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Full-text available
In this paper, we establish lower bounds for n-term approximations in the metric of L 2(I 2 ) of characteristic functions of plane convex subsets of the square I 2 with respect to arbitrary orthogonal systems. It is shown that, as n1/n1/n .
Article
We write f = Ω(g) if f(x) ≥ cg(x) with some positive constant c for all x from the domain of functions f and g. We show that at least Ω(n 2/r) entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank below r. This improves the previously known bound Ω(n 2/r 2). If we additionally know that the changes are bo...