Wyniki wyszukiwania

1

Isomorphism of some anisotropic Besov and sequence spaces

100%

Kamont A.

Studia Mathematica

|

1994

|

tom 110

|

nr 2

169-189

EN

An isomorphism between some anisotropic Besov and sequence spaces is established, and the continuity of a Stieltjes-type integral operator, acting on some of these spaces, is proved.

2

ε-Entropy and moduli of smoothness in $L^{p}$-spaces

100%

Kamont A.

Studia Mathematica

|

1992

|

tom 102

|

nr 3

277-302

EN

The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in $L^p(𝕀^d)$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in $L^p(ℝ^d)$ whose tail function decreases as $O(λ^{-γ})$ is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on $ℝ^d$ and the minimax risk for that class is discussed.

3

The Lebesgue constants for the Franklin orthogonal system

63%

Ciesielski Z., Kamont A.

Studia Mathematica

|

2004

|

tom 164

|

nr 1

55-73

EN

To each set of knots $t_{i} = i/2n$ for i = 0,...,2ν and $t_{i} = (i-ν)/n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space $𝓢_{ν,n}$ of all piecewise linear and continuous functions on I = [0,1] with knots $t_{i}$ and the orthogonal projection $P_{ν,n}$ of L²(I) onto $𝓢_{ν,n}$. The main result is $lim_{(n-ν)∧ ν → ∞} ||P_{ν,n}||₁ = sup_{ν,n : 1 ≤ ν ≤ n} ||P_{ν,n}||₁ = 2 + (2 - √3)²$. This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

4

Greedy approximation and the multivariate Haar system

63%

Kamont A., Temlyakov V.

Studia Mathematica

|

2004

|

tom 161

|

nr 3

199-223

EN

We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis 𝓗 in $L_{p}([0,1])$, 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis $𝓗^{d} = 𝓗 × ... × 𝓗$ in $L_{p}([0,1]^{d})$, 1 < p < ∞, p ≠ 2). We prove some convergence results and also some results on convergence rate of weak type greedy algorithms. Our results are expressed in terms of properties of the basis with respect to a given weakness sequence.

Ograniczanie wyników

4 Studia Mathematica

4 Kamont A.

1 Ciesielski Z.

1 Temlyakov V.

2 2004

1 1994

1 1992

Isomorphism of some anisotropic Besov and sequence spaces

ε-Entropy and moduli of smoothness in $L^{p}$-spaces

The Lebesgue constants for the Franklin orthogonal system

Greedy approximation and the multivariate Haar system