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1
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General Haar systems and greedy approximation

100%
Kamont A.
Studia Mathematica
|
2001
|
tom 145
|
nr 2
165-184
EN
We show that each general Haar system is permutatively equivalent in $L^{p}([0,1])$, 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^{p}([0,1])$, 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^{p}([0,1]^{d})$, 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases in $L^{p}([0,1]^{d})$ for 1 < p < ∞, p ≠ 2 and d ≥ 2. We also note that the above-mentioned general Haar system is not permutatively equivalent to the whole dyadic Haar system in any $L^{p}([0,1])$, 1 < p < ∞, p ≠ 2.
2
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Asymptotic behaviour of Besov norms via wavelet type basic expansions

100%
Kamont A.
Annales Polonici Mathematici
|
2016
|
tom 116
|
nr 2
101-144
EN
J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $Ω ⊂ ℝ^d$ is a smooth bounded domain, 1 ≤ p < ∞ and $f ∈ W^{1,p}(Ω)$, then $lim_{s↗1} (1-s)∫_{Ω} ∫_{Ω} (|f(x)-f(y)|^p)/(||x-y||^{d+sp}) dxdy = K∫_{Ω} |∇f(x)|^p dx$, where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_p^{s,p}(Ω)$. The purpose of this paper is to obtain analogous asymptotic formulae for some other equivalent seminorms, defined using coefficients of the expansion of f with respect to a wavelet or wavelet type basis. We cover both the case of the usual (isotropic) Besov and Sobolev spaces, and the Besov and Sobolev spaces with dominating mixed smoothness.
3
Content available remote

General Franklin systems as bases in H¹[0,1]

64%
Gevorkyan G., Kamont A.
Studia Mathematica
|
2005
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tom 167
|
nr 3
259-292
EN
By a general Franklin system corresponding to a dense sequence of knots 𝓣 = (tₙ, n ≥ 0) in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots 𝓣, that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is a characterization of sequences 𝓣 for which the corresponding general Franklin system is a basis or an unconditional basis in H¹[0,1].
4
Content available remote

Combinatorics of Dyadic Intervals: Consistent Colourings

64%
Kamont A., Müller P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2014
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tom 62
|
nr 2
101-115
EN
We study the problem of consistent and homogeneous colourings for increasing families of dyadic intervals. We determine when this problem can be solved and when it cannot.
5
Content available remote

Unconditionality of general Franklin systems in $L^{p}[0,1]$, 1 < p < ∞

64%
Gevorkyan G., Kamont A.
Studia Mathematica
|
2004
|
tom 164
|
nr 2
161-204
EN
By a general Franklin system corresponding to a dense sequence 𝓣 = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots 𝓣, that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^{p}[0,1]$, 1 < p < ∞.
6
Content available remote

On the trigonometric conjugate to the general Franklin system

64%
Gevorkyan G., Kamont A.
Studia Mathematica
|
2009
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tom 193
|
nr 3
203-239
EN
We investigate when the trigonometric conjugate to the periodic general Franklin system is a basis in C(𝕋). For this, we find some necessary and some sufficient conditions.
7
Content available remote

A martingale approach to general Franklin systems

64%
Kamont A., Müller P.
Studia Mathematica
|
2006
|
tom 177
|
nr 3
251-275
EN
We prove unconditionality of general Franklin systems in $L^{p}(X)$, where X is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.
8
Content available remote

Unconditionality of orthogonal spline systems in H¹

51%
Gevorkyan G., Kamont A., Keryan K., Passenbrunner M.
Studia Mathematica
|
2015
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tom 226
|
nr 2
123-154
EN
We give a simple geometric characterization of knot sequences for which the corresponding orthonormal spline system of arbitrary order k is an unconditional basis in the atomic Hardy space H¹[0,1].
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