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Nonlinear Analysis: Theory, Methods & Applications, 2011
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1997
Let n ≥ 0 and πn be a rearrangement of {0,1} if n = 0 and {1, 2, ..., 2 n } if n ≥ 1.
Igor Novikov, Evgenij Semenov
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Let n ≥ 0 and πn be a rearrangement of {0,1} if n = 0 and {1, 2, ..., 2 n } if n ≥ 1.
Igor Novikov, Evgenij Semenov
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Journal of Contemporary Mathematical Analysis, 2007
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Navasardyan, K. A., Stepanyan, A. A.
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Navasardyan, K. A., Stepanyan, A. A.
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On Perturbations of the Haar System
Mathematical Notes, 2004This paper discusses the basis properties in \(L_p[0,1]\), \(1 \leq p < \infty\), of sequences of the form \[ \{e,\phi_{k,j};\;k=0, 1\dots,\;j= 0,\dots, 2^k -1\} \tag{1} \] which are close, in a certain sense, to the classical system of Haar functions \[ \{e, \chi_{k,j};\;k=0, 1, \dots,\;j= 0, \dots, 2^k -1\}, \] where \(e = \chi_{[0,1)}\), \(h_{k,j}(t)
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Reproducibility of the Haar system
1997Let {xn} 1 ∞ be a normalized basis of a Banach space X,x n tends weakly to 0, and X is isomorphic to a subspace of some space Y, with a basis {yk} 1 ∞ . It is known (see Theorem 1.b.3) that there exists a subsequence {xnj} j=1 ∞ of {xn} 1 ∞ which is equivalent to a block basis of {yk} 1 ∞ .
Igor Novikov, Evgenij Semenov
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1998
The Haar basis is known since 1910. Here we consider the Haar basis on the real line IR and describe some of its properties which are useful for the construction of general wavelet systems. Let L2 (IR) be the space of all complex valued functions f on IR such that their L2-norm is finite: $$ \left\| {f\left\| {2 = \left( {\int_{ - \infty }^\infty {\
Wolfgang Härdle +3 more
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The Haar basis is known since 1910. Here we consider the Haar basis on the real line IR and describe some of its properties which are useful for the construction of general wavelet systems. Let L2 (IR) be the space of all complex valued functions f on IR such that their L2-norm is finite: $$ \left\| {f\left\| {2 = \left( {\int_{ - \infty }^\infty {\
Wolfgang Härdle +3 more
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Subsequences of the Haar system
1997If the H.s. is an unconditional basis of an r.i. space E, then the spaces spanned by subsequences of the H.s. are complemented in E. These spaces can be characterized in the following form.
Igor Novikov, Evgenij Semenov
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The Haar system and martingales
1997Let β 1 ⊂ β 2 ⊂ ... ⊂ β n ⊂ ... be an increasing sequence of σ-subalgebras of Σ. A sequence {x n } n=1 ∞ of integrable functions is said to be a martingale with respect to {β n } n=1 ∞ if for all n.
Igor Novikov, Evgenij Semenov
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2004
In this chapter we will present an example of an orthonormal system on [0,1] known as the Haar system. The Haar basis is the simplest and historically the first example of an orthonormal wavelet basis. Many of its properties stand in sharp contrast to the corresponding properties of the trigonometric basis (Definition 2.5).
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In this chapter we will present an example of an orthonormal system on [0,1] known as the Haar system. The Haar basis is the simplest and historically the first example of an orthonormal wavelet basis. Many of its properties stand in sharp contrast to the corresponding properties of the trigonometric basis (Definition 2.5).
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Rearrangements of the Haar system
Mathematical Notes of the Academy of Sciences of the USSR, 1974It is proved that any fixed rearrangement of the Haar system either is or is not a system of convergence almost everywhere simultaneously for all classes Lp[0, 1] (1 ≤ p ≤ ∞).
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