Haar Series and Linear Operators

Novikov, Igor; Семенов, Э. М.

doi:10.1007/978-94-017-1726-7

Cited by 35 publications

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“…Much is known (cf., e.g., [19] and references therein) about the boundedness of such Fourier Haar multiplier operator from L p ([0, 1], R) to L q ([0, 1], R , (1.3) where 1 r = 1 q − 1 p . In both cases, the equivalence constants depend only on p and q.…”

Section: Introductionmentioning

confidence: 98%

Operator-valued Fourier Haar multipliers

Girardi

2007

Journal of Mathematical Analysis and Applications

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Section: Introductionmentioning

confidence: 98%

Operator-valued Fourier Haar multipliers

Girardi

2007

Journal of Mathematical Analysis and Applications

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“…In this case there are no Haar spaces of dimension greater than one. For refinements of this important result see [16] For > 1, the Haar space of order of least dimension is yet to be determined and only be known for a few special cases.…”

Section: Introduction To the Multivariate Case Definition 2211 Thementioning

confidence: 99%

On Bivariate Haar Functions And Interpolation Polynomial

Dehghan¹,

Fard²

2014

J. Math. Computer Sci.

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In this paper we consider bivariate Haar series in general case, where bivariate Haar functions are defined on the plane. Here we define a new bivariate Haar function that is included two independent variables. Indeed we presented the new function that is not in previous researches. Mathematicians have applied bivariate Haar function based on tensor product that is a special case of bivariate case. In this research we define the Haar functions by applying another way. Therefore, we define the Haar function differently. And also, the interpolation polynomial with two variables is explained. Then we compare two methods for calculating the approximating function. Namely, we consider a numerical example for comparing the new approximation to bivariate interpolation polynomial. In this example we compute interpolation polynomial by points with Newton lattice form. The calculations indicate that the accuracy of the obtained solutions is acceptable when the number of calculation points is small.

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“…The above information on rearrangement invariant spaces and the Haar system is presented in [2,8] and [1,2,5,6], respectively.…”

mentioning

confidence: 99%

“…This article studies the spaces of multipliers in some pairs of function spaces. The multipliers with respect to the Haar system were studied in [3][4][5][6][7] and many other articles.…”

mentioning

confidence: 99%

The space of Fourier-Haar multipliers

2005

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The Haar system constitutes an unconditional basis for a separable rearrangement invariant (symmetric) space E if and only if the multiplier determined by the sequence λ nk = (−1) n , k = 0, 1, for n = 0 and k = 0, 1, . . . , 2 n for n ≥ 1, is bounded in E. If the Lorentz space Λ(ϕ) differs from L 1 and L ∞ then there is a multiplier with respect to the Haar system which is bounded in Λ(ϕ) and unbounded in L ∞ and L 1 .1. The Haar system is the following system of functions: χ 0 0 (t) = 1 andwhere 1 ≤ k ≤ 2 n , n = 0, 1, . . . . We denote by Ω the set of indices (n, k) corresponding to the Haar functions. The formula m = 2 n + k establishes a one-to-one correspondence between Ω and the set N of naturals and enables us to use the single index Haar system {χ m }. Every sequence λ = (λ 1 , λ 2 , . . . ) generates the multiplier Λ that is defined at the polynomials with respect to the Haar system as follows: Λ m c m χ m = m λ m c m χ m . By the classical Paley-Marcinkiewicz Theorem [1, Theorem 3 and Corollary 2] or [2, Section 2, p. 5] the multiplier Λ is bounded in L p (1 < p < ∞) if sup m |λ m | < ∞. Moreover, if 1 < p < ∞ then Λ Lp ≈ λ l∞ = sup m |λ m |,where the sign ≈ stands for two-sided estimates with constants independent of λ. This article studies the spaces of multipliers in some pairs of function spaces. The multipliers with respect to the Haar system were studied in [3-7] and many other articles. 2.We now give the relevant definitions. A Banach function space E on [0, 1] with the Lebesgue measure is called rearrangement invariant or symmetric if (1) |x(t)| ≤ |y(t)| and y ∈ E implies x ∈ E and x E ≤ y E ;(2) equimeasurability of x(t) and y(t) with y ∈ E implies x ∈ E and x E = y E . We suppose henceforth that E is separable or dual to a separable space. Denote the characteristic function of a measurable set e ⊂ [0, 1] by κ e (t). For every τ > 0 the operatoris bounded in a rearrangement invariant space E and σ τ E ≤ max(1, τ ). The numberslog σ τ E log τ , β E = lim τ →∞ log σ τ E log τ Tol yatti; Voronezh.

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Haar Series and Linear Operators

Cited by 35 publications

References 0 publications

Operator-valued Fourier Haar multipliers

Operator-valued Fourier Haar multipliers

On Bivariate Haar Functions And Interpolation Polynomial

The space of Fourier-Haar multipliers

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