Unconditionality of orthogonal spline systems in $L^{p}$

Markus Passenbrunner

Displaying similar documents to “Unconditionality of orthogonal spline systems in $L^{p}$ ”

Remark on spline unconditional bases in $H^{1} (D)$

Leszek Skrzypczak (1989)

Banach Center Publications

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Semiorthogonal linear prewavelets on irregular meshes

Peter Oswald (2006)

Banach Center Publications

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We extend results on constructing semiorthogonal linear spline prewavelet systems in one and two dimensions to the case of irregular dyadic refinement. In the one-dimensional case, we obtain sharp two-sided inequalities for the $L_{p}$ -condition, 1 < p < ∞, of such systems.

A quadratic spline-wavelet basis on the interval

Černá, Dana, Finěk, Václav, Šimůnková, Martina

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In signal and image processing as well as in numerical solution of differential equations, wavelets with short support and with vanishing moments are important because they have good approximation properties and enable fast algorithms. A B-spline of order $m$ is a spline function that has minimal support among all compactly supported refinable functions with respect to a given smoothness. And recently, B. Han and Z. Shen constructed Riesz wavelet bases of $L_{2} (ℝ)$ with $m$ vanishing moments based...

Contribution to construction of global cubic $C^{1}$ or $C^{2}$ -spline on equidistant knotset

Kobza, Jiří

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On approximation by Chebyshevian box splines

Zygmunt Wronicz (2002)

Annales Polonici Mathematici

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Chebyshevian box splines were introduced in [5]. The purpose of this paper is to show some new properties of them in the case when the weight functions $w_{j}$ are of the form $w_{j} (x) = W_{j} (v_{n + j} \cdot x)$ , where the functions $W_{j}$ are periodic functions of one variable. Then we consider the problem of approximation of continuous functions by Chebyshevian box splines.

Kolmogorov problem in $W^{r} H^{ω} [0, 1]$ and extremal Zolotarev ω-splines

Bagdasarov Sergey K.

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AbstractThe main result of the paper, based on the Borsuk Antipodality Theorem, describes extremal functions of the Kolmogorov-Landau problem(*) $f^{(m)} (ξ) \to s u p$ , $f \in W^{r} H^{ω} [ξ, b]$ , ${| | f | |}_{ℂ [a, b]} \leq B$ ,for all 0 < m ≤ r, ξ ≤ a or ξ = (a+b)/2, all B > 0 and concave moduli of continuity ω on ℝ₊. It is shown that any extremal function $=_{B, r, m, ω, ξ}$ of the problem (*) enjoys the following two characteristic properties. First, the function $^{(r)} (\cdot) -^{(r)} (ξ)$ is extremal for the problem(**) $\int_{ξ}^{b} h (t) ψ (t) d t \to s u p$ , $h \in H^{ω} [ξ, b]$ , h(ξ) = 0,for an appropriate choice of the kernel ψ with a finite...

On the Fejér means of bounded Ciesielski systems

Ferenc Weisz (2001)

Studia Mathematica

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We investigate the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system arises from the Haar system. It is shown that the maximal operator of the Fejér means of the Ciesielski-Fourier series is bounded from the Hardy space $H_{p}$ to $L_{p}$ if 1/2 < p < ∞ and m ≥ 0, |k| ≤ m + 1. Moreover, it is of weak type (1,1). As a consequence, the Fejér means of the Ciesielski-Fourier series of a function f converges to f a.e. if...

A new algorithm for approximating the least concave majorant

Martin Franců, Ron Kerman, Gord Sinnamon (2017)

Czechoslovak Mathematical Journal

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The least concave majorant, $\hat{F}$ , of a continuous function $F$ on a closed interval, $I$ , is defined by $\hat{F} (x) = inf {G (x) : G \geq F, G concave}, x \in I .$ We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$ . Given any function $F \in 𝒞^{4} (I)$ , it can be well-approximated on $I$ by a clamped cubic spline $S$ . We show that $\hat{S}$ is then a good approximation to $\hat{F}$ . We give two examples, one to illustrate, the other to apply our algorithm. ...

Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces

Aydin Sh. Shukurov (2012)

Colloquium Mathematicae

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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_{p}$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_{p}$ (1 ≤ p < +∞).

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let ${R ₙ}_{n = 1}^{\infty}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$ -space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset C ()$ and $L_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset L ₁ ()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

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

Gegham G. Gevorkyan, Anna Kamont (2004)

Studia Mathematica

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By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , 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 < ∞.

The universal Banach space with a $K$ -suppression unconditional basis

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2018)

Commentationes Mathematicae Universitatis Carolinae

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Using the technique of Fraïssé theory, for every constant $K \geq 1$ , we construct a universal object $𝕌_{K}$ in the class of Banach spaces possessing a normalized $K$ -suppression unconditional Schauder basis.

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

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It is well known that if φ(t) ≡ t, then the system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is a basis in some Lebesgue space $L_{p}$ . The aim of this short note is to show that the answer to this question is negative.

On extending $C^{k}$ functions from an open set to $ℝ$ with applications

Walter D. Burgess, Robert M. Raphael (2023)

Czechoslovak Mathematical Journal

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For $k \in ℕ \cup {\infty}$ and $U$ open in $ℝ$ , let $C^{k} (U)$ be the ring of real valued functions on $U$ with the first $k$ derivatives continuous. It is shown that for $f \in C^{k} (U)$ there is $g \in C^{\infty} (ℝ)$ with $U \subseteq coz g$ and $h \in C^{k} (ℝ)$ with ${f g |}_{U} {= h |}_{U}$ . The function $f$ and its $k$ derivatives are not assumed to be bounded on $U$ . The function $g$ is constructed using splines based on the Mollifier function. Some consequences about the ring $C^{k} (ℝ)$ are deduced from this, in particular that $Q_{cl} (C^{k} (ℝ)) = Q (C^{k} (ℝ))$ .

Corrigendum to the paper “The universal Banach space with a $K$ -suppression unconditional basis”

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2020)

Commentationes Mathematicae Universitatis Carolinae

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We observe that the notion of an almost ${𝔉ℑ}_{K}$ -universal based Banach space, introduced in our earlier paper [1]: Banakh T., Garbulińska-Wegrzyn J., The universal Banach space with a $K$ -suppression unconditional basis, Comment. Math. Univ. Carolin. 59 (2018), no. 2, 195–206, is vacuous for $K = 1$ . Taking into account this discovery, we reformulate Theorem 5.2 from [1] in order to guarantee that the main results of [1] remain valid.

Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

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If $E = e_{i}$ and $F = f_{i}$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i, j}$ with norm $| | a_{i, j} {| |}_{E (F)} = | | \sum_{k} | | \sum_{l} a_{k, l} f_{l} | | e_{k} | |$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

Uniqueness of unconditional basis of $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ , 0 < p < 1

F. Albiac, C. Leránoz (2002)

Studia Mathematica

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We prove that the quasi-Banach spaces $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $ℓ ₁ (c ₀)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

General Haar systems and greedy approximation

Anna Kamont (2001)

Studia Mathematica

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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...

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

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Let $σ_{A} (n) = | (a, a^{'}) \in A ² : a + a^{'} = n |$ , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_{A} (n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_{A} (n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_{A} (n ̅) \leq 768$ for all n̅ ∈ Zₘ.

The Lebesgue constants for the Franklin orthogonal system

Z. Ciesielski, A. Kamont (2004)

Studia Mathematica

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To each set of knots $t_{i} = i / 2 n$ for i = 0,...,2ν and $t_{i} = (i - ν) / n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space $_{ν, n}$ 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 $_{ν, n}$ . The main result is $l i m_{(n - ν) \land ν \to \infty} | | P_{ν, n} | | ₁ = s u p_{ν, n : 1 \leq ν \leq n} | | P_{ν, n} | | ₁ = 2 + (2 - \sqrt 3) ²$ . This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

On the non-equivalence of rearranged Walsh and trigonometric systems in $L_{p}$

Aicke Hinrichs, Jörg Wenzel (2003)

Studia Mathematica

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We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_{p}$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

Displaying similar documents to “Unconditionality of orthogonal spline systems in L p ”

Displaying similar documents to “Unconditionality of orthogonal spline systems in $L^{p}$ ”