Unconditionality of general Franklin systems in L<sup>p</sup>[0,1], 1&lt;p&lt;∞

Gevorkyan, G. G.; Kamont, Anna

doi:10.4064/sm164-2-4

Cited by 32 publications

(25 citation statements)

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“…the case k = 2) forms an unconditional basis in L p [0, 1], 1 < p < ∞. Here we obtain an estimate for general periodic orthonormal spline functions, which combined with the methods developed in [10] yield the unconditionality of periodic orthonormal spline systems in L p (T).…”

Section: Introductionmentioning

confidence: 94%

“…Considerable effort has been made in the past to weaken the restriction to dyadic knot sequences. In the series of papers [9,11,10] this restriction was removed step-by-step for general Franklin systems, with the final result that it was shown for each admissible point sequence (t n ) n≥0 with parameter k = 2, the associated general Franklin system forms an unconditional basis in L p [0, 1], 1 < p < ∞. Combining the methods used in [11,10] with some new inequalities from [15] it was proved in [13] that non-periodic orthonormal spline systems are unconditional bases in L p [0, 1], 1 < p < ∞, for any spline order k and any admissible point sequence (t n ).…”

Section: Introductionmentioning

confidence: 99%

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Unconditionality of periodic orthonormal spline systems in $L^p$

Keryan

Passenbrunner

2019

Studia Math.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Unconditionality of periodic orthonormal spline systems in $L^p$

Keryan

Passenbrunner

2019

Studia Math.

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“…It is clear that if the sequence T is weakly regular, then |J n | ∼ γ |Δ n |. Therefore, in the estimates obtained in papers [2] and [5] for function f n we can replace J n by Δ n . Now we list some estimates for Franklin function, obtained in [2] and [5].…”

Section: Some Properties Of a General Franklin Systemmentioning

confidence: 99%

“…Therefore, in the estimates obtained in papers [2] and [5] for function f n we can replace J n by Δ n . Now we list some estimates for Franklin function, obtained in [2] and [5]. First, there exists a constant C γ , such that for all n and x…”

Section: Some Properties Of a General Franklin Systemmentioning

confidence: 99%

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On absolute convergence of series by a general Franklin system

Gevorkyan

2014

J. Contemp. Mathemat. Anal.

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The paper considers general Franklin systems generated by strong regular partitions of the segment [0; 1]. For such systems we prove the following assertions: 1) the absolute convergence of a Fourier-Franklin series at a point is a local property; 2) the Fourier-Franklin series of a function of bounded variation absolute converges almost everywhere; 3) any almost everywhere finite measurable function can be represented by an almost everywhere absolutely convergent series by a general Franklin system. MSC2010 numbers : 42C15; 42C25

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Properties of local orthonormal systems Part I: Unconditionality in Lp$L^p$, 1<p<∞$1<p<\infty$

Gulgowski,

Kamont,

Passenbrunner

2024

Mathematische Nachrichten

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Assume that we are given a filtration on a probability space of the form that each is generated by the partition of one atom of into two atoms of having positive measure. Additionally, assume that we are given a finite‐dimensional linear space S of ‐measurable, bounded functions on Ω so that on each atom A of any σ‐algebra , all ‐norms of functions in S are comparable independently of n or A. Denote by the space of functions that are given locally, on atoms of , by functions in S and by the orthoprojector (with respect to the inner product in ) onto . Since satisfies the above assumption and is then the conditional expectation with respect to , for such filtrations, martingales are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences form an unconditionally convergent series and are democratic in for . This implies that those differences form a greedy basis in ‐spaces for .

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

Cited by 32 publications

References 6 publications

Unconditionality of periodic orthonormal spline systems in $L^p$

Unconditionality of periodic orthonormal spline systems in $L^p$

On absolute convergence of series by a general Franklin system

Properties of local orthonormal systems Part I: Unconditionality in Lp$L^p$, 1<p<∞$1<p<\infty$

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Unconditionality of general Franklin systems in Lp[0,1], 1<p<∞

Cited by 32 publications

References 6 publications

Unconditionality of periodic orthonormal spline systems in $L^p$

Unconditionality of periodic orthonormal spline systems in $L^p$

On absolute convergence of series by a general Franklin system

Properties of local orthonormal systems Part I: Unconditionality in Lp$L^p$, 1<p<∞$1&lt;p&lt;\infty$

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

Properties of local orthonormal systems Part I: Unconditionality in Lp$L^p$, 1<p<∞$1<p<\infty$