The Cantor function

Dovgoshey, Oleksiy; Мартио, Олли; Ryazanov, Vladimir; Вуоринен, Матти

doi:10.1016/j.exmath.2005.05.002

Cited by 127 publications

(71 citation statements)

References 37 publications

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“…We now derive a recurrence relation for the moments of the weighted Cantor measure. We note that the relation exhibits the approximation in (6). While the relation was shown in [2], the proof of Theorem 3.2 as presented in this paper is original.…”

Section: Moments Of the Weighted Cantor Measurementioning

confidence: 66%

Moments of the weighted Cantor measures

Harding

Riasanovsky

2019

Demonstratio Mathematica

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Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure µ α is the unique Borel probability measure onIn Sections 1 and 2 we examine several general properties of the measure µ α and the associated Legendre polynomials in L 2 µ α [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of µ α , (2) characterize precisely when the moments I m = [0,1] x m dµ α exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O((log log(1/ε)) · m log m). We also state analogous results in the natural case where α is palindromic for the measure ν α attained by shifting µ α to [−1/2, 1/2]. N weight vectors. We now paraphrase Hutchinson's result.

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Section: Moments Of the Weighted Cantor Measurementioning

confidence: 66%

Moments of the weighted Cantor measures

Harding

Riasanovsky

2019

Demonstratio Mathematica

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“…It is manifest that C c (X) = R, where X = [0, 1]. But the Cantor function f is a monotonic nonconstant continuous function, and C c f = [0, 1]\C = [0, 1], where C is the Cantor set, see [9], [5]. Therefore the Cantor function f belongs to…”

Section: It Is Manifest Thatmentioning

confidence: 99%

On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

2015

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Let Cc(X) = {f ∈ C(X) : |f (X)| ≤ ℵ0}, C F (X) = {f ∈ C(X) : |f (X)| < ∞}, and Lc(X) = {f ∈ C(X) : C f = X}, where C f is the union of all open subsets U ⊆ X such that |f (U )| ≤ ℵ0, and CF (X) be the socle of C(X) (i.e., the sum of minimal ideals of C(X)). It is shown that if X is a locally compact space, then Lc(X) = C(X) if and only if X is locally scattered. We observe that Lc(X) enjoys most of the important properties which are shared by C(X) and Cc(X). Spaces X such that Lc(X) is regular (von Neumann) are characterized. Similarly to C(X) and Cc(X), it is shown that Lc(X) is a regular ring if and only if it is ℵ0-selfinjective. We also determine spaces X such that Soc Lc(X) = CF (X) (resp.,Ri, where Ri = R for each i, and X has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces X for which CF (X) is a prime ideal in Lc(X) are characterized and consequently for these spaces, we infer that Lc(X) can not be isomorphic to any C(Y ).

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“…The corresponding Cantor type function has the symmetric Lebesgue derivative C1 at every point in C except ends, see e.g. the survey [22]. Then by the Fatou theorem, see e.g.…”

Section: On Dimension Of Spaces Of Solutionsmentioning

confidence: 99%

On the Riemann-Hilbert problem in multiply connected domains

Ryazanov

2016

Open Mathematics

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Abstract:We proved the existence of multivalent solutions with the infinite number of branches for the RiemannHilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthermore, it is shown that the dimension of the spaces of these solutions is infinite.

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The Cantor function

Cited by 127 publications

References 37 publications

Moments of the weighted Cantor measures

Moments of the weighted Cantor measures

On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

On the Riemann-Hilbert problem in multiply connected domains

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