$L^1$ convergence of the reconstruction formula for the potential function

Yating, Chen; Cheng, Yan-Hsiou; Law, Chun‐Kong; Tsay, Jhishen

doi:10.1090/s0002-9939-02-06297-4

Cited by 38 publications

(11 citation statements)

References 14 publications

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“…Let Σ 1 ⊂ Σ be the subspace of all asymptotically equivalent nodal sequences and let Σ * 2 , then the corresponding function F n converges to q in L 1 (0, 1) as well as pointwisely a.e. It would be interesting to know whether the space Σ 1 can be expanded to include all sequences that converge to q in some way, which seems to be a more natural space.…”

Section: Introductionmentioning

confidence: 99%

On the quasi-nodal map for the Sturm–Liouville problem

Cheng¹,

Law²

2006

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We show that the space of Sturm–Liouville operators characterized by H = (q, α, β) ∈ L1 (0, 1) × [0, π)2 such that is homeomorphic to the partition set of the space of all admissible sequences which form sequences that converge to q, α, and β individually. This space, Γ, of quasi-nodal sequences is a superset of, and is more natural than, the space of asymptotically nodal sequences defined in Law and Tsay (On the well-posedness of the inverse nodal problem. Inv. Probl.17 (2001), 1493–1512). The definition of Γ relies on the L1 convergence of the reconstruction formula for q by the exactly nodal sequence.

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Section: Introductionmentioning

confidence: 99%

On the quasi-nodal map for the Sturm–Liouville problem

Cheng¹,

Law²

2006

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…This issue is well studied now and called the inverse nodal problem (cf. [9,10,16,24,27,35,41] etc.). For the problem (1.5), we have the following Theorem 1.3 (Uniqueness).…”

Section: Introductionmentioning

confidence: 99%

On Nodal Properties for Some Nonlinear Conformable Fractional Differential Equations

Wang

Cheng

2022

Taiwanese J. Math.

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A class of conformable fractional differential equationsis considered. We first give a sufficient condition for the existence of sign-changing solutions with the prescribed number of zeros to this problem. On the basis of this result, we turn to a specific case of the above problem and give a uniqueness theorem related to the function ω. Essentially, the main methods using in this work are properties of conformable fractional calculus, the scaling argument and Prüfer-type substitutions.

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“…Later on, some remarkable results were obtained by some authors. For example, Chen, Law, Koyunbakan, Pinasco, etc have reconstructed the potential function and its derivatives from nodal points (zeros of eigenfunctions) in previous studies . In addition to the above, inverse nodal problem was extensively studied for some different problems in other studies .…”

Section: Introductionmentioning

confidence: 99%

Inverse nodal problem for polynomial pencil of Sturm‐Liouville operator

Goktas

Koyunbakan

Gulsen

2018

Math Methods in App Sciences

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The paper is about boundary value problem for polynomial pencil of Sturm‐Liouville operators. Especially, we find all coefficients of the operator by using nodal points (zeros of eigenfunctions). Regularly, we find eigenvalues, nodal points, and nodal lengths by Prüfer substitution. These results are used to give a reconstruction formula for all complex functions qd(x), d=true0,n−1‾, which are known potentials in the theory. However, method is similar with some papers; our results more general then because of including many potential functions.

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$L^1$ convergence of the reconstruction formula for the potential function

Abstract: Abstract. It is known that the potential function of the Sturm-Liouville problem can be reconstructed from the nodal data by a pointwise limit. We show that this convergence is in fact L 1 .

Cited by 38 publications

References 14 publications

On the quasi-nodal map for the Sturm–Liouville problem

On the quasi-nodal map for the Sturm–Liouville problem

On Nodal Properties for Some Nonlinear Conformable Fractional Differential Equations

Inverse nodal problem for polynomial pencil of Sturm‐Liouville operator

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