Injectivity sets for spherical means on ℝ n and on symmetric spacesand on symmetric spaces

Rawat, Rama; Sitaram, Alladi

doi:10.1007/bf02511160

Cited by 18 publications

(9 citation statements)

References 6 publications

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“…We would like to mention that Heisenberg uniqueness pair up to a certain extent is similar to an annihilating pair of Borel measurable sets of positive measure as described by Havin and Joricke [6]. Further, the notion of Heisenberg uniqueness pair has a sharp contrast to the known results about determining sets for measures by Sitaram et al [3,14], due to the fact that the determining set Λ for the functionμ has also been considered a thin set.…”

Section: Introductionmentioning

confidence: 95%

Heisenberg uniqueness pairs for some algebraic curves in the plane

Giri

Srivastava

2017

Advances in Mathematics

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Abstract. A Heisenberg uniqueness pair is a pair (Γ, Λ), where Γ is a curve and Λ is a set in R 2 such that whenever a finite Borel measure µ having support on Γ which is absolutely continuous with respect to the arc length on Γ satisfiesμ| Λ = 0, then it is identically 0. In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials.

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Section: Introductionmentioning

confidence: 95%

Heisenberg uniqueness pairs for some algebraic curves in the plane

Giri

Srivastava

2017

Advances in Mathematics

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“…Further, the concept of Heisenberg uniqueness pair has a sharp contrast to the known results about determining sets for measures by Sitaram et al [3,18], due to the fact that the determining set Λ for the functionμ has also been considered a thin set. In particular, if Γ is compact, thenμ is a real analytic function having exponential growth and it can vanish on a very delicate set.…”

Section: Introductionmentioning

confidence: 98%

Heisenberg uniqueness pairs for some algebraic curves and surfaces

Giri¹,

Srivastava²

2017

Trans. Amer. Math. Soc.

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Abstract. Let X(Γ) be the space of all finite Borel measure µ in R 2 which is supported on the curve Γ and absolutely continuous with respect to the arc length of Γ. For Λ ⊂ R 2 , the pair (Γ, Λ) is called a Heisenberg uniqueness pair for X(Γ) if any µ ∈ X(Γ) satisfiesμ| Λ = 0, implies µ = 0. We explore the Heisenberg uniqueness pairs corresponding to the cross, exponential curves, and surfaces. Then, we prove a characterization of the Heisenberg uniqueness pairs corresponding to finitely many parallel lines. We observe that the size of the determining sets Λ for X(Γ) depends on the number of lines and their irregular distribution that further relates to a phenomenon of interlacing of certain trigonometric polynomials.

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“…If the univariate test is distribution-free, then so is the multivariate test. In Section 2 we show that a result by Rawat and Sitaram (2000) in integral geometry implies that if H 0 is false, then applying a univariate consistent test on distances from a single center point will result, for almost every center point, in a multivariate test with power increasing to one as the sample size increases.…”

Section: Introductionmentioning

confidence: 99%

Multivariate tests of association based on univariate tests

Heller,

Heller

2016

Preprint

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For testing two random vectors for independence, we consider testing whether the distance of one vector from a center point is independent from the distance of the other vector from a center point by a univariate test. In this paper we provide conditions under which it is enough to have a consistent univariate test of independence on the distances to guarantee that the power to detect dependence between the random vectors increases to one, as the sample size increases. These conditions turn out to be minimal. If the univariate test is distribution-free, the multivariate test will also be distribution-free. If we consider multiple center points and aggregate the center-specific univariate tests, the power may be further improved, and the resulting multivariate test may be distribution-free for specific aggregation methods (if the univariate test is distribution-free). We show that several multivariate tests recently proposed in the literature can be viewed as instances of this general approach.

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Injectivity sets for spherical means on ℝ n and on symmetric spacesand on symmetric spaces

Cited by 18 publications

References 6 publications

Heisenberg uniqueness pairs for some algebraic curves in the plane

Heisenberg uniqueness pairs for some algebraic curves in the plane

Heisenberg uniqueness pairs for some algebraic curves and surfaces

Multivariate tests of association based on univariate tests

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