On the Localization of Rectangular Partial Sums for Multiple Fourier Series

Liu, Fon-Che

doi:10.2307/2037904

Cited by 2 publications

(3 citation statements)

References 3 publications

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“…This follows directly from either an inequality of Sangwine-Yager or one of Brannen; see Theorem 1 or Corollary 2 of [13], respectively. The estimate (35) both generalizes and strengthens [23, Lemma 4.2], which concerns the case n = 2. The authors of the latter paper were unaware that an even stronger estimate for n = 2 was found earlier by Matheron [39].…”

Section: N and Letsupporting

confidence: 77%

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Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

Bianchi¹,

Gardner²,

Kiderlen³

2011

J. Amer. Math. Soc.

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We propose strongly consistent algorithms for reconstructing the characteristic function 1 K of an unknown convex body K in R n from possibly noisy measurements of the modulus of its Fourier transform 1 K . This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k n ) noisy covariogram measurements to construct a convex polytope P k that approximates K or its reflection −K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all threedimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between P k and ±K tends to zero as k tends to infinity.

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Section: N and Letsupporting

confidence: 77%

“…If g is also Lipschitz, then by [35,Theorem 3], the square partial sums z∈Z n k c z e iπz•x/L of the Fourier series of g converge uniformly to g. Therefore, if g is also an even function, we can write (79)…”

Section: Phase Retrieval: Framework and Technical Lemmasmentioning

confidence: 99%

Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

Bianchi¹,

Gardner²,

Kiderlen³

2011

J. Amer. Math. Soc.

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show abstract

“…The absolute convergence and summability of multiple Fourier series and integrals and also of more general expansions have been investigated in [54,56,85,98,125,128,157,164,181,198,206,221,222,223,234,237,327,[362][363][364][365][366]376,383,384,419,446,462]. (0 = @ l f + ty) as,v (y), 9 9 N while the integration is taken over the surface of the unit sphere S N, then N--1…”

Section: N4-omentioning

confidence: 99%

Multiple fourier series and integrals

Голубов¹

1984

J Math Sci

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On the Localization of Rectangular Partial Sums for Multiple Fourier Series

Abstract: Abstract.The question of the localization for rectangular partial sums of the multiple Fourier series for functions of Sobolev spaces is settled.

Cited by 2 publications

References 3 publications

Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

Multiple fourier series and integrals

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