Functions with bounded spectrum

Bang, Ha Huy

doi:10.1090/s0002-9947-1995-1283539-1

Cited by 30 publications

(11 citation statements)

References 6 publications

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“…Here we investigate the support of a function only in terms of its Mellin (or inverse Mellin) transform without passing to the complexification. Similar "real variable" results have been obtained for the Fourier transform [1,2,7,8] , the Hankel transform [10], the Y-transform [9], and the Airy transform [11].…”

Section: Introductionsupporting

confidence: 71%

“…for the inverse Mellin transform (2). Similarly, the Hausdorff-Young inequality has the form M γ f Lp(R n ) ≤ C τ γ− 1 q f (τ )…”

Section: Introductionmentioning

confidence: 97%

“…for the inverse Mellin transform (2). Hereafter q is assumed to be the conjugate exponent of p (i.e., p −1 + q −1 = 1), and C is a universal constant.…”

Section: Introductionmentioning

confidence: 99%

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New type Paley-Wiener theorems for the modified multidimensional Mellin transform

Tuan

1998

The Journal of Fourier Analysis and Applications

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Section: Introductionsupporting

confidence: 71%

“…for the inverse Mellin transform (2). Similarly, the Hausdorff-Young inequality has the form M γ f Lp(R n ) ≤ C τ γ− 1 q f (τ )…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

New type Paley-Wiener theorems for the modified multidimensional Mellin transform

Tuan

1998

The Journal of Fourier Analysis and Applications

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“…Bang, on his own and with coauthors, has since generalized (4.2) to the Fourier transform in Euclidean space in higher dimensions, as well as to Orlicz spaces and Lorentz spaces (with the appropriate associated operators); see [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32] and [33]. In [19], Bang also covers the case of L p -functions on the one-dimensional torus for P (x) = x, again using a complex Paley-Wiener theorem.…”

Section: Literature Reviewmentioning

confidence: 99%

Real Paley–Wiener theorems and local spectral radius formulas

Andersen¹,

Jeu²

2010

Trans. Amer. Math. Soc.

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Abstract. We systematically develop real Paley-Wiener theory for the Fourier transform on R d for Schwartz functions, L p -functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given as to why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley-Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein, and furthermore we give evidence that a number of real PaleyWiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real PaleyWiener theory is included.

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“…The classical Paley-Wiener theorem [10] describes functions of compact spectrum through its analytic extension into the complex space and its growth at infinity. In [1,11] functions with compact spectrum were described through the norms of differential operators of infinite order. So by a Paley-Wiener-type theorem we understand any result describing functions with spectrum of a given configuration.…”

Section: Introductionmentioning

confidence: 99%