Sidon sets and -sets
David Grow (1987)
Colloquium Mathematicae
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David Grow (1987)
Colloquium Mathematicae
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E. Ferreyra, T. Godoy, M. Urciuolo (2004)
Studia Mathematica
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Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from to for certain p,q. For m ≥ 6 the results are sharp except for some border points.
B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)
Studia Mathematica
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We determine the norm in , 1 < p < ∞, of the operator , where and are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real...
Hans Wallin (1965)
Annales de l'institut Fourier
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Soit un sous-ensemble compact de ayant des points intérieurs et soit la distribution d’équilibre sur de masse totale 1 par rapport au noyau avec pour , et pour . La restriction de à l’intérieur de est absolument continue et a pour densité . On donne une formule explicite pour et, pour une classe générale d’ensembles , on démontre que , définie en réalité sur un ensemble de mesure de Lebesgue nulle, croît comme la distance à la frontière de élevée à la puissance...
In Sook Park (2005)
Studia Mathematica
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Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform if admits a continuous extension . Let denote the collection of such T’s. We show that for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then...
J. P. Schreiber (1971)
Colloquium Mathematicae
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Fabio Nicola (2010)
Studia Mathematica
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We study Fourier integral operators of Hörmander’s type acting on the spaces , 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in . We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on if the mapping is constant on the fibres, of codimension r,...
Saifallah Ghobber, Philippe Jaming (2014)
Studia Mathematica
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The aim of this paper is to prove new uncertainty principles for integral operators with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris’s local uncertainty principle which states that if a nonzero function is highly localized near a single point then (f) cannot be concentrated in a set of finite measure. The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a nonzero function and...
Giancarlo Travaglini (1987)
Colloquium Mathematicae
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Gero Fendler (1985)
Annales de l'institut Fourier
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Let be a locally compact group, for let denote the closure of in the convolution operators on . Denote the dual of which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space . It is shown that on the unit sphere of the topology and the strong -multiplier topology coincide.
Michael Langenbruch (2007)
Studia Mathematica
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We characterize the partial differential operators P(D) admitting a continuous linear right inverse in the space of Fourier hyperfunctions by means of a dual (Ω̅)-type estimate valid for the bounded holomorphic functions on the characteristic variety near . The estimate can be transferred to plurisubharmonic functions and is equivalent to a uniform (local) Phragmén-Lindelöf-type condition.
Earl Berkson (2014)
Studia Mathematica
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Let , where, for 1 ≤ r < ∞, (resp., ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values,...
Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)
Journal of the European Mathematical Society
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We strengthen the Carleson-Hunt theorem by proving estimates for the -variation of the partial sum operators for Fourier series and integrals, for . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.
Abdellatif Chahbi, Brahim Fadli, Samir Kabbaj (2015)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let be a compact group, let be a fixed element and let be a continuous automorphism on such that . Using the non-abelian Fourier transform, we determine the non-zero continuous solutions of the functional equation in terms of unitary characters of .
Andrejs Dunkels (1976)
Annales de l'institut Fourier
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For potentials , where and are certain Schwartz distributions, an inversion formula for is derived. Convolutions and Fourier transforms of distributions in -spaces are used. It is shown that the equilibrium distribution with respect to the Riesz kernel of order , , of a compact subset of has the following property: its restriction to the interior of is an absolutely continuous measure with analytic density which is expressed by an explicit formula.
S. Thangavelu (2002)
Colloquium Mathematicae
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Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let be the heat kernel associated to the Laplace-Beltrami operator and let be the Kostant polynomials. We establish the following...
J. J. Guadalupe, V. I. Kolyada (2001)
Studia Mathematica
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We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients of -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any the series is the Fourier series of some function φ ∈ ReH¹ with . ...
T. Muraleedharan, K. Parthasarathy (1996)
Colloquium Mathematicae
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In the study of spectral synthesis S-sets and C-sets (see Rudin [3]; Reiter [2] uses the terminology Wiener sets and Wiener-Ditkin sets respectively) have been discussed extensively. A new concept of Ditkin sets was introduced and studied by Stegeman in [4] so that, in Reiter’s terminology, Wiener-Ditkin sets are precisely sets which are both Wiener sets and Ditkin sets. The importance of such sets in spectral synthesis and their connection to the C-set-S-set problem (see Rudin [3])...