Sidon sets and -sets
David Grow (1987)
Colloquium Mathematicae
Similarity:
David Grow (1987)
Colloquium Mathematicae
Similarity:
Yuichi Kanjin (2001)
Studia Mathematica
Similarity:
We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space , 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on , 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of .
J. J. Guadalupe, V. I. Kolyada (2001)
Studia Mathematica
Similarity:
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 . ...
S. Thangavelu (2002)
Colloquium Mathematicae
Similarity:
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...
Fabio Nicola (2010)
Studia Mathematica
Similarity:
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,...
José García-Cuerva
Similarity:
CONTENTSIntroduction.......................................................................................................................................................... 5Chapter I. Some preliminary lemmas............................................................................................................ 8Chapter II. Weighted spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary..........................................................................................................................
E. Ferreyra, T. Godoy (2006)
Colloquium Mathematicae
Similarity:
Let φ:ℝ ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let μ be the Borel measure on ℝ ³ defined by with D = x ∈ ℝ ²:|x| ≤ 1 and let be the convolution operator with the measure μ. Let be the decomposition of φ into irreducible factors. We show that if for each of degree 1, then the type set can be explicitly described as a closed polygonal region.
Abdellatif Chahbi, Brahim Fadli, Samir Kabbaj (2015)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
Similarity:
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 .
B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)
Studia Mathematica
Similarity:
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...
Epperson Jay, Frazier Michael
Similarity:
Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form , in which each function and is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...
S. V. Konyagin, V. N. Temlyakov (2003)
Studia Mathematica
Similarity:
We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form , where is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...
Ferenc Móricz (2002)
Studia Mathematica
Similarity:
The harmonic Cesàro operator is defined for a function f in for some 1 ≤ p < ∞ by setting for x > 0 and for x < 0; the harmonic Copson operator ℂ* is defined for a function f in by setting for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If for some 1 ≤ p ≤ 2, then a.e., where f̂ denotes the Fourier transform of f. (ii) If for some 1 < p ≤ 2, then...
Guanghui Lu, Shuangping Tao (2017)
Open Mathematics
Similarity:
The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ generated by the Littlewood-Paley operator [...] Mκ∗ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from...
Dachun Yang (2005)
Studia Mathematica
Similarity:
Let be a space of homogeneous type, i.e. X is a set, ϱ is a quasi-metric on X with the property that there are constants θ ∈ (0,1] and C₀ > 0 such that for all x,x’,y ∈ X, , and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, . Let ε ∈ (0,θ], |s| < ε and maxd/(d+ε),d/(d+s+ε) < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces and establishes their frame characterizations by first establishing a Plancherel-Pólya-type...
Earl Berkson (2014)
Studia Mathematica
Similarity:
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,...
Dansheng Yu, Ping Zhou, Songping Zhou (2007)
Studia Mathematica
Similarity:
We first give a necessary and sufficient condition for , 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either or , under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x)...
Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)
Journal of the European Mathematical Society
Similarity:
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.
Abdellah Youssfi (1995)
Annales de l'institut Fourier
Similarity:
In this paper we consider the regularity problem for the commutators where is a locally integrable function and are the Riesz transforms in the -dimensional euclidean space . More precisely, we prove that these commutators are bounded from into the Besov space for and if and only if is in the -Triebel-Lizorkin space . The reduction of our result to the case gives in particular that the commutators are bounded form into the Sobolev space if and only if ...
István Blahota, György Gát, Ushangi Goginava (2007)
Colloquium Mathematicae
Similarity:
The main aim of this paper is to prove that the maximal operator of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space to the space weak-.
Artūras Dubickas (2012)
Annales Polonici Mathematici
Similarity:
Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let be the mth coefficient of the square f(x)² of...
In Sook Park (2005)
Studia Mathematica
Similarity:
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
Similarity:
Frédéric Bayart, Yanick Heurteaux (2012)
Annales scientifiques de l'École Normale Supérieure
Similarity:
A famous theorem of Carleson says that, given any function , , its Fourier series converges for almost every . Beside this property, the series may diverge at some point, without exceeding . We define the divergence index at as the infimum of the positive real numbers such that and we are interested in the size of the exceptional sets , namely the sets of with divergence index equal to . We show that quasi-all functions in have a multifractal behavior with respect to...
Sichun Wang (2006)
Studia Mathematica
Similarity:
Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator and its associated global maximal operator by , f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, , f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...
Rémi Arcangéli, Juan José Torrens (2013)
Studia Mathematica
Similarity:
We collect and extend results on the limit of as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and is the intrinsic seminorm of order l+σ in the Sobolev space . In general, the above limit is equal to , where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.
Chike Obi (1979)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
Similarity:
Si dimostra che l'equazione considerata nel titolo ammette soluzioni periodiche rappresentabili da una serie di Fourier con un numero finito di termini solo se è lineare.
Finbarr Holland, David Walsh (1995)
Studia Mathematica
Similarity:
Let 1 < p < ∞, q = p/(p-1) and for define , x > 0. Moser’s Inequality states that there is a constant such that where is the unit ball of . Moreover, the value a = 1 is sharp. We observe that f where the integral operator has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...
E. Ferreyra, T. Godoy, M. Urciuolo (2002)
Colloquium Mathematicae
Similarity:
Let , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let , and . Let φ₁,...,φₙ be real functions in such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on given by , where and dx denotes the Lebesgue measure on ℝⁿ. Let and let be the operator norm of from into , where the spaces are taken with respect to the Lebesgue measure. The type set is defined by . In the case for 1 ≤ i,k ≤ n we characterize the...
T. Godoy, P. Rocha (2013)
Colloquium Mathematicae
Similarity:
We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by , where , w = (w₁,...,wₙ) ∈ ℂⁿ, , and η(w) = η₀(|w|²) with . We characterize the set of pairs (p,q) such that the convolution operator with ν is bounded. We also obtain -improving properties of measures supported on the graph of the function .
Richard Lechner
Similarity:
We prove an interpolatory estimate linking the directional Haar projection to the Riesz transform in the context of Bochner-Lebesgue spaces , 1 < p < ∞, provided X is a UMD-space. If , the result is the inequality , (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of . In order to obtain the interpolatory result (1) we analyze stripe operators , λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....
Yayuan Xiao (2017)
Czechoslovak Mathematical Journal
Similarity:
We obtain the boundedness of Calderón-Zygmund singular integral operators of non-convolution type on Hardy spaces for , where is a space of homogeneous type in the sense of Coifman and Weiss (1971), and is the regularity exponent of the kernel of the singular integral operator . Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast...