Displaying similar documents to “Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series”

Norm convergence of some power series of operators in L p with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

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Let X be a closed subspace of L p ( μ ) , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that s u p n | | U | | < . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like n 1 ( U f ) / n 1 - α , 0 ≤ α < 1, in terms of | | f + + U n - 1 f | | p , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

An M q ( ) -functional calculus for power-bounded operators on certain UMD spaces

Earl Berkson, T. A. Gillespie (2005)

Studia Mathematica

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For 1 ≤ q < ∞, let q ( ) denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes q ( ) as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction...

Generalized absolute convergence of single and double Vilenkin-Fourier series and related results

Nayna Govindbhai Kalsariya, Bhikha Lila Ghodadra (2024)

Mathematica Bohemica

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We consider the Vilenkin orthonormal system on a Vilenkin group G and the Vilenkin-Fourier coefficients f ^ ( n ) , n , of functions f L p ( G ) for some 1 < p 2 . We obtain certain sufficient conditions for the finiteness of the series n = 1 a n | f ^ ( n ) | r , where { a n } is a given sequence of positive real numbers satisfying a mild assumption and 0 < r < 2 . We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of f and give multiplicative...

On the order of magnitude of Walsh-Fourier transform

Bhikha Lila Ghodadra, Vanda Fülöp (2020)

Mathematica Bohemica

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For a Lebesgue integrable complex-valued function f defined on + : = [ 0 , ) let f ^ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that f ^ ( y ) 0 as y . But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of L 1 ( + ) there is a definite rate at which the Walsh-Fourier transform tends...

Multiparameter ergodic Cesàro-α averages

A. L. Bernardis, R. Crescimbeni, C. Ferrari Freire (2015)

Colloquium Mathematicae

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Net (X,ℱ,ν) be a σ-finite measure space. Associated with k Lamperti operators on L p ( ν ) , T , . . . , T k , n ̅ = ( n , . . . , n k ) k and α ̅ = ( α , . . . , α k ) with 0 < α j 1 , we define the ergodic Cesàro-α̅ averages n ̅ , α ̅ f = 1 / ( j = 1 k A n j α j ) i k = 0 n k i = 0 n j = 1 k A n j - i j α j - 1 T k i k T i f . For these averages we prove the almost everywhere convergence on X and the convergence in the L p ( ν ) norm, when n , . . . , n k independently, for all f L p ( d ν ) with p > 1/α⁎ where α = m i n 1 j k α j . In the limit case p = 1/α⁎, we prove that the averages n ̅ , α ̅ f converge almost everywhere on X for all f in the Orlicz-Lorentz space Λ ( 1 / α , φ m - 1 ) with φ ( t ) = t ( 1 + l o g t ) m . To obtain the result in the limit case we need...

Transference of weak type bounds of multiparameter ergodic and geometric maximal operators

Paul Hagelstein, Alexander Stokolos (2012)

Fundamenta Mathematicae

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Let U , . . . , U d be a non-periodic collection of commuting measure preserving transformations on a probability space (Ω,Σ,μ). Also let Γ be a nonempty subset of d and the associated collection of rectangular parallelepipeds in d with sides parallel to the axes and dimensions of the form n × × n d with ( n , . . . , n d ) Γ . The associated multiparameter geometric and ergodic maximal operators M and M Γ are defined respectively on L ¹ ( d ) and L¹(Ω) by M g ( x ) = s u p x R 1 / | R | R | g ( y ) | d y and M Γ f ( ω ) = s u p ( n , . . . , n d ) Γ 1 / n n d j = 0 n - 1 j d = 0 n d - 1 | f ( U j U d j d ω ) | . Given a Young function Φ, it is shown that M satisfies the weak type estimate ...

Spectral radius of weighted composition operators in L p -spaces

Krzysztof Zajkowski (2010)

Studia Mathematica

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We prove that for the spectral radius of a weighted composition operator a T α , acting in the space L p ( X , , μ ) , the following variational principle holds: l n r ( a T α ) = m a x ν M ¹ α , e X l n | a | d ν , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, M ¹ α , e is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and -measurable function, where = n = 0 α - n ( ) . This considerably extends the range of validity of the above formula, which was previously known...

On L p integrability and convergence of trigonometric series

Dansheng Yu, Ping Zhou, Songping Zhou (2007)

Studia Mathematica

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We first give a necessary and sufficient condition for x - γ ϕ ( x ) L p , 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either k = 1 a k c o s k x or k = 1 b k s i n k x , 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)...

On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi (2009)

Colloquium Mathematicae

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Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure m G . We consider the map τ φ defined on X × G by τ φ : ( x , g ) ( τ x , φ ( x ) g ) and the cocycle ( φ ) n generated by φ. Using a characterization of the ergodic invariant measures for τ φ , we give the form of the ergodic decomposition of μ ( d x ) m G ( d g ) or more generally of the τ φ -invariant measures μ χ ( d x ) χ ( g ) m G ( d g ) , where μ χ ( d x ) is χ∘φ-conformal for an exponential χ on G.

A note on average behaviour of the Fourier coefficients of j th symmetric power L -function over certain sparse sequence of positive integers

Youjun Wang (2024)

Czechoslovak Mathematical Journal

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Let j 2 be a given integer. Let H k * be the set of all normalized primitive holomorphic cusp forms of even integral weight k 2 for the full modulo group SL ( 2 , ) . For f H k * , denote by λ sym j f ( n ) the n th normalized Fourier coefficient of j th symmetric power L -function ( L ( s , sym j f ) ) attached to f . We are interested in the average behaviour of the sum n = a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 x ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) 6 λ sym j f 2 ( n ) , where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

Pointwise Fourier inversion of distributions on spheres

Francisco Javier González Vieli (2017)

Czechoslovak Mathematical Journal

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Given a distribution T on the sphere we define, in analogy to the work of Łojasiewicz, the value of T at a point ξ of the sphere and we show that if T has the value τ at ξ , then the Fourier-Laplace series of T at ξ is Abel-summable to τ .

The harmonic Cesáro and Copson operators on the spaces L p ( ) , 1 ≤ p ≤ 2

Ferenc Móricz (2002)

Studia Mathematica

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The harmonic Cesàro operator is defined for a function f in L p ( ) for some 1 ≤ p < ∞ by setting ( f ) ( x ) : = x ( f ( u ) / u ) d u for x > 0 and ( f ) ( x ) : = - - x ( f ( u ) / u ) d u for x < 0; the harmonic Copson operator ℂ* is defined for a function f in L ¹ l o c ( ) by setting * ( f ) ( x ) : = ( 1 / x ) x f ( u ) d u 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 f L p ( ) for some 1 ≤ p ≤ 2, then ( ( f ) ) ( t ) = * ( f ̂ ) ( t ) a.e., where f̂ denotes the Fourier transform of f. (ii) If f L p ( ) for some 1 < p ≤ 2, then...

On the convergence to 0 of mₙξmod 1

Bassam Fayad, Jean-Paul Thouvenot (2014)

Acta Arithmetica

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We show that for any irrational number α and a sequence m l l of integers such that l i m l | | | m l α | | | = 0 , there exists a continuous measure μ on the circle such that l i m l | | | m l θ | | | d μ ( θ ) = 0 . This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence m l l of integers such that | | | m l α | | | 0 and such that m l θ [ 1 ] is dense on the circle if and only if θ ∉ ℚα + ℚ.

Solution of a functional equation on compact groups using Fourier analysis

Abdellatif Chahbi, Brahim Fadli, Samir Kabbaj (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let G be a compact group, let n N { 0 , 1 } be a fixed element and let σ be a continuous automorphism on G such that σ n = I . Using the non-abelian Fourier transform, we determine the non-zero continuous solutions f : G C of the functional equation f ( x y ) + k = 1 n - 1 f ( σ k ( y ) x ) = n f ( x ) f ( y ) , x , y G , in terms of unitary characters of G .

Moving averages

S. V. Butler, J. M. Rosenblatt (2008)

Colloquium Mathematicae

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In ergodic theory, certain sequences of averages A k f may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence A m k f of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are A k f ( x ) = 1 / ( 2 k ) j = 4 k + 1 4 k + 2 k f ( T j x ) , then the subsequence A k ² f will not be pointwise good even on L , but the subsequence A 2 k f will be pointwise good on L¹. Understanding when the hyperexponential...

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart, Yanick Heurteaux (2012)

Annales scientifiques de l'École Normale Supérieure

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A famous theorem of Carleson says that, given any function f L p ( 𝕋 ) , p ( 1 , + ) , its Fourier series ( S n f ( x ) ) converges for almost every x 𝕋 . Beside this property, the series may diverge at some point, without exceeding O ( n 1 / p ) . We define the divergence index at  x as the infimum of the positive real numbers β such that S n f ( x ) = O ( n β ) and we are interested in the size of the exceptional sets E β , namely the sets of  x 𝕋 with divergence index equal to  β . We show that quasi-all functions in  L p ( 𝕋 ) have a multifractal behavior with respect to...

Characterization of surjective convolution operators on Sato's hyperfunctions

Michael Langenbruch (2010)

Banach Center Publications

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Let μ ( d ) ' be an analytic functional and let T μ be the corresponding convolution operator on Sato’s space ( d ) of hyperfunctions. We show that T μ is surjective iff T μ admits an elementary solution in ( d ) iff the Fourier transform μ̂ satisfies Kawai’s slowly decreasing condition (S). We also show that there are 0 μ ( d ) ' such that T μ is not surjective on ( d ) .

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

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Let f = n = 1 a ( n ) q n S k + 1 / 2 ( N , χ 0 ) be a nonzero cuspidal Hecke eigenform of weight k + 1 2 and the trivial nebentypus χ 0 , where the Fourier coefficients a ( n ) are real. Bruinier and Kohnen conjectured that the signs of a ( n ) are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies { a ( t n 2 ) } n , where t is a squarefree integer such that a ( t ) 0 . Let q and d be natural numbers such that ( d , q ) = 1 . In this work, we show that { a ( t n 2 ) } n is equidistributed over any arithmetic progression n d mod q .

A transplantation theorem for ultraspherical polynomials at critical index

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 λ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients c ( λ ) ( f ) of λ -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any f λ the series n = 1 c ( λ ) ( f ) c o s n θ is the Fourier series of some function φ ∈ ReH¹ with | | φ | | R e H ¹ c | | f | | λ . ...

Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

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Let G : = SO ( n , 1 ) and Γ ( n - 1 ) / 2 for n = 2 , 3 and when δ > n - 2 for n 4 , we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family { T H G } of compact subsets, there exists η > 0 such that # [ e ] Γ T = ( T ) + O ( ( T ) 1 - η ) for an explicit measure on H G which depends on Γ . We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic...

Truncation and Duality Results for Hopf Image Algebras

Teodor Banica (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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Associated to an Hadamard matrix H M N ( ) is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with G S N . We study a certain family of discrete measures μ r [ 0 , N ] , coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type 0 N ( x / N ) p d μ r ( x ) = 0 N ( x / N ) r d ν p ( x ) , where μ r , ν r are the truncations of the spectral measures μ,ν associated to H , H t . We also prove, using these truncations μ r , ν r , that for any deformed Fourier matrix H = F M Q F N we have μ = ν.

A variation norm Carleson theorem

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 L p estimates for the r -variation of the partial sum operators for Fourier series and integrals, for r > 𝚖𝚊𝚡 { p ' , 2 } . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

On the higher power moments of cusp form coefficients over sums of two squares

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = SL ( 2 , ) . Denote by λ f ( n ) the n th normalized Fourier coefficient of f . We are interested in the average behaviour of the sum a 2 + b 2 x λ f j ( a 2 + b 2 ) for x 1 , where a , b and j 9 is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power L -functions and Rankin-Selberg L -functions.

Perron-Frobenius operators and the Klein-Gordon equation

Francisco Canto-Martín, Håkan Hedenmalm, Alfonso Montes-Rodríguez (2014)

Journal of the European Mathematical Society

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For a smooth curve Γ and a set Λ in the plane 2 , let A C ( Γ ; Λ ) be the space of finite Borel measures in the plane supported on Γ , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ . Following [12], we say that ( Γ , Λ ) is a Heisenberg uniqueness pair if A C ( Γ ; Λ ) = { 0 } . In the context of a hyperbola Γ , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly...

H calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the H calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if - A generates a bounded analytic C 0 semigroup ( T t ) on a UMD space, then the H calculus of A is bounded if and only if ( T t ) has a dilation to a bounded group on L 2 ( [ 0 , 1 ] , X ) . This generalises a Hilbert space result of C.LeMerdy. If X is an L p space we can choose another L p space in place of L 2 ( [ 0 , 1 ] , X ) .