Displaying similar documents to “Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations”

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...

The Fourier transform in Lebesgue spaces

Erik Talvila (2025)

Czechoslovak Mathematical Journal

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For each f L p ( ) ( 1 p < ) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each p , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to L p ( ) . There is an exchange theorem and inversion in norm.

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...

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...

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 . ...

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)...

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 .

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 .

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).

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...

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 .

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...

Isomorphic properties in spaces of compact operators

Ioana Ghenciu (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the definition of p -limited completely continuous operators, 1 p < . The question of whether a space of operators has the property that every p -limited subset is relative compact when the dual of the domain and the codomain have this property is studied using p -limited completely continuous evaluation operators.

Multiplication operators on L ( L p ) and p -strictly singular operators

William Johnson, Gideon Schechtman (2008)

Journal of the European Mathematical Society

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A classification of weakly compact multiplication operators on L ( L p ) , 1<p< , i s g i v e n . T h i s a n s w e r s a q u e s t i o n r a i s e d b y S a k s m a n a n d T y l l i i n 1992 . T h e c l a s s i f i c a t i o n i n v o l v e s t h e c o n c e p t o f p - s t r i c t l y s i n g u l a r o p e r a t o r s , a n d w e a l s o i n v e s t i g a t e t h e s t r u c t u r e o f g e n e r a l p - s t r i c t l y s i n g u l a r o p e r a t o r s o n Lp . T h e m a i n r e s u l t i s t h a t i f a n o p e r a t o r T o n Lp , 1<p<2 , i s p - s t r i c t l y s i n g u l a r a n d T|X i s a n i s o m o r p h i s m f o r s o m e s u b s p a c e X o f Lp , t h e n X e m b e d s i n t o Lr f o r a l l r<2 , b u t X n e e d n o t b e i s o m o r p h i c t o a H i l b e r t s p a c e . It is also shown that if T is convolution by a biased coin on L p of the Cantor group, 1 p < 2 , and T | X is an isomorphism for some reflexive subspace X of L p , then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.

On the persistence of decorrelation in the theory of wave turbulence

Anne-Sophie de Suzzoni (2013)

Journées Équations aux dérivées partielles

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We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable u 0 with values in the Sobolev space H s with s big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by e i θ for all θ . We investigate about the persistence of the decorrelation...

Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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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 G ( f ) : = k Λ f ̂ ( k ) e i ( k , x ) , where Λ d 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...

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 ) .

On hyponormal operators in Krein spaces

Kevin Esmeral, Osmin Ferrer, Jorge Jalk, Boris Lora Castro (2019)

Archivum Mathematicum

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In this paper the hyponormal operators on Krein spaces are introduced. We state conditions for the hyponormality of bounded operators focusing, in particular, on those operators T for which there exists a fundamental decomposition 𝕂 = 𝕂 + 𝕂 - of the Krein space 𝕂 with 𝕂 + and 𝕂 - invariant under T .

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

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CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into W m , p ( Ω ) into L S ( Ω ) (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding W m , p ( Ω ) into L φ ( Ω ) ...............................................................

On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let f , g and h be three distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 and k 3 for the full modular group Γ = SL ( 2 , ) , respectively, and let λ f ( n ) , λ g ( n ) and λ h ( n ) denote the n th normalized Fourier coefficients of f , g and h , respectively. We consider the cancellations of sums related to arithmetic functions λ g ( n ) , λ h ( n ) twisted by λ f ( n ) and establish the following results: n x λ f ( n ) λ g ( n ) i λ h ( n ) j f , g , h , ε x 1 - 1 / 2 i + j + ε for any ε > 0 , where 1 i 2 , j 5 are any fixed positive integers.