Displaying similar documents to “On the Nörlund means of Vilenkin-Fourier series”

Multi-dimensional Fejér summability and local Hardy spaces

Ferenc Weisz (2009)

Studia Mathematica

Similarity:

It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space W ( h p , ) to W ( L p , ) . This implies the almost everywhere convergence of the Fejér means in a cone for all f W ( L , ) , which is larger than L ( d ) .

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

Nayna Govindbhai Kalsariya, Bhikha Lila Ghodadra (2024)

Mathematica Bohemica

Similarity:

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...

Maximal operators of Fejér means of double Vilenkin-Fourier series

István Blahota, György Gát, Ushangi Goginava (2007)

Colloquium Mathematicae

Similarity:

The main aim of this paper is to prove that the maximal operator σ * : = s u p | σ n , n | of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space H 1 / 2 to the space weak- L 1 / 2 .

Convergence of greedy approximation II. The trigonometric system

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 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...

On L p integrability and convergence of trigonometric series

Dansheng Yu, Ping Zhou, Songping Zhou (2007)

Studia Mathematica

Similarity:

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)...

A transplantation theorem for ultraspherical polynomials at critical index

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 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 . ...

Pointwise Fourier inversion of distributions on spheres

Francisco Javier González Vieli (2017)

Czechoslovak Mathematical Journal

Similarity:

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 weak type inequality for the Walsh system

Ushangi Goginava (2008)

Studia Mathematica

Similarity:

The main aim of this paper is to prove that the maximal operator σ is bounded from the Hardy space H 1 / 2 to weak- L 1 / 2 and is not bounded from H 1 / 2 to L 1 / 2 .

Some weighted norm inequalities for a one-sided version of g * λ

L. de Rosa, C. Segovia (2006)

Studia Mathematica

Similarity:

We study the boundedness of the one-sided operator g λ , φ between the weighted spaces L p ( M ¯ w ) and L p ( w ) for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of g λ , φ . For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of g λ , φ from L p ( ( M ¯ ) [ p / 2 ] + 1 w ) to L p ( w ) , where ( M ¯ ) k denotes the operator M¯ iterated k times.

Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein (2001)

Studia Mathematica

Similarity:

Let M S denote the strong maximal operator. Let M x and M y denote the one-dimensional Hardy-Littlewood maximal operators in the horizontal and vertical directions in ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that Q M y M x h < but Q M x M y h = . It is shown that if f is a function supported on Q such that Q M y M x f < but Q M x M y f = , then there exists a set A of finite measure in ℝ² such that A M S f = .

Restricted weak type inequalities for the one-sided Hardy-Littlewood maximal operator in higher dimensions

Fabio Berra (2022)

Czechoslovak Mathematical Journal

Similarity:

We give a quantitative characterization of the pairs of weights ( w , v ) for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak ( p , p ) type inequality for 1 p < . More precisely, given any measurable set E 0 , the estimate w ( { x n : M + , d ( 𝒳 E 0 ) ( x ) > t } ) C [ ( w , v ) ] A p + , d ( ) p t p v ( E 0 ) holds if and only if the pair ( w , v ) belongs to A p + , d ( ) , that is, | E | | Q | [ ( w , v ) ] A p + , d ( ) v ( E ) w ( Q ) 1 / p for every dyadic cube Q and every measurable set E Q + . The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the...

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

Similarity:

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).

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

Kristóf Szarvas, Ferenc Weisz (2016)

Czechoslovak Mathematical Journal

Similarity:

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L p ( d ) (in the case p > 1 ), but (in the case when 1 / p ( · ) is log-Hölder continuous and p - = inf { p ( x ) : x d } > 1 ) on the variable Lebesgue spaces L p ( · ) ( d ) , too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type ( 1 , 1 ) . In the present note we generalize Besicovitch’s covering theorem for the so-called γ -rectangles. We introduce a general maximal operator M s γ , δ and with the help of generalized Φ -functions, the strong-...

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

Ferenc Móricz (2002)

Studia Mathematica

Similarity:

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...

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

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

Similarity:

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 .

Multifractal analysis of the divergence of Fourier series

Frédéric Bayart, Yanick Heurteaux (2012)

Annales scientifiques de l'École Normale Supérieure

Similarity:

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...

Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

Imed Feki, Ameni Massoudi (2024)

Czechoslovak Mathematical Journal

Similarity:

We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces H k , p ( G ) , where k * , 1 p and G is either the open unit disk 𝔻 or the annular domain G s , 0 < s < 1 of the complex space . More precisely, we study the behavior on the interior of G of any function f belonging to the unit ball of the Hardy-Sobolev spaces H k , p ( G ) from its behavior on any open connected subset I of the boundary G of G with respect to the L 1 -norm. Our results can be viewed as an improvement and generalization of...

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Suying Liu, Minghua Yang (2018)

Czechoslovak Mathematical Journal

Similarity:

Let L be a non-negative self-adjoint operator acting on L 2 ( n ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A r weight on n × n , 1 < r < . In this article we obtain a weighted atomic decomposition for the weighted Hardy space H L , w p ( n × n ) , 0 < p 1 associated to L . Based on the atomic decomposition, we show the dual relationship between H L , w 1 ( n × n ) and BMO L , w ( n × n ) .

Generalized Cesàro operators on certain function spaces

Sunanda Naik (2010)

Annales Polonici Mathematici

Similarity:

Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators b , c is bounded on the Dirichlet spaces p , a . We also give a short and direct proof of boundedness of b , c on the Hardy space H p for 1 < p < ∞.

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

Similarity:

Let A α f ( x ) = | B ( 0 , | x | ) | - α / n B ( 0 , | x | ) f ( t ) d t be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that A α is bounded from L p to L p α with p α = n p / ( α p - n p + n ) when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space S α , Y , which is strictly larger than X, and a ’target’ space T Y , which is strictly smaller than Y, under the assumption that A α is bounded from X into Y and the Hardy-Littlewood...