Displaying similar documents to “On L p integrability and convergence of trigonometric series”

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...

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).

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...

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 .

Weighted integrability of double cosine series with nonnegative coefficients

Chang-Pao Chen, Ming-Chuan Chen (2003)

Studia Mathematica

Similarity:

Let f c ( x , y ) j = 1 k = 1 a j k ( 1 - c o s j x ) ( 1 - c o s k y ) with a j k 0 for all j,k ≥ 1. We estimate the integral 0 π 0 π x α - 1 y β - 1 ϕ ( f c ( x , y ) ) d x d y in terms of the coefficients a j k , where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that for a normed linear space X , if f 1 : X is continuous and semiconvex with modulus ω , f 2 : X is continuous and semiconcave with modulus ω and f 1 f 2 , then there exists f C 1 , ω ( X ) such that f 1 f f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua (2022)

Czechoslovak Mathematical Journal

Similarity:

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.

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

Similarity:

For an integer k 2 , let ( n ) n be the k - generalized Pell sequence which starts with 0 , ... , 0 , 1 ( k terms) and each term afterwards is given by the linear recurrence n = 2 n - 1 + n - 2 + + n - k . In this paper, we find all k -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ( P n ( 2 ) ) n . ...

On the average behavior of the Fourier coefficients of j th symmetric power L -function over certain sequences of positive integers

Anubhav Sharma, Ayyadurai Sankaranarayanan (2023)

Czechoslovak Mathematical Journal

Similarity:

We investigate the average behavior of the n th normalized Fourier coefficients of the j th ( j 2 be any fixed integer) symmetric power L -function (i.e., L ( s , sym j f ) ), attached to a primitive holomorphic cusp form f of weight k for the full modular group S L ( 2 , ) over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum S j * : = 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 ( a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 ) , where x is sufficiently large, and L ( s , sym j f ) : = n = 1 λ sym j f ( n ) n s . When j = 2 , the error term which we obtain improves the earlier known result.

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

Similarity:

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then a sequence x A , where A is a finite alphabet, is both α -substitutive and β -substitutive if and only if x is ultimately...

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 the distribution of ( k , r ) -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

Similarity:

A natural number n is said to be a ( k , r ) -integer if n = a k b , where k > r > 1 and b is not divisible by the r th power of any prime. We study the distribution of such ( k , r ) -integers in the Piatetski-Shapiro sequence { n c } with c > 1 . As a corollary, we also obtain similar results for semi- r -free integers.

On the structure of sequences with forbidden zero-sum subsequences

W. D. Gao, R. Thangadurai (2003)

Colloquium Mathematicae

Similarity:

We study the structure of longest sequences in d which have no zero-sum subsequence of length n (or less). We prove, among other results, that for n = 2 a and d arbitrary, or n = 3 a and d = 3, every sequence of c(n,d)(n-1) elements in d which has no zero-sum subsequence of length n consists of c(n,d) distinct elements each appearing n-1 times, where c ( 2 a , d ) = 2 d and c ( 3 a , 3 ) = 9 .

A localization property for B p q s and F p q s spaces

Hans Triebel (1994)

Studia Mathematica

Similarity:

Let f j = k a k f ( 2 j + 1 x - 2 k ) , where the sum is taken over the lattice of all points k in n having integer-valued components, j∈ℕ and a k . Let A p q s be either B p q s or F p q s (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on n . The aim of the paper is to clarify under what conditions f j | A p q s is equivalent to 2 j ( s - n / p ) ( k | a k | p ) 1 / p f | A p q s .

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

Similarity:

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 ) ...............................................................

Weak convergence of mutually independent X B and X A under weak convergence of X X B - X A

W. Szczotka (2006)

Applicationes Mathematicae

Similarity:

For each n ≥ 1, let v n , k , k 1 and u n , k , k 1 be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let X B ( t ) = ( 1 / c ) j = 1 [ n t ] ( v n , j - v ̅ ) , X A ( t ) = ( 1 / c ) j = 1 [ n t ] ( u n , j - u ̅ ̅ ) , t ≥ 0, and X = X B - X A . The main result gives conditions under which the weak convergence X X , where X is a Lévy process, implies X B X B and X A X A , where X B and X A are mutually independent Lévy processes and X = X B - X A .

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

Guodong Hua (2022)

Czechoslovak Mathematical Journal

Similarity:

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.

The topology of the space of ℋ𝒦 integrable functions in n

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

Similarity:

It is known that there is no natural Banach norm on the space ℋ𝒦 of n -dimensional Henstock-Kurzweil integrable functions on [ a , b ] . We show that the ℋ𝒦 space is the uncountable union of Fréchet spaces ℋ𝒦 ( X ) . On each ℋ𝒦 ( X ) space, an F -norm · X is defined. A · X -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an F -norm is also defined for a · X -continuous linear operator. Hence, many important results in functional analysis hold for the ℋ𝒦 ( X ) space. It is well-known that every...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

Similarity:

Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Some theorems of Korovkin type

Tomoko Hachiro, Takateru Okayasu (2003)

Studia Mathematica

Similarity:

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C ( X ) ) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., C ( Y ) ), and ϕ a linear isometry from M into C(Y) (resp., C ( Y ) ). We show, under the assumption that Π N Π T , where Π N is...

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.