L p -norm convergence of series in compact, totally disconnected groups

Gát, György; Toledo, Rodolfo

doi:10.1007/bf02342335

Cited by 17 publications

(21 citation statements)

References 6 publications

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“…We remark that both the the a.e. convergence of E n f for functions in L 1 (G), and the norm convergence of E n f for functions in L p (G), 1 ≤ p < ∞, as well as in C(G), are proved in [4]. By Lemma 2.4, this also implies Propositions 2.5 and 2.6.…”

Section: Therefore the Right (And Left) Shift Invariency Of The Haar mentioning

confidence: 62%

“…The method to prove this lemma is similar to the usual way which gives the Dirichlet kernels with respect to the Walsh [9], Vilenkin [1] systems. Moreover, we refer to [4]. This completes the proof of Lemma 2.7.…”

Section: Therefore the Right (And Left) Shift Invariency Of The Haar mentioning

confidence: 86%

“…These density arguments [6,4] can be proved directly, by using the fact that the characteristic function of a conjugacy class is also a polynomial. This is exactly the statement of Lemma 2.2.…”

Section: The L 2 Casementioning

confidence: 98%

“…In [4] and [5] the author with a coauthor studied the complete product of finite (not neccesarily Abelian) (CPF) groups and the product system ψ as the generalization of the Vilenkin and Walsh systems. In fact, the structure of these groups are similar.…”

Section: Introduction and The Theoremsmentioning

confidence: 99%

“…The definition of CPF groups was given in [4] in the following form: Let N denote the set of nonnegative integers, P the set of positive integers, and C the set of complex numbers. Denote by m := (m k : k ∈ N) a sequence of positive integers such that m k ≥ 2, k ∈ N, and G k a finite group with order m k with number of conjugacy classes p k , k ∈ N. For simplicity, we will use the same notation (+) for the group operation of G k (− is the inverse operation), k ∈ N, and denote by e the identity of these groups.…”

Section: Introduction and The Theoremsmentioning

confidence: 99%

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Fejér means of functions on noncommutative Vilenkin groups with respect to the character system

Gát

2006

Anal Math

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A b s t r a c t . It is known that the Fejér means -with respect to the character system of the Walsh, and bounded Vilenkin groups -of an integrable function converge to the function a.e. In this work we discuss analogous problems on the complete product of finite, not necessarily Abelian groups with respect to the character system for functions that are constant on the conjugacy classes. We find that the nonabelian case differs from the commutative case. We prove the a.e. convergence of the (C, 1) means of the Fourier series of square integrable functions. We also prove the existence of a function f ∈ L q for some q > 1 such that sup |σnf | = +∞ a.e. This is a sharp contrast between the Abelian and the nonabelian cases.

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Section: Therefore the Right (And Left) Shift Invariency Of The Haar mentioning

confidence: 62%

Section: Therefore the Right (And Left) Shift Invariency Of The Haar mentioning

confidence: 86%

Section: The L 2 Casementioning

confidence: 98%

Section: Introduction and The Theoremsmentioning

confidence: 99%

Section: Introduction and The Theoremsmentioning

confidence: 99%

See 3 more Smart Citations

Fejér means of functions on noncommutative Vilenkin groups with respect to the character system

Gát

2006

Anal Math

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Convergence in L p -norm of Fourier series on the complete product of quaternion groups with bounded orders

Gát

Toledo

2014

Acta. Math. Sin.-English Ser.

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On the convergence in L 1-norm of Cesàro means with respect to representative product systems

Gát

Toledo

2008

Acta Math Hung

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The aim of this paper is to prove that the Cesàro means of order α (0 < α < 1) of the Fourier series with respect to representative product systems converge to the function in L 1 -norm, only for certain values of α which depend on some parameter of the representative product system. IntroductionSeveral results in Fourier analysis with respect to Walsh functions are obtained viewing them as the characters of the dyadic group, i.e., the complete product of the discrete cyclic group of order 2 with the product of topologies and measures. Then we often order the Walsh functions in Paley's sense writing them as a nite product of the Rademacher functions. It is named the WalshPaley system. The above structure was generalized by Vilenkin [7] in 1947 studying the complete product of arbitrary cyclic groups. The construction of the system here is similar, taking the nite product of the characters of the cyclic groups as it Paley did.A natural generalization of the Vilenkin group is the complete product of arbitrary, not necessarily commutative groups. In this case we use representation theory in order to obtain orthonormal systems. These systems are named representative product systems and Section 1 deals with their denition and properties. In [2] the authors proved the fact that the Fejér means of the Fourier series with respect to representative product systems on bounded groups converge to the function in L p -norm (1 p < ∞), although we can not state the same for the Fourier series in general. We attempt to extend this statement to Cesàro means of order α where 0 < α < 1.In Section 2 we dene the basic concept and properties concerning Cesàro means and prove the lemmas we need for our main result in Section 3: There is an 0 α 0 < 1 2 such that the Cesàro means of order α of the Fourier series with respect to representative product systems on bounded groups converge to the function in L p -norm (1 p < ∞) for all α 0 < α < 1. This result was proved by Skvortsov (see [5]) for the commutative cases and for all 0 < α < 1. Corollary 1 generalizes this result for monomial representative product systems. In Section 3 we also deal with the divergence of the Cesàro means of order α and we obtain an 0 α 1 α 0 such that for all 0 < α < α 1 there exists an f ∈ L 1 (G) for which σ α n f does not converge to the function f in L 1 -norm.Throughout this work we denote by N, P, C the set of nonnegative, positive integers and complex numbers, respectively. In order to simplify notation we always use multiplication to denote the group operation and use the symbol e to denote the identity of the groups. If in our notations the indexset of a sum or a product is empty then the value of the sum and the product is 0 and 1 respectively. Moreover throughout this paper we often use constants C which have dierent values in dierent occurrences and they only depend on the value of α. The notation which we used in this paper is similar to the one in [3]. Representative product systemsLet m := (m k , k ∈ N) be a sequence of positive integers such tha...

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L p -norm convergence of series in compact, totally disconnected groups

Cited by 17 publications

References 6 publications

Fejér means of functions on noncommutative Vilenkin groups with respect to the character system

Fejér means of functions on noncommutative Vilenkin groups with respect to the character system

Convergence in L p -norm of Fourier series on the complete product of quaternion groups with bounded orders

On the convergence in L 1-norm of Cesàro means with respect to representative product systems

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