Topics on N¨ orlund logarithmic means Topics on N¨ orlund logarithmic means Nacima Memi´ c Topics on N¨ orlund logarithmic means Nacima Memi´ c University of Sarajevo, Bosnia and Herzegovina 28.08.2017 Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 1 / 19
Topics on N¨ orlund logarithmic means Topics on Content N¨ orlund logarithmic means Nacima Memi´ c Almost everywhere convergence of some subsequences ( t m n f ) n of N¨ orlund logarithmic means of Walsh Fourier coefficients for every integrable function f and divergence for other classes of subsequences. Convergence and divergence in norm of N¨ orlund logarithmic means of generalized Walsh Fourier coefficients on some unbounded Vilenkin groups. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 2 / 19
Topics on N¨ orlund logarithmic means Topics on Motivation for Walsh-Paley system N¨ orlund logarithmic means Nacima Memi´ c The Riesz logarithmic means of Walsh or trigonometric Fourier � n − 1 1 S k f series of any integrable function f converges k =1 log n k almost everywhere to the original function f . This is not true for the Walsh Fourier series which diverges everywhere for some integrable function f satisfying ϕ ( | f | ) < ∞ , where ϕ ( u ) = o ( u √ log u ). � As G´ at and Goginava mentioned the following results show a similarity of N¨ orlund logarithmic means with Walsh Fourier series rather than classical logarithmic means. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 3 / 19
Topics on N¨ orlund logarithmic means Topics on Goginava’s results for Walsh-Paley system N¨ orlund logarithmic means Nacima Memi´ c (Goginava 2005) Theorem Let { ( m n ) n : n ≥ 1 } be a sequence of positive integers for which ∞ log 2 ( m n − 2 ⌊ log m n ⌋ + 1) � < ∞ . log m n n =1 Then, the operator t ∗ f := sup | t m n f | is of weak type (1 , 1) . n ≥ 1 Corollary Let { ( m n ) n : n ≥ 1 } be the sequence defined in the theorem above and f an integrable function. Then, t m n f → f , a.e. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 4 / 19
Topics on N¨ orlund logarithmic means Topics on Goginava’s results for Walsh-Paley system N¨ orlund logarithmic means Nacima Memi´ c Corollary For all integrable function f , we have t 2 n f → f , a.e. I. Blahota [1] proved the validity of the same results on the 2-adic group Bijection between 2-adic and dyadic integers Important difference between the two systems of characters Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 5 / 19
Topics on N¨ orlund logarithmic means Topics on Motivation for Walsh-Paley system N¨ orlund logarithmic means Nacima Memi´ c (G´ at-Goginava 2009): Theorem Let ϕ : [0 , ∞ ) → [0 , ∞ ) be a function such that ϕ ( u ) is nondecreasing and ϕ ( u ) = o ( u √ log u ) . Then there exist a function u f ∈ L and a measurable set E with positive measure for which � ϕ ( | f | ) < ∞ and lim sup t n f ( x ) = ∞ , ∀ x ∈ E . Remark Due to the corollary above it is impossible to replace lim sup t n f ( x ) = ∞ by lim t n f ( x ) = ∞ , in this theorem. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 6 / 19
Topics on N¨ orlund logarithmic means Topics on Notations and definitions N¨ orlund logarithmic means Nacima Memi´ c The N¨ orlund logarithmic means are defined by n − 1 n − 1 t n f := 1 S k f 1 � � n − k , l n := k . l n k =1 k =1 n − 1 F n := 1 D k � n − k , l n k =1 t n f = F n ∗ f . Define the function ϕ : N \ { 0 } → N by ϕ ( n ) = n − 2 [log 2 n ] . Set ϕ 1 ( n ) = ϕ ( n ), ϕ 0 ( n ) = n and ϕ i ( n ) = ϕ ◦ ϕ i − 1 ( n ) when i ≥ 2. For every n ∈ N \ { 0 } , i ≥ 0, such that ϕ i ( n ) > 0, define the functions α i ( n ) = [log 2 ( ϕ i ( n ))] and β i ( n ) = l ϕ i ( n ) . Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 7 / 19
Topics on N¨ orlund logarithmic means Topics on Main results N¨ orlund logarithmic means Nacima Memi´ c Theorem Let ( m n ) n be an increasing sequence of positive integers. Suppose that β i ( m n ) � = O (1) . l m n i : ϕ i ( m n ) > 0 Then, t m n f → f , a.e. The condition of [4, Theorem 1] from which Goginava proves that t m n f → f , a.e. formulated in our notations is ∞ α 2 1 ( m n ) � α 0 ( m n ) < + ∞ . n =1 Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 8 / 19
Topics on N¨ orlund logarithmic means Topics on Main results N¨ orlund logarithmic means Since ♯ { i : ϕ i ( n ) > 0 } ≤ α 1 ( n ), it follows that Nacima Memi´ c α 0 ( m n ) < α 2 α i ( m n ) 1 ( m n ) � α 0 ( m n ) . i : ϕ i ( m n ) > 0 If the sequence ( m n ) n satisfies the condition of [4, Theorem 1], then α 2 1 ( m n ) = o ( α 0 ( m n )) , which implies that � α i ( m n ) = o ( α 0 ( m n )) , i : ϕ i ( m n ) > 0 or equivalently, � β i ( m n ) = o ( β 0 ( m n )) = o ( l m n ) . i : ϕ i ( m n ) > 0 Therefore, this theorem is a generalization of [4, Theorem 1]. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 9 / 19
Topics on N¨ orlund logarithmic means Topics on Main results N¨ orlund logarithmic means Nacima Memi´ c Theorem Let ( m n ) n and ( s n ) n be increasing sequences of positive integers for which: the sequence ( ϕ s n ( m n )) n is increasing, 1 l m n = o ( β s n ( m n ) √ s n ) , when n → ∞ , 2 then there exists an integrable function f such that t m n f � f on a subset of positive measure. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 10 / 19
Topics on N¨ orlund logarithmic means Convergence in norm of logarithmic means for Topics on N¨ orlund logarithmic Walsh-Fourier coefficients means Nacima Memi´ c (F. Schipp, W.R. Wade, P. Simon, and J. P´ al, 1990) Theorem � � 1 Let f ∈ C and ω ( δ, f ) ∞ = o , then � S n f − f � ∞ → 0 . log( 1 δ ) (G´ at-Goginava 2006) Theorem � � 1 Let f ∈ C and ω ( δ, f ) ∞ = o , then � t n f − f � ∞ → 0 . log( 1 δ ) Theorem � � 1 There exists a function g ∈ C such that ω ( δ, g ) ∞ = O , and log( 1 δ ) t n g (0) diverges. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 11 / 19
Topics on N¨ orlund logarithmic means Convergence in norm of logarithmic means for Topics on N¨ orlund logarithmic Walsh-Fourier coefficients means Nacima Memi´ c (B.I. Golubov, A.V. Efimov, and V.A. Skvortsov, 1991) Theorem Let f ∈ L 1 and ω ( δ, f ) L 1 = o � � 1 , then � S n f − f � 1 → 0 . log( 1 δ ) (G´ at-Goginava 2006) Theorem Let f ∈ L 1 and ω ( δ, f ) L 1 = o � � 1 , then � t n f − f � 1 → 0 . log( 1 δ ) Theorem � � There exists a function g ∈ L 1 such that ω ( δ, g ) L 1 = O 1 , and log( 1 δ ) � t n g − g � 1 � 0 . Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 12 / 19
Topics on N¨ orlund logarithmic means Topics on Fourier series, Fej´ er means-Convergence in norm N¨ orlund logarithmic means Nacima Memi´ c For Vilenkin systems � S n f − f � p → 0 , 1 < p < ∞ (P. Simon 1976). � S M n f − f � 1 → 0 , (G. H. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, A. I. Rubinstein, 1981). For bounded groups � σ n f − f � p → 0 , 1 ≤ p < ∞ (Simon, P., P´ al, J., 1977). On arbitrary groups as a trivial consequence of the convergence of the partial sums: � σ n f − f � p → 0 , 1 < p < ∞ . On every unbounded group there exists an integrable function f such that � σ M n f − f � 1 � 0 (Price, J., 1957). For every Vilenkin system and every integrable function σ M n f → f a.e (G´ at 2003). Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 13 / 19
Topics on N¨ orlund logarithmic means Topics on Motivations for Part 2 N¨ orlund logarithmic means Nacima Memi´ c In general, the Fej´ er (C, 1) means have better properties, than the logarithmic ones. In the case of some unbounded Vilenkin systems the situation may change. In their paper [2] the authors have proved a convergence result of the subsequence ( t M n f ) n to the integrable function f in the L 1 norm for some unbounded Vilenkin groups. The main tool was the boundedness of the sequence ( � F M n � 1 ) n . Paradoxically, this is the reason for the divergence of the whole sequence ( t n f ) n . Therefore, in order to construct unbounded groups on which the sequence ( t n f ) n converges in the L 1 norm, the property of uniform boundedness should be avoided. Nacima Memi´ c (University of Sarajevo) Topics on N¨ orlund logarithmic means 28.08.2017 14 / 19
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