Displaying similar documents to “Complexity of Hartman sequences”

Construction techniques for some thin sets in duals of compact abelian groups

D. J. Hajela (1986)

Annales de l'institut Fourier

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Various techniques are presented for constructing Λ (p) sets which are not Λ ( p + ϵ ) for all ϵ > 0 . The main result is that there is a Λ (4) set in the dual of any compact abelian group which is not Λ ( 4 + ϵ ) for all ϵ > 0 . Along the way to proving this, new constructions are given in dual groups in which constructions were already known of Λ (p) not Λ ( p + ϵ ) sets, for certain values of p . The main new constructions in specific dual groups are: – there is a Λ (2k) set which is not Λ ( 2 k + ϵ ) in Z ( 2 ) Z ( 2 ) for all 2 k , k N and...

An almost-sure estimate for the mean of generalized Q -multiplicative functions of modulus 1

Jean-Loup Mauclaire (2000)

Journal de théorie des nombres de Bordeaux

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Let Q = ( Q k ) k 0 , Q 0 = 1 , Q k + 1 = q k Q k , q k 2 , be a Cantor scale, 𝐙 Q the compact projective limit group of the groups 𝐙 / Q k 𝐙 , identified to 0 j k - 1 𝐙 / q j 𝐙 , and let μ be its normalized Haar measure. To an element x = { a 0 , a 1 , a 2 , } , 0 a k q k + 1 - 1 , of 𝐙 Q we associate the sequence of integral valued random variables x k = 0 j k a j Q j . The main result of this article is that, given a complex 𝐐 -multiplicative function g of modulus 1 , we have lim x k x ( 1 x k n x k - 1 g ( n ) - 0 j k 1 q j 0 a q j g ( a Q j ) ) = 0 μ -a.e .

Harmonic interpolating sequences, L p and BMO

John B. Garnett (1978)

Annales de l'institut Fourier

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Let ( z ν ) be a sequence in the upper half plane. If 1 < p and if y ν 1 / p f ( z ν ) = a ν , ν = 1 , 2 , ... ( * ) has solution f ( z ) in the class of Poisson integrals of L p functions for any sequence ( a ν ) p , then we show that ( z ν ) is an interpolating sequence for H . If f ( z ν ) = a ν , ν = 1 , 2 , ... has solution in the class of Poisson integrals of BMO functions whenever ( a ν ) , then ( z ν ) is again an interpolating sequence for H . A somewhat more general theorem is also proved and a counterexample for the case p 1 is described.

On a generalization of de Rham lemma

Kyoji Saito (1976)

Annales de l'institut Fourier

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Let M be a free module over a noetherian ring. For ω 1 , ... , ω k M , let 𝒜 be the ideal generated by coefficients of ω 1 ... ω k . For an element ω p M with p < prof . 𝒜 , if ω ω 1 ... ω k = 0 , there exists η 1 , ... , η k p - 1 M such that ω = i = 1 k η i ω i . This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.

Oscillation conditions for difference equations with several variable arguments

George E. Chatzarakis, Takaŝi Kusano, Ioannis P. Stavroulakis (2015)

Mathematica Bohemica

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Consider the difference equation Δ x ( n ) + i = 1 m p i ( n ) x ( τ i ( n ) ) = 0 , n 0 x ( n ) - i = 1 m p i ( n ) x ( σ i ( n ) ) = 0 , n 1 , where ( p i ( n ) ) , 1 i m are sequences of nonnegative real numbers, τ i ( n ) [ σ i ( n ) ], 1 i m are general retarded (advanced) arguments and Δ [ ] denotes the forward (backward) difference operator Δ x ( n ) = x ( n + 1 ) - x ( n ) [ x ( n ) = x ( n ) - x ( n - 1 ) ]. New oscillation criteria are established when the well-known oscillation conditions lim sup n i = 1 m j = τ ( n ) n p i ( j ) > 1 lim sup n i = 1 m j = n σ ( n ) p i ( j ) > 1 and lim inf n i = 1 m j = τ i ( n ) n - 1 p i ( j ) > 1 e lim inf n i = 1 m j = n + 1 σ i ( n ) p i ( j ) > 1 e are not satisfied. Here τ ( n ) = max 1 i m τ i ( n ) [ σ ( n ) = min 1 i m σ i ( n ) ] . The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.