Displaying similar documents to “Interpolation by bounded functions”

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.

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 .

Algebraic independence over p

Peter Bundschuh, Kumiko Nishioka (2004)

Journal de Théorie des Nombres de Bordeaux

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Let f ( x ) be a power series n 1 ζ ( n ) x e ( n ) , where ( e ( n ) ) is a strictly increasing linear recurrence sequence of non-negative integers, and ( ζ ( n ) ) a sequence of roots of unity in ¯ p satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over p of the elements f ( α 1 ) , ... , f ( α t ) from p in terms of the distinct α 1 , ... , α t p satisfying 0 < | α τ | p < 1 for τ = 1 , ... , t . A striking application of our basic result says that, in the case e ( n ) = n , the set { f ( α ) | α p , 0 < | α | p < 1 } is algebraically independent over p if...

On the representation of functions by orthogonal series in weighted L p spaces

M. Grigorian (1999)

Studia Mathematica

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It is proved that if φ n is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form k = 1 c k φ k ( x ) , where c k l q for all q>2, with the following properties: 1. For any p ∈ [1,2) and f L μ p [ 0 , 1 ] = f : ʃ 0 1 | f ( x ) | p μ ( x ) d x < there are numbers ɛ k , k=1,2,…, ɛ k = 1 or 0, such that l i m n ʃ 0 1 | k = 1 n ɛ k c k φ k ( x ) - f ( x ) | p μ ( x ) d x = 0 . 2. For every p ∈ [1,2) and f L μ p [ 0 , 1 ] there are a function g L 1 [ 0 , 1 ] with g(x) = f(x) on E and numbers δ k , k=1,2,…, δ k = 1 or 0,...

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 &lt; 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.

On the discrepancy of sequences associated with the sum-of-digits function

Gerhard Larcher, N. Kopecek, R. F. Tichy, G. Turnwald (1987)

Annales de l'institut Fourier

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If w = ( q k ) k N denotes the sequence of best approximation denominators to a real α , and s α ( n ) denotes the sum of digits of n in the digit representation of n to base w , then for all x irrational, the sequence ( s α ( n ) · x ) n N is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if α has bounded continued fraction coefficients.