Displaying similar documents to “Partial differential operators depending analytically on a parameter”

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,...

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 &lt; | α τ | p &lt; 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 &lt; | α | p &lt; 1 } is algebraically independent over p if...

Some characterizations of ultrabornological spaces

Manuel Valdivia (1974)

Annales de l'institut Fourier

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Let U be an infinite-dimensional separable Fréchet space with a topology defined by a family of norms. Let F be an infinite-dimensional Banach space. Then F is the inductive limit of a family of spaces equal to E . The choice of suitable classes of Fréchet spaces allows to give characterizations of ultrabornological spaces using the result above.. Let Ω be a non-empty open set in the euclidean n -dimensional space R n . Then F is the inductive limit of a family of spaces equal to D ( Ω ) . ...

On the closure of spaces of sums of ridge functions and the range of the X -ray transform

Jan Boman (1984)

Annales de l'institut Fourier

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For a R n { 0 } and Ω an open bounded subset of R n definie L p ( Ω , a ) as the closed subset of L p ( Ω ) consisting of all functions that are constant almost everywhere on almost all lines parallel to a . For a given set of directions a ν R n { 0 } , ν = 1 , ... , m , we study for which Ω it is true that the vector space ( * ) L p ( Ω , a 1 ) + + L p ( Ω , a m ) is a closed subspace of L p ( Ω ) . This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator...

Estimates of Fourier transforms in Sobolev spaces

V. Kolyada (1997)

Studia Mathematica

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We investigate the Fourier transforms of functions in the Sobolev spaces W 1 r 1 , . . . , r n . It is proved that for any function f W 1 r 1 , . . . , r n the Fourier transform f̂ belongs to the Lorentz space L n / r , 1 , where r = n ( j = 1 n 1 / r j ) - 1 n . Furthermore, we derive from this result that for any mixed derivative D s f ( f C 0 , s = ( s 1 , . . . , s n ) ) the weighted norm ( D s f ) L 1 ( w ) ( w ( ξ ) = | ξ | - n ) can be estimated by the sum of L 1 -norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

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.