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Displaying similar documents to “Two-sided estimates of the approximation numbers of certain Volterra integral operators”

Two-sided estimates for the approximation numbers of Hardy-type operators in L and L¹

W. Evans, D. Harris, J. Lang (1998)

Studia Mathematica

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In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator T : L p ( + ) L p ( + ) defined by ( T f ) ( x ) v ( x ) ʃ 0 u ( t ) f ( t ) d t when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

Partial differential operators depending analytically on a parameter

Frank Mantlik (1991)

Annales de l'institut Fourier

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Let P ( λ , D ) = | α | m a α ( λ ) D α be a differential operator with constant coefficients a α depending analytically on a parameter λ . Assume that the family { P( λ ,D) } is of constant strength. We investigate the equation P ( λ , D ) 𝔣 λ g λ where 𝔤 λ is a given analytic function of λ with values in some space of distributions and the solution 𝔣 λ is required to depend analytically on λ , too. As a special case we obtain a regular fundamental solution of P( λ ,D) which depends analytically on λ . This result answers a question of L. Hörmander. ...

On the best constant in the Khinchin-Kahane inequality

Rafał Latała, Krzysztof Oleszkiewicz (1994)

Studia Mathematica

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We prove that if r i is the Rademacher system of functions then ( ʃ i = 1 n x i r i ( t ) 2 d t ) 1 / 2 2 ʃ i = 1 n x i r i ( t ) d t for any sequence of vectors x i in any normed linear space F.

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 some singular integral operatorsclose to the Hilbert transform

T. Godoy, L. Saal, M. Urciuolo (1997)

Colloquium Mathematicae

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Let m: ℝ → ℝ be a function of bounded variation. We prove the L p ( ) -boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by T m f ( x ) = p . v . k ( x - y ) m ( x + y ) f ( y ) d y where k ( x ) = j 2 j φ j ( 2 j x ) for a family of functions φ j j satisfying conditions (1.1)-(1.3) given below.

Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

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We show that, if the coefficients (an) in a series a 0 / 2 + n = 1 a n c o s ( n t ) tend to 0 as n → ∞ and satisfy the regularity condition that m = 0 j = 1 [ n = j 2 m ( j + 1 ) 2 m - 1 | a n - a n + 1 | ] ² 1 / 2 < , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series n = 1 b n s i n ( n t ) tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if n = 1 | b n | / n < . These conclusions were previously known to hold under stronger restrictions on the sizes...

Isomorphism of certain weak L p spaces

Denny Leung (1993)

Studia Mathematica

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It is shown that the weak L p spaces p , , L p , [ 0 , 1 ] , and L p , [ 0 , ) are isomorphic as Banach spaces.

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.

( H p , L p ) -type inequalities for the two-dimensional dyadic derivative

Ferenc Weisz (1996)

Studia Mathematica

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It is shown that the restricted maximal operator of the two-dimensional dyadic derivative of the dyadic integral is bounded from the two-dimensional dyadic Hardy-Lorentz space H p , q to L p , q (2/3 < p < ∞, 0 < q ≤ ∞) and is of weak type ( L 1 , L 1 ) . As a consequence we show that the dyadic integral of a ∞ function f L 1 is dyadically differentiable and its derivative is f a.e.

Geometric study of the beta-integers for a Perron number and mathematical quasicrystals

Jean-Pierre Gazeau, Jean-Louis Verger-Gaugry (2004)

Journal de Théorie des Nombres de Bordeaux

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We investigate in a geometrical way the point sets of     obtained by the   β -numeration that are the   β -integers   β [ β ]   where   β   is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the   β -numeration, allowing to lift up the   β -integers to some points of the lattice   m   ( m =   degree of   β ) lying about the dominant eigenspace of the companion matrix of   β  . When   β   is in particular a Pisot number, this framework gives another proof of the fact...

Boundedness of Marcinkiewicz functions.

Minako Sakamoto, Kôzô Yabuta (1999)

Studia Mathematica

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The L p boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s g * λ -functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts μ S ϱ and μ λ * , ϱ to S and g * λ . The definition of μ S ϱ ( f ) is as follows: μ S ϱ ( f ) ( x ) = ( ʃ | y - x | < t | 1 / t ϱ ʃ | z | t Ω ( z ) / ( | z | n - ϱ ) f ( y - z ) d z | 2 ( d y d t ) / ( t n + 1 ) ) 1 / 2 , where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere S n - 1 , and...

Tauberian theorems for Cesàro summable double sequences

Ferenc Móricz (1994)

Studia Mathematica

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( s j k : j , k = 0 , 1 , . . . ) be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which ( s j k ) converges in Pringsheim’s sense. These conditions are satisfied if ( s j k ) is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If ( s j k ) is summable (C,1,1) to a finite limit and there exist constants n 1 > 0 and H such that j k ( s j k - s j - 1 , k - s j - 1 , k + s j - 1 , k - 1 ) - H , j ( s j k - s j - 1 , k ) - H and k ( s j k - s j , k - 1 ) - H whenever j , k > n 1 , then...