Displaying similar documents to “Weighted integrability and L¹-convergence of multiple trigonometric series”

On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin (2001)

Journal de théorie des nombres de Bordeaux

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Let q , q 1 , ⋯ , q s ≥ 2 be integers, and let α 1 , α 2 , ⋯ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ( α m q n , ⋯ , α m + s - 1 q n ) m , n = 1 M N coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ( x n ) n = 1 M N in s -dimensional unit cube ( s , M , N = 1 , 2 , ⋯ ) . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ( α 1 q 1 n , ⋯ , α s q s n ) n = 1 N (Korobov’s problem).

Order functions of plurisubharmonic functions

Halil Celik, Evgeny Poletsky (1997)

Studia Mathematica

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We consider the following problem: find on ℂ 2 a plurisubharmonic function with a given order function. In particular, we prove that any positive ambiguous function on ℂ ℙ 1 which is constant outside a polar set is the order function of a plurisubharmonic function.

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...

On the L 1 -convergence of Fourier series

S. Fridli (1997)

Studia Mathematica

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Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier...

On partitions without small parts

J.-L. Nicolas, A. Sárközy (2000)

Journal de théorie des nombres de Bordeaux

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Let r ( n , m ) denote the number of partitions of n into parts, each of which is at least m . By applying the saddle point method to the generating series, an asymptotic estimate is given for r ( n , m ) , which holds for n → ∞ , and 1 ≤ m ≤ c 1 n log n c 2 .

Two-parameter Hardy-Littlewood inequality and its variants

Chang-Pao Chen, Dah-Chin Luor (2000)

Studia Mathematica

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Let s* denote the maximal function associated with the rectangular partial sums s m n ( x , y ) of a given double function series with coefficients c j k . The following generalized Hardy-Littlewood inequality is investigated: | | s * | | p , μ ≤ C p , α , β Σ j = 0 ∞ Σ k = 0 ∞ ( j ̅ ) p - α - 2 ( k ̅ ) p - β - 2 | c j k | p 1 / p , where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on c j k and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property...