Wyniki wyszukiwania

1

Weighted integrability and L¹-convergence of multiple trigonometric series

100%

Chen C.-P.

Studia Mathematica

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1994

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tom 108

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nr 2

177-190

EN

We prove that if $c_{jk} → 0$ as max(|j|,|k|) → ∞, and $∑^∞_{|j|=0±} ∑^∞_{|k|=0±} θ(|j|^⊤)ϑ(|k|^⊤)|Δ_{12}c_{jk}| < ∞$, then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and $∬_{T²} |s_{mn}(x,y) - f(x,y)|·|ϕ(x)ψ(y)|dxdy → 0$ as min(m,n) → ∞, where f(x,y) is the limiting function of the rectangular partial sums $s_{mn}(x,y)$, (ϕ,θ) and (ψ,ϑ) are pairs of type I. A generalization of this result concerning L¹-convergence is also established. Extensions of these results to double series of orthogonal functions are also considered. These results can be extended to n-dimensional case. The aforementioned results generalize work of Balashov [1], Boas [2], Chen [3,4,5], Marzuq [9], Móricz [11], Móricz-Schipp-Wade [14], and Young [16].

2

Two-parameter Hardy-Littlewood inequality and its variants

64%

Chen C.-P., Luor D.-C.

Studia Mathematica

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2000

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tom 139

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nr 1

9-27

EN

Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: $||s*||_{p,μ}≤C_{p,α,β} {Σ_{j=0}^∞Σ_{k=0}^∞(j̅ )^{p-α-2}(k̅)^{p-β-2}|c_{jk}|^p }^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{jk}$ 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 of $s_{mn}(x,y)$ is established. These results generalize the work of Askey-Wainger [1], Balashov [2], Boas [3], Chen [5], [6], [8], [9], Marzug [15], Móricz [16]-[18], [19], Móricz-Schipp-Wade [20], Ram-Bhatia [22], Stechkin [24], Weisz [26]-[28], and Young [30].

3

Weighted integrability of double cosine series with nonnegative coefficients

64%

Chen C.-P., Chen M.-C.

Studia Mathematica

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2003

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tom 156

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nr 2

133-141

EN

Let $f_{c}(x,y) ≡ ∑_{j=1}^{∞} ∑_{k=1}^{∞} a_{jk}(1 - cos jx)(1 - cos ky)$ with $a_{jk} ≥ 0$ for all j,k ≥ 1. We estimate the integral $∫_{0}^{π}∫_{0}^{π} x^{α-1} y^{β-1} ϕ(f_{c}(x,y)) dxdy$ in terms of the coefficients $a_{jk}$, where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].

4

Almost everywhere convergence of Laguerre series

64%

Chen C.-P., Lin C.-C.

Studia Mathematica

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1994

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tom 109

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nr 3

291-301

EN

Let $a ∈ ℤ^+$ and $f ∈ L^p (ℝ^+), 1 ≤ p ≤ ∞ $. Denote by $c_j$ the inner product of f and the Laguerre function $ℒ^a_j$. We prove that if ${c_j}$ satisfies $lim_{λ↓1} \overline lim_{n→∞} ∑_{n

5

Uniform convergence of double trigonometric series

64%

Chen C.-P., Chen G.-B.

Studia Mathematica

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1996

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tom 118

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nr 3

245-259

EN

It is shown that under certain conditions on ${c_{jk}}$, the rectangular partial sums $s_{mn}(x,y)$ converge uniformly on $T^2$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is $∑_{|k|= n}^∞ |Δc_k| = o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $nc_{n} = o(1)$ as n → ∞. As a consequence, our result generalizes those of Chaundy-Jolliffe [CJ], Jolliffe [J], Nurcombe [N], and Xie-Zhou [XZ].

Ograniczanie wyników

5 Studia Mathematica

5 Chen C.-P.

1 Chen G.-B.

1 Chen M.-C.

1 Lin C.-C.

1 Luor D.-C.

1 2003

1 2000

1 1996

2 1994

Weighted integrability and L¹-convergence of multiple trigonometric series

Two-parameter Hardy-Littlewood inequality and its variants

Weighted integrability of double cosine series with nonnegative coefficients

Almost everywhere convergence of Laguerre series

Uniform convergence of double trigonometric series