Pointwise convergence of cone-like restricted two-dimensional Fejér means of Walsh-Fourier series

2016

Mediterr. J. Math.

A general summability method of multi-dimensional Fourier transforms, the so called θ-summability is investigated. Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f ∈ W (L1, ∞)(R d ) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of f ∈ W (Lp, ∞)(R d ), whenever 1 < p < ∞. As an application we generalize the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya for f ∈ W (L1, ∞)(R) and for strong θ-summability.Mathematics Subject Classification. Primary 42B08; Secondary 42A38, 42A24, 42B25.

Section: Introductionmentioning

confidence: 99%

Multi-dimensional Fourier Transforms, Lebesgue Points and Strong Summability

2016

Mediterr. J. Math.

“…The condition on γ j seems to be very natural, because Gát [8] proved in the two-dimensional case that to each cone-like set with respect to the first dimension there exists a larger cone-like set with respect to the second dimension and reversely, if and only if (5) holds.…”

Section: Hardy-littlewood Inequality and Cone-like Setsmentioning

confidence: 97%

“…The unrestricted convergence (i.e. if n ∈ N d , n → ∞) does not hold for all f ∈ L 1 (T d ) (see Gát [8]), it holds only for all f ∈ L 1 (log L) d−1 (T d ) (see Zygmund [25]). Recently Gát [8] generalized the convergence result in the two-dimensional case.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Herz spaces and restricted summability of Fourier transforms and Fourier series

Journal of Mathematical Analysis and Applications

2008

A general summability method, the so-called θ -summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1, 1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σ θ T f → f over a cone-like set a.e. for all f ∈ L 1 (R d ). Moreover, σ θ T f (x) converges to f (x) over a cone-like set at each Lebesgue point of f ∈ L 1 (R d ) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.

“…summability methods, is studied intensively in the literature (see e.g. Butzer and Nessel [2], Trigub and Belinsky [26], Gát [9,10], Goginava [12,13,14], Simon [24] and Weisz [29,32]). The Marcinkiewicz θ-means are defined by…”

Section: Introductionmentioning

confidence: 99%

Lebesgue points of double Fourier series and strong summability

Journal of Mathematical Analysis and Applications

2015

A general summability method of double Fourier series is given with the help of a double sequence θ(k, n). Under some conditions on θ we show that the Marcinkiewicz-θ-means of a function f ∈ L 1 (T 2 ) converge to f at each modified strong Lebesgue point. The same holds for a weaker version of Lebesgue points, for the so called modified Lebesgue points of f ∈ L p (T 2 ), whenever 1 < p < ∞. The sufficient conditions of θ are proved for the Fejér, Abel and Cesàro summations. As an application we give simple proofs for the classical one-dimensional strong summability results of Hardy and Littlewood, Marcinkiewicz, Zygmund and Gabisoniya and generalize them for strong θ-summability. Some other special cases of the θ-summation are considered as well, such as the Weierstrass, Picar, Bessel, de La Vallée-Poussin, Rogosinski and Riesz summations.