Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series
We discuss the Norlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in $L_p$. We investigate the rate of approximation by this means, in particular, in $\text{Lip}(\alpha , p)$, where $\alpha > 0$ and $1 \leq p \leq \infty$. For $p = \infty$, by $L...
Gespeichert in:
| Datum: | 2016 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1824 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We discuss the Norlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in $L_p$. We investigate the rate of approximation by this means, in particular, in $\text{Lip}(\alpha , p)$, where $\alpha > 0$ and $1 \leq p \leq \infty$. For $p = \infty$, by $L_p$, we mean $C$, i.e., the collection of continuous functions.
Our main theorem states that the approximation behavior of this two-dimensional Walsh – Kaczmarz –Norlund means is as good as the approximation behavior of the one-dimensional Walsh– and Walsh – Kaczmarz –Norlund means. Earlier results for one-dimensional N¨orlund means of the Walsh – Fourier series was given by M´oricz and Siddiqi [J.
Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] and Fridli, Manchanda and Siddiqi [Acta Sci. Math. (Szeged). – 2008. – 74. – P. 593 – 608], for one-dimensional Walsh – Kaczmarz –N¨orlund means by the author [Georg. Math. J. –2011. – 18. – P. 147 – 162] and for two-dimensional trigonometric system by M´oricz and Rhoades [J. Approxim. Theory. –
1987. – 50. – P. 341 – 358]. |
|---|