<abstract xmlns="http://www.w3.org/1999/xhtml">

We show the uniformly boundedness of the L1 norm of general matrix transform kernel functions with respect to the Walsh-Paley system. Special such matrix means are the well-known Cesàro, Riesz, Bohner-Riesz means. Under some conditions, we verify that the kernels <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0013_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mtext>K</mtext></mrow><mtext>n</mtext><mtext>T</mtext></msubsup><mo>=</mo><msubsup><mo>∑</mo><mrow><mtext>k</mtext><mo>=</mo><mn>1</mn></mrow><mtext>n</mtext></msubsup><mrow><msub><mrow><mtext>t</mtext></mrow><mrow><mtext>k</mtext><mo>,</mo><mtext>n</mtext></mrow></msub><msub><mrow><mtext>D</mtext></mrow><mtext>k</mtext></msub></mrow></mrow></math>
<tex-math>{\rm{K}}_{\rm{n}}^{\rm{T}} = \sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{D}}_{\rm{k}}}}</tex-math>
</alternatives>
</inline-formula>, (where Dk is the kth Dirichlet kernel) satisfy
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0013_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msub><mrow><mrow><mrow><mo>‖</mo><mrow><msubsup><mrow><mtext>K</mtext></mrow><mtext>n</mtext><mtext>T</mtext></msubsup></mrow><mo>‖</mo></mrow></mrow></mrow><mn>1</mn></msub><mo>≤</mo><mtext>c</mtext><mtext>.</mtext></mrow></math>
<tex-math>{\left\| {{\rm{K}}_{\rm{n}}^{\rm{T}}} \right\|_1} \le {\rm{c}}{\rm{.}}</tex-math>
</alternatives>
</disp-formula>
As a result of this we prove that for any 1 <italic>≤</italic> p &lt; ∞ and f ∈ Lp the Lp-norm convergence <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0013_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mo>∑</mo><mrow><mtext>k</mtext><mo>=</mo><mn>1</mn></mrow><mtext>n</mtext></msubsup><mrow><msub><mrow><mtext>t</mtext></mrow><mrow><mtext>k</mtext><mo>,</mo><mtext>n</mtext></mrow></msub><msub><mrow><mtext>S</mtext></mrow><mtext>k</mtext></msub><mrow><mo>(</mo><mtext>f</mtext><mo>)</mo></mrow></mrow><mo>→</mo><mtext>f</mtext></mrow></math>
<tex-math>\sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{S}}_{\rm{k}}}\left( {\rm{f}} \right)} \to {\rm{f}}</tex-math>
</alternatives>
</inline-formula> holds. Besides, for each integrable function f we have that these means converge to f almost everywhere.
</abstract>

We show the uniformly boundedness of the L1 norm of general matrix transform kernel functions with respect to the Walsh-Paley system. Special such matrix means are the well-known Cesàro, Riesz, Bohner-Riesz means. Under some conditions, we verify that the kernels 


KnT=∑k=1ntk,nDk
{\rm{K}}_{\rm{n}}^{\rm{T}} = \sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{D}}_{\rm{k}}}}

, (where Dk is the kth Dirichlet kernel) satisfy



‖KnT‖1≤c.
{\left\| {{\rm{K}}_{\rm{n}}^{\rm{T}}} \right\|_1} \le {\rm{c}}{\rm{.}}


As a result of this we prove that for any 1 ≤ p < ∞ and f ∈ Lp the Lp-norm convergence 


∑k=1ntk,nSk(f)→f
\sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{S}}_{\rm{k}}}\left( {\rm{f}} \right)} \to {\rm{f}}

 holds. Besides, for each integrable function f we have that these means converge to f almost everywhere.

Norm and almost everywhere convergence of matrix transform means of Walsh-Fourier series

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

We show the uniformly boundedness of the L1 norm of general matrix transform kernel functions with respect to the Walsh-Paley system. Special such matrix means...

Norm and almost everywhere convergence of matrix transform means of Walsh-Fourier series

Published Online: Dec 26, 2023

Page range: 244 - 258

Received: Jun 22, 2022

DOI: https://doi.org/10.2478/ausm-2023-0013

Keywords
character system, Fourierseries, Walsh-Paley system, kernel function, matrix transform, almost everywhere convergence

© 2023 István Blahota et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Norm and almost everywhere convergence of matrix transform means of Walsh-Fourier series

Published Online: Dec 26, 2023

Page range: 244 - 258

Received: Jun 22, 2022

DOI: https://doi.org/10.2478/ausm-2023-0013

Keywordscharacter system, Fourierseries, Walsh-Paley system, kernel function, matrix transform, almost everywhere convergence

© 2023 István Blahota et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
character system, Fourierseries, Walsh-Paley system, kernel function, matrix transform, almost everywhere convergence