On double cosine, sine, and Walsh series with monotone coefficients
On double cosine, sine, and Walsh series with monotone coefficients
We extend from one-dimensional to two-dimensional series the results by Hardy and Littlewood [6] on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript r"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-integrability of the sum <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> …