It is proved that the product method <mml:math alttext="$A\left( {C,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>C</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, where <mml:math alttext="$\left( {C,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>C</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> is the Ces&#224;ro arithmetic mean matrix, is totally effective under certain conditions concerning the matrix <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>. This general result is applied to study absolute N&#246;rlund summability of Fourier series and other related series.

International Journal of Mathematics and Mathematical Sciences

Some absolutely effective product methods

Abstract

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