1989/01/01 by S. Thangavelu, Sundaram Thangavelu · 7 citations
Mathematics · #Mathematical functions and polynomials #Approximation Theory and Sequence Spaces #Mathematical Approximation and Integration
paper · pdf · doi:10.1090/s0002-9947-1989-0958904-7
We study the summability of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -dimensional Hermite expansions where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n > 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that the critical index for the Riesz summability is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n minus 1 right-parenthesis slash 2"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">(n - 1)/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also prove analogues of the Fejér-Lebesgue theorem and Riemann’s localisation principle when the index <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">α</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Riesz means is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than left-parenthesis 3 n minus 2 right-parenthesis slash 6"> <mml:semantics> <mml:mrow> <mml:mo>></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>6</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">> (3n - 2)/6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .