A Decomposition of the Limit q-Bernstein Type Operators Via a Universal Factor

Abstract The focus of this work is on the properties of the unifying operator $$U_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> on C [0, 1], which serves as a universal left factor in a decomposition of the limit q -Bernstein type operators, $$L_{\infty ,q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:math> . More precisely, the factorization $$L_{\infty ,q}= U_q\circ T_L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>∘</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where $$T_L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>L</mml:mi> </mml:msub> </mml:math> is a linear operator on C [0, 1] depending on L , holds. It is shown that this factorization facilitates the derivation of new results and/or the simplification of proofs for the known ones.