Phillips-Type q -Bernstein Operators on Triangles

<jats:p>The purpose of the paper is to introduce a new analogue of Phillips-type Bernstein operators <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mfenced open="(" close=")"> <mrow> <msubsup> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi>m</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>u</mi> </mrow> </msubsup> <mi>f</mi> </mrow> </mfenced> <mfenced open="(" close=")"> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </mfenced> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mfenced open="(" close=")"> <mrow> <msubsup> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> <mrow> <mi>v</mi> </mrow> </msubsup> <mi>f</mi> </mrow> </mfenced> <mfenced open="(" close=")"> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </mfenced> </math> </jats:inline-formula>, their products <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mi>m</mi> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mi>f</mi> </mrow> </mfenced> <mfenced open="(" close=")"> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </mfenced> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi mathvariant="script">Q</mi> </mrow> <mrow> <mi>n</mi> <mi>m</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mi>f</mi> </mrow> </mfenced> <mfenced open="(" close=")"> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </mfenced> </math> </jats:inline-formula>, their Boolean sums <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi mathvariant="script">S</mi> </mrow> <mrow> <mi>m</mi> <mi>n</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mi>f</mi> </mrow> </mfenced> <mfenced open="(" close=")"> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </mfenced> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mfenced open="(" close=")"> <mrow> <msub> <mrow> <mi mathvariant="script">T</mi> </mrow> <mrow> <mi>n</mi> <mi>m</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> <mi>f</mi> </mrow> </mfenced> <mfenced open="(" close=")"> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </mfenced> </math> </jats:inline-formula> on triangle <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <msub> <mrow> <mi mathvariant="script">T</mi> </mrow> <mrow> <mi>h</mi> </mrow> </msub> </math> </jats:inline-formula>, which interpolate a given function on the edges, respectively, at the vertices of triangle using quantum analogue. Based on Peano’s theorem and using modulus of continuity, the remainders of the approximation formula of corresponding operators are evaluated. Graphical representations are added to demonstrate consistency to theoretical findings. It has been shown that parameter <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <mi>q</mi> </math> </jats:inline-formula> provides flexibility for approximation and reduces to its classical case for <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <mi>q</mi> <mo>=</mo> <mn>1</mn> </math> </jats:inline-formula>.</jats:p>

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