Approximation via a product ( <i>P</i> , <i>K</i> _<i>a</i> )-convergence method on <i>H</i> _ω ( <i>K</i> )

Abstract This paper presents a Korovkin-type approximation theorem for positive linear operators defined on the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mi>ω</m:mi> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:mi>K</m:mi> </m:mrow> </m:mfenced> </m:math> ${H}_{\omega }\left(K\right)$ with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:mfenced close=")" open="["> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:mfenced> </m:math> $K=\left[0,\infty \right)$ . The main result is formulated using the concept of ( P , K a )-convergence, which is defined as the product of P -statistical convergence and K a -convergence. We provide a constructive example of a sequence of operators that satisfies the conditions of the theorem. The behaviour of the operator in the example has been further illustrated through graphs. Furthermore, we extend our results to the multidimensional case. Finally, we use a modulus of smoothness to calculate the rate of ( P , K a )-convergence for this sequence of operators.