| Summary: | The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mi>q</mi></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes.
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