| Summary: | It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be constructed. In this paper, we construct a discrete analogue <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>h</mi><mi>β</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><msup><mi>d</mi><mrow><mn>2</mn><mi>m</mi></mrow></msup><mrow><mi>d</mi><msup><mi>x</mi><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></mfrac><mo>+</mo><mn>2</mn><mfrac><msup><mi>d</mi><mi>m</mi></msup><mrow><mi>d</mi><msup><mi>x</mi><mi>m</mi></msup></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> in the Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mn>2</mn><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>. We develop an algorithm for constructing optimal quadrature formulas exact on exponential-trigonometric functions using a discrete operator. Based on this algorithm, in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, we give an optimal quadrature formula exact for trigonometric functions. Finally, we present the rate of convergence of the optimal quadrature formula in the Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> for the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>.
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