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An optimal set of quadrature formulas with an odd number of nodes for  trigonometric polynomials in Borges’ sense [Numer. Math. 67 (1994), 271-288],  as well as trigonometric multiple orthogonal polynomials of semi-integer  degree are defined and studied. The main properties of such a kind of  orthogonality are proved. Also, an optimal set of quadrature rules is  characterized by trigonometric multiple orthogonal polynomials of semiinteger  degree. Finally, theoretical results are illustrated by some numerical  examples. [Projekat Ministarstva nauke Republike Srbije, br. 174015 i br.  44006]

Trigonometric multiple orthogonal polynomials of semi-integer degree and the corresponding quadrature formulas