L-optimal designs for a trigonometric Fourier regression model with no intercept

This paper is devoted to the problem of constructing L-optimal designs for a trigonometric Fourier regression model with no intercept. The paper considers diagonal matrices $L$ with a combination of zeros and ones on the main diagonal. It is shown that in the case when $L = I$(i. e., when the unit matrix is chosen as the matrix $L$), the L-optimal design coincides with the D-optimal one. In the more general case (when some diagonal elements are equal to zero), it is shown that the dimension of the problem can be reduced if the optimal design is symmetric. The obtained results are illustrated by the example of the problem of constructing two L-optimal designs for the trigonometric model of order 12, which is reduced to the problem of constructing designs for models of order 3 and 4 correspondingly.