On Turan-type inequalities for trigonometric polynomials of half-integer order

O.V. Polyakov (Public Education Institution "Dnipropetrovsk Regional Physical and Mathematical Boarding Lyceum")

Abstract


Some exact inequalities of the Turan type are obtained in the paper for trigonometric polynomials $$$h(x)$$$ of half-interger order $$$n+\frac {1}{2}$$$, $$$n=1, 2, ...$$$, such that all $$$2n+1$$$ their zeros are real and located on a segment $$$[0;2\pi )$$$. Namely, the inequality that relates the norms in the space $$$C$$$ of the  trigonometric polynomials $$$h(x)$$$ of half-integer order $$$n+\frac {1}{2}$$$, $$$n=1, 2, ...$$$, and its second derivative $$$h''(x)$$$, $$$\|h''\|_c\ge \frac {2n+1}{4}\|h\|_c$$$, that is the inequalities that connect the norms in the space $$$L_2$$$ of the  trigonometric polynomials $$$h(x)$$$ of half-interger order $$$n+\frac {1}{2}$$$, $$$n=1, 2, ...$$$, and its first derivative $$$h'(x)$$$, that is $$$\|h'\|_{L_2}\ge \sqrt {\frac {2n+1}{8}}\|h\|_{L_2}$$$. The resulting inequalities cannot be improved. In proving the theorems, we use the method that was developed by V.F. Babenko and S.A. Pichugov for trigonometric polynomials, all of whose roots are real.

Keywords


trigonometric polynomials; norms; inequalities

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References


Turan P. "Uber die Abietung von polynomen", Composito Math, 1939; 7: pp. 89-95. (in German)

Korneichuk N.P., Babenko V.F., Ligun A.A. Extremum properties of polynomials and splines, Kyiv, 1992; 304 p. (in Russian)

Babenko V.F., Pichugov S.A. "On inequalities for derivatives of polynomials with real zeros", Ukrainian Math. J., 1986; 38(4): pp. 411-416. (in Russian)

Turetzkij A.Kh. Interpolation theory in problems, Vyshejshaya shkola, 1968; 318 p. (in Russian)




DOI: https://dx.doi.org/10.15421/241912

  

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