<p>For the class of sine polynomials <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$b_1\sin t+b_2\sin 2t+...+b_N\sin Nt,\; (b_N\not = 0),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>sin</mo> <mi>t</mi> <mo>+</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>sin</mo> <mn>2</mn> <mi>t</mi> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>b</mi> <mi>N</mi> </msub> <mo>sin</mo> <mi>N</mi> <mi>t</mi> <mo>,</mo> <mspace width="0.277778em" /> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mi>N</mi> </msub> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> which are nonnegative on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(0,\pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, W. Rogosinski and G. Szegő derived, among other things, exact bounds for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|b_2|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> via the Lukács presentation of nonnegative algebraic polynomials and a variational type argument for exact bounds, but they did not find the extremizers. Within this algebraic framework, we construct explicit polynomials which attain these bounds and prove their uniqueness. The proof uses the Fejér - Riesz representation of nonnegative trigonometric polynomials, a 7-band Toeplitz matrix of arbitrary finite dimension, and Chebyshev polynomials of the second kind and their derivatives.</p>

Extremizers for the Rogosinski - Szegő Estimate of the Second Coefficient in Nonnegative Sine Polynomials

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发表日期：

2026年5月9日

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Dmitriy Dmitrishin,
Alexander Stokolos,
Walter Trebels

摘要

For the class of sine polynomials $b_1\sin t+b_2\sin 2t+...+b_N\sin Nt,\; (b_N\not = 0),$ $b_{1} sin t + b_{2} sin 2 t + . . . + b_{N} sin N t, (b_{N} \neq 0),$ which are nonnegative on $(0,\pi )$ $(0, π)$ , W. Rogosinski and G. Szegő derived, among other things, exact bounds for $|b_2|$ $| b_{2} |$ via the Lukács presentation of nonnegative algebraic polynomials and a variational type argument for exact bounds, but they did not find the extremizers. Within this algebraic framework, we construct explicit polynomials which attain these bounds and prove their uniqueness. The proof uses the Fejér - Riesz representation of nonnegative trigonometric polynomials, a 7-band Toeplitz matrix of arbitrary finite dimension, and Chebyshev polynomials of the second kind and their derivatives.