Nikol’skii Inequality Between the Uniform Norm and $$\varvec{L_q}$$ L q -Norm with Ultraspherical Weight of Algebraic Polynomials on an Interval

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We study the Nikol’skii inequality for algebraic polynomials on the interval \([-1,1]\) between the uniform norm and the norm of the space \(L^{\phi }_q,\)\({1\le q<\infty },\) with the ultraspherical weight \(\phi (x)=\phi ^{(\alpha )}(x)=(1-x^2)^\alpha ,\)\({\alpha \ge -1/2.}\) We prove that the polynomial with unit leading coefficient that deviates least from zero in the space \(L_q^\psi \) with the Jacobi weight \(\psi (x)=\phi ^{(\alpha +1,\alpha )}(x)=(1-x)^{\alpha +1}(1+x)^{\alpha }\) is an extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation generated by the ultraspherical weight.