Sharp Remez-Type Inequalities Estimating the L_q-Norm of a Function Via Its L_p-Norm

For any q ≥ p > 0, 𝛼 = (r + 1/q)/(r + 1/p), f_p ∈ [0,∞], and β ∈ [0, 2𝜋), we prove a sharp Remez-type inequality

$$ {\left\Vert x\right\Vert}_q\le \frac{{\left\Vert {\varphi}_r+c\right\Vert}_q}{{\left\Vert {\varphi}_r+c\right\Vert}_{L_p\left(\left[0,2\uppi \right]/{B}_y\left(\beta \right)\right)}^{\alpha }}{\left\Vert {x}^{(r)}\right\Vert}_{L_p\left(\left[0,2\uppi \right]/B\right)}^{\alpha }{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{1-\alpha } $$

for 2𝜋-periodic functions x ∈ L^r_∞, which have zeros and satisfy the condition

$$ {\left\Vert {x}_{+}\right\Vert}_p\kern0.5em {\left\Vert {x}_{-}\right\Vert}_p^{-1}={f}_p,\kern10em (1) $$

where 𝜑r is Euler’s perfect spline of order r, the number c is such that the function x = 𝜑_r +c satisfies condition (1), B is an arbitrary Lebesgue-measurable set such that

$$ \mu B\le \beta {\left({\left\Vert {\varphi}_r+c\right\Vert}_p{\left\Vert {x}^{(r)}\right\Vert}_{\infty }{\left\Vert x\right\Vert}_p^{-1}\right)}^{-1/\left(r+1/p\right)}, $$

the set B_y(β) is defined by B_y(β) := {t ∈ [0, 2𝜋] : |𝜑_r(t) + c| > y(β)}, and moreover, μB_y(β) = β. We also establish sharp Remez-type inequalities of various metrics for trigonometric polynomials and polynomial splines satisfying relation (1).