Sharp Remez-Type Inequalities Estimating the Lq-Norm of a Function Via Its Lp-Norm

• Published in: Ukrainian mathematical journal Vol. 74; no. 5; pp. 726 - 742
• Main Authors: Kofanov, V. A., Olexandrova, T. V.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2022
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Statistics
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-022-02097-z

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 x q ≤ φ r + c q φ r + c L p 0 2 π / B y β α x r L p 0 2 π / B α x r ∞ 1 − α for 2𝜋-periodic functions x ∈ L r ∞ , which have zeros and satisfy the condition x + p x − p − 1 = f p , 1 where 𝜑 r 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 μB ≤ β φ r + c p x r ∞ x p − 1 − 1 / r + 1 / p , 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).