Bounds for Orthogonal Expansion Coefficients via Weighted Primitives

Abstract

For trigonometric and polynomial orthogonal expansions, we present coefficient bounds with an explicit numerator determined by the total variation of the function and the largest cumulative signed area of the weighted basis element. This area, denoted by $M_n$, is the sharp constant in the corresponding weighted integral estimate. We calculate or bound $M_n$ for several classical systems, thereby obtaining quantitative inequalities beyond an asymptotic statement of coefficient decay. Integration between consecutive zeros gives a geometric interpretation of the estimate. The comparison includes Chebyshev polynomials of the first three kinds, Legendre, Jacobi, Gegenbauer, and ordinary Laguerre polynomials, with normalization and boundary conditions stated explicitly. Classical estimates and their elementary consequences are organized as Fourier analysis in inequalities. In the controlled distortion application, ideal curves have selected coefficients absent, and restrictions on the variation, corridor width, and number of extrema of admissible departures give explicit guarantees for those coefficients.