[Paper Review] On the best approximation of certain classes of periodic functions by trigonometric polynomials

A. S. Serdyuk, Ie. Yu. Ovsii|arXiv (Cornell University)|May 30, 2010

Mathematical Approximation and Integration8 references21 citations

TL;DR

This paper establishes asymptotically exact estimates for the best trigonometric polynomial approximation of periodic functions in the uniform norm, specifically for classes $ C^{eta}_{eta}H_{ ho} $ defined via $(\psi,\beta)$-derivatives with a given modulus of continuity $\omega$. The key result provides a precise asymptotic formula for the best approximation error $ E_n(C^{eta}_{eta}H_{\omega}) $, involving an integral of $ \psi(1/t)\omega(t)/t $, under conditions on $ \psi \in \mathfrak{M}_0' $ and $ \beta \notin 2\mathbb{Z} $, extending known order estimates to exact asymptotic expressions.

ABSTRACT

We obtain the estimates for the best approximation in the uniform metric of the classes of $2π$-periodic functions whose $(ψ,β)$-derivatives have a given majorant $ω$ of the modulus of continuity. It is shown that the estimates obtained here are asymptotically exact under some natural conditions on the parameters $ψ,$ $ω$ and $β$ defining the classes

Motivation & Objective

• To derive asymptotically exact estimates for the best approximation of classes of $ 2\pi $-periodic functions by trigonometric polynomials of degree at most $ n-1 $ in the uniform metric.
• To extend known order estimates for $ E_n(C^\psi_\beta H_\omega) $ to exact asymptotic formulas under general conditions on the majorant $ \omega $, the sequence $ \psi $, and the parameter $ \beta $.
• To characterize the rate of decay of the best approximation error when $ \psi \in \mathfrak{M}_0' $, a class of functions with controlled half-decay modulus, and $ \beta \in \mathbb{R} $, including non-integer $ \beta $.

Proposed method

• The authors define the class $ C^\psi_\beta H_\omega $ as the set of $ 2\pi $-periodic continuous functions whose $ (\psi,\beta) $-derivative belongs to the modulus of continuity class $ H_\omega $, using a generalized derivative based on Fourier series with phase shifts.
• They construct a test function $ f^* $ whose $ (\psi,\beta) $-derivative is an odd, $ 2\pi/n $-periodic function $ \varphi_n $, designed to maximize the approximation error at specific points.
• Using the de la Vallée Poussin theorem, they establish a lower bound for the best approximation error $ E_n(f^*) $ by analyzing the alternating sign behavior of the error at $ 2n $ equally spaced points on the period.
• The method involves transforming the error expression into an integral involving $ \psi(nu) $, $ \sin(ut) $, and the modulus of continuity $ \omega $, leveraging the symmetry and decay properties of $ \psi \in \mathfrak{M}_0' $.
• They derive a precise asymptotic formula for $ E_n(C^\psi_\beta H_\omega) $ by combining the lower bound with an upper bound derived from known estimates, leading to an exact expression involving $ \int_0^{1/n} \psi(1/t) \omega(t)/t \, dt $.
• The final formula includes a constant $ \theta_n(\omega) \in [2/3,1] $, reflecting the oscillatory nature of the extremal function, and an error term $ O(1)\psi(n)\omega(1/n) $.

Experimental results

Research questions

• RQ1 What is the exact asymptotic behavior of the best approximation error $ E_n(C^\psi_\beta H_\omega) $ for $ \psi \in \mathfrak{M}_0' $ and $ \beta \in \mathbb{R} $, beyond known order estimates?
• RQ2 How does the approximation error depend on the modulus of continuity $ \omega $, the sequence $ \psi $, and the parameter $ \beta $, particularly when $ \sin(\beta\pi/2) \neq 0 $?
• RQ3 Can the best approximation error be expressed as an integral involving $ \psi(1/t) \omega(t)/t $, and what is the role of the half-decay modulus $ \mu(t) $ in this context?
• RQ4 Under what conditions on $ \psi $ and $ \omega $ is the approximation error asymptotically equivalent to $ \int_0^{1/n} \psi(1/t) \omega(t)/t \, dt $?
• RQ5 How do the results generalize known exact estimates for Weyl-Nagy and Weyl classes, especially for $ \psi(k) = k^{-r} $ and $ \beta = r $?

Key findings

• The best approximation error $ E_n(C^\psi_\beta H_\omega) $ satisfies the asymptotically exact formula: $$ E_n(C^\psi_\beta H_\omega) = \frac{\theta_n(\omega)}{\pi}\left|\sin\frac{\beta\pi}{2}\right|\int_0^{1/n}\psi\left(\frac{1}{t}\right)\frac{\omega(t)}{t}\,dt + O(1)\psi(n)\omega(1/n), $$ where $ \theta_n(\omega) \in [2/3,1] $, for $ \psi \in \mathfrak{M}_0' $ and $ \beta \notin 2\mathbb{Z} $.
• The integral term $ \int_0^{1/n}\psi(1/t)\omega(t)/t\,dt $ dominates the error rate when $ \psi \in \mathfrak{M}_0' $, capturing the precise decay behavior of the approximation error.
• For $ \psi(k) = k^{-r} $, the formula recovers known order estimates, but now with an exact constant factor $ \theta_n(\omega) $, improving upon previous order-only results.
• The error term $ O(1)\psi(n)\omega(1/n) $ is uniformly bounded and becomes negligible compared to the main term under the given conditions.
• The result is sharp in the sense that the lower bound derived via the de la Vallée Poussin theorem matches the upper bound up to a constant factor, confirming asymptotic exactness.
• The dependence on $ \sin(\beta\pi/2) $ reflects that the approximation error vanishes when $ \beta \in 2\mathbb{Z} $, consistent with symmetry properties of the Fourier series.