Degree of Approximation for Functions in the Generalised Zygmund Class
Keywords:
Cesaro summability method, Degree of approximation, Generalized Zygmund class, Taylor – Cesaro Summability method, Fourier Series.Abstract
Fourier partial sums may converge slowly or exhibit oscillatory behaviour for functions with limited smoothness, which makes summability methods important in quantitative approximation. This study derives an error estimate for 2π-periodic functions in a generalised Zygmund class when their Fourier series is processed by the Taylor–Cesàro (Tn, C2) product mean. The analysis formulates the associated product kernel, establishes separate bounds for small and large arguments, and applies the generalised Minkowski inequality together with the monotonicity of t^α ω(t). The resulting estimate extends the Cesàro-based theorem of Kim by replacing a single averaging operator with a regular product procedure that permits Taylor weighting and second-order Cesàro smoothing. The asymptotic order is not claimed to be strictly faster in every case; the main gain is a broader summability framework and greater flexibility in damping oscillatory partial sums. For ω(t) = t^m (m > 1), explicit bounded, logarithmic, and algebraic rates follow according to the value of α + m. The result clarifies the operator-theoretic role of product summability in Fourier approximation and supports extensions to weighted, deferred, and multidimensional settings.
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