### ORDER EQUALITIES IN DIFFERENT METRICS FOR MODULI OF SMOOTHNESS OF VARIOUS ORDERS

#### Abstract

In this paper, we obtain order equalities for the \(k\)th order \(L_{q}(T)\)-moduli of smoothness \(\omega_{k}(f;\delta)_{q}\) in terms of expressions that contain the \(l\)th order \(L_{p}(T)\)-moduli of smoothness \(\omega_{ l }(f;\delta)_{p}\) on the class of periodic functions \(f\in L_{p}(T)\) with monotonically decreasing Fourier coefficients, where \(1<p<q<\infty,\) \(k,l \in \mathbb{N},\) and \(T=(-\pi,\pi].\)

#### Keywords

Inequalities of different metrics for moduli of smoothness, Order equality, Trigonometric Fourier series with monotonic coefficients

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