Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs

Abstract

For a function f continuous on a closed interval, its modulus of fractality ν(f, ε) is defined as the function that maps any ε > 0 to the smallest number of squares of size ε that cover the graph of f. The following condition for the uniform convergence of the Fourier series of f is obtained in terms of the modulus of fractality and the modulus of continuity ω(f, δ): if

$$\begin{array}{*{20}{c}} {\omega (f,\pi /n)\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right) \to 0}&{\text{as}}&{n \to + \infty ,} \end{array}$$

then the Fourier series of f converges uniformly. This condition refines the known Dini–Lipschitz test. In addition, for the growth order of the partial sums S_n(f, x) of a continuous function f, we derive an estimate that is uniform in x ∈ [0, 2π]:

$${S_n}(f,x) = o\left( {\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right)} \right).$$

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The optimality of this estimate is shown.