This paper deals with the uniform approximation by Fourier sums and the best uniform approximation (by trigonometric polynomials) of some classes of functions that arise as convolutions with appropriate kernels.
Let \(\psi(k)\) be an arbitrary fixed sequence of real numbers and let \(\beta\) be a fixed real number. Then a function \(f\) in \(L^{1}\) with Fourier series \[ f\sim \frac{a_{0}}{2}+\sum^{\infty}_{k=1} (a_{k}\cos kx+ b_{k}\sin kx) \] is said to belong to the class \(L_{\beta}^{\psi}\) if the series \[ \sum^{\infty}_{k=1}\frac{1}{\psi(k)} \left(a_{k}\cos \left(kx + \frac{\beta\pi}{2}\right)+b_{k}\sin \left(kx+\frac{\beta\pi}{2}\right)\right) \] is the Fourier series of a summable function \(f_{\beta}^{\psi}\). For \(1\leq p\leq\infty\) the class \(C^{\psi}_{\beta,p}\) is formed by the \(2\pi\)-periodic continuous functions \(f\) such that \(f_{\beta}^{\psi}\) exists, \(f^{\psi}_{\beta}\in L^{p}\) and moreover \(\|f_{\beta}^{\psi}\|_{p}\leq 1\), \(f_{\beta}^{\psi}\perp 1\). It is known that functions in the class \(C_{\beta,p}^{\psi}\) are convolutions of functions in the unit ball of \(L^{p}\) with the kernel \[ \psi_{\beta}(t)=\sum^{\infty}_{k=1}\psi(k)\cos \left(kt-\frac{\beta\pi}{2}\right). \]
Assume now that the sequence \(\{\psi(k)\}\) is given by the restriction to the set of natural numbers of a convex continuous function \(\psi(t)\), \(t\geq 1\), with \(\lim\limits_{t\to\infty}\psi(t)=0\) satisfying the additional property that there is a constant \(k>0\) such that \[ 0<\frac{t}{\psi^{-1}(\psi(t)/2)-t}\leq k,\quad t\geq 1. \] The class of such convex functions will be denoted by \(\mathfrak{M}_{0}\). As to the approximation by Fourier series, consider the quantity \[ \mathcal{E}_{n}(C^{\psi}_{\beta,p})_{C}=\sup_{f\in C^{\psi}_{\beta,p}}\|f(\cdot)-S_{n-1}(f;\cdot)\|_{\infty},\quad 1\leq p\leq\infty, \] where \(S_{n-1}(f;\cdot)\) is the partial Fourier sum of \(f\) of order \(n-1\). The following result is then proved.
Theorem 1. Let \(\psi(t)t^{1/p}\in\mathfrak{M}_{0}\) and let \[ \sum^{\infty}_{k=1}\psi^{p'}(k) k^{p'-2}<\infty,\quad 1<p<\infty,\quad \frac{1}{p}+\frac{1}{p'}=1. \] Then, for any \(n\in\mathbb{N}\) and \(\beta\in\mathbb{R}\), the following relations are true: \[ K^{(1)}_{\psi,p}\left(\sum^{\infty}_{k=n}\psi^{p'}(k)k^{p'-2}\right)^{1/p'} \leq \mathcal{E}_{n}(C^{\psi}_{\beta,p})_{C}\leq K^{(2)}_{\psi,p}\left(\sum^{\infty}_{k=n}\psi^{p'}(k)k^{p'-2}\right)^{1/p'}, \] where \(K^{(1)}_{\psi,p}\) and \(K^{(2)}_{\psi,p}\) are positive constants that depend only on \(\psi\) and \(p\).
A similar result can be stated for the best uniform approximation. For this purpose consider now the quantity \[ E_{n}(C^{\psi}_{\beta,p})_{C}=\sup_{f\in C^{\psi}_{\beta,p}}\inf_{t_{n-1}\in\mathcal{T}_{2n-1}} \|f(\cdot)-t_{n-1}(\cdot)\|_{\infty},\quad 1\leq p\leq\infty, \] where \(\mathcal{T}_{2n-1}\) is the space of all trigonometric polynomials of degree not greater than \(n-1\).
To evaluate \(E_{n}(C^{\psi}_{\beta,p})_{C}\) one needs an additional assumption on the function \(\psi\) which is given in terms of the characteristic \[ \alpha(\psi;t)=\frac{\psi(t)}{t|\psi'(t)|} \quad (\psi'(t)=\psi'(t+0). \] Then the following result is proved.
Theorem 2. Let \[ \sum^{\infty}_{k=1}\psi^{p'}(k) k^{p'-2}<\infty \] and let \(\psi(t)=g_{p}(t)t^{-1/p}\), where \(g_{p}\in\mathfrak{M}_{0}\), \(1<p<\infty\), \(\dfrac{1}{p}+\dfrac{1}{p'}=1\) and \[ \inf_{t\geq 1}\alpha (g_{p};t)>\frac{p'}{2}. \] Then, for any \(n\in\mathbb{N}\) and \(\beta\in\mathbb{R}\), the following relations are true: \[ K^{(3)}_{\psi,p}\left(\sum^{\infty}_{k=n}\psi^{p'}(k)k^{p'-2}\right)^{1/p'} \leq E_{n}(C^{\psi}_{\beta,p})_{C}\leq \mathcal{E}_{n}(C^{\psi}_{\beta,p})_{C}\leq K^{(2)}_{\psi,p}\left(\sum^{\infty}_{k=n}\psi^{p'}(k)k^{p'-2}\right)^{1/p'}, \] where \(K^{(2)}_{\psi,p}\) and \(K^{(3)}_{\psi,p}\) are positive constants that depend only on \(\psi\) and \(p\).