Let B denote the Banach space under the sup-norm \(\| \cdot \|\) of real-valued bounded functions on the real line \({\mathbb{R}}\) or \([-\pi,\pi]\) and let \(H_{\alpha}\) denote the space of Hölder continuous functions on \({\mathbb{R}}\) or \([-\pi,\pi].\) Suppose \(P(f;\zeta;x),\) \(Q(f;\zeta;x)\) and \(W(f;\zeta;x)\) denote respectively Picard, Poisson-Cauchy and Gauss- Weierstrass singular integrals of f. Recently, E. Deeba and the authors [Rend. Math., Appl., VII. Ser. 8, 345-355 (1988; Zbl 0677.42015)] have applied these integrals to study error bounds for approximation of functions of class \(L_ p\). In Theorem 1 of the present paper, the authors have studied the rate of convergence of functions \(f\in B\) by these integrals. Precisely, they have proved the following theorem: Let for \(f\in B,\omega (\delta)=\sup_{| t| \leq \delta}| f(x+h)- f(x)| \quad (\delta >0)\) and let \(\omega (t)/t\) be a non-increasing function of t. Then, as \(\zeta \to 0+,\) the following hold: \[ (i)\quad \| f-P(f;\zeta)\| =O\{\omega (\zeta)\},\quad (ii)\quad \| f- Q(f;\zeta)\| =O\{\omega (\zeta)| \log (1/\zeta)| \}, \]
\[ (iii)\quad \| f-W(f;\zeta)\| =O\{\omega (\zeta)/\sqrt{\zeta}\}. \] In the second theorem, the authors have obtained error bounds for \(f\in H_{\alpha}\quad (0<\alpha \leq 1)\) in the Hölder metric \(\| \cdot \|_{\beta}\quad (0\leq \beta <\alpha \leq 1)\).