On summability methods for Fourier series and Fourier integrals

In the problem of summability at a point at which the derivative of indefinite
integral exists for Fourier series and Fourier integrals of integrable functions
a new sufficient condition is obtained. In the case of "arithmetic means" the
corresponding condition is also necessary.
Exact rates of approximation by the classical Gauss-Weierstrass, Bochner-Riesz,
and Marcinkiewicz-Riesz means, as well as by non-classical Bernstein-Stechkin means
are found.
These problems are related to the representability of a function as an absolutely
convergent Fourier integral. For this, new conditions are obtained, while for radial functions
even a criterion.