Approximation properties of summable functions on sets of full measure

Estimates are obtained of the rate of approximation almost everywhere as a function of the modulus of continuity of the approximated functions in $L^p$, and of the set from which the approximating functions are chosen. From this point of view the author studies the approximation of functions by Steklov means, partial sums of Fourier–Haar series, arbitrary sequences of polynomials in the Haar and Faber–Schauder systems, and piecewise monotone functions with variable intervals of monotonicity. The estimates of the rate of approximation almost everywhere that are obtained are distinguished from approximation estimates in an integral metric (i.e. from estimates of the type of Jackson's theorem in $L^p$) by unbounded factors depending on the modulus of continuity and the approximating functions. Estimates of the growth of these factors are obtained, and it is established that in a number of cases these estimates are best possible, or almost so.
Bibliography: 17 titles.