Whitney's theorem in the $L^p$-metric, $0<p<\infty$

In the original proof of the theorem on local approximation of functions by algebraic polynomials, Whitney estimated the deviation of the function from an interpolating polynomial with equally spaced nodes by means of finite differences. In this article we show how Whitney's idea of choosing nodes depending on a parameter with subsequent averaging can be applied to functions in $L^p$. The methods indicated allow one to obtain an estimate for the speed of approximating functions given on au arc of the unit circle by trigonometric polynomials or splines.