Mixed series by classical orthogonal polynomials

This work is dedicated to the foundations of the rapidly developing theory of special (mixed) series with the property of sticking of their partial sums by classical polynomials orthogonal either on the intervals or on uniform grids. It is shown that partial sums of special series compare favorably by approximative properties with corresponding partial sums of Fourier series by the same orthogonal polynomials. For example, the partial sums of mixed series can be successfully used to solve the problem of simultaneous approximation of a differentiable function and its multiple derivatives, while the partial sums of the Fourier series by orthogonal polynomials are not suitable for this task.