Discrete Transform with Stick Property Based on $\{\sin x\sin kx\}$ and Second Kind Chebyshev Polynomials Systems

In this paper we introduce the discrete series with the «sticking»-property of the periodic ($\{\sin x \sin kx\}$ system) and non-periodic (using the system of the second kind of Chebyshev polynomials $U_k(x)$) cases. It is shown that series of the system $\{\sin x \sin kx\}$ have an advantage over cosine Fourier series because they have better approximation properties near the bounds of the $[0,\pi]$ segment. Similarly discrete series of the system $U_k(x)$ near the bound of the $[-1,1]$ approximates given function significantly better than Fouries sums of Chebyshev polynomials.