Best approximations of functions in the $L_p$ metric by Haar and Walsh polynomials

In this work the modulus of continuity of functions in the $L_p$ metric $(1\leqslant p<\nobreak\infty)$ is estimated through its best approximations in this metric by Haar and Walsh polynomials. Besides, estimates of best approximations of functions by Haar and Walsh polynomials in the $L_q$ metric are obtained by the same approximations in the $L_p$ metric $(1\leqslant p<q\leqslant\infty)$. In the last case, the results are analogous to those which were proved for approximations by trigonometric polynomials by P. L. Ul'yanov and also by S. B. Stechkin and A. A. Konyushkov.
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