Lebesgue inequality

An estimate of the deviation of the partial sums of a Fourier series using best approximations. In the case of the trigonometric system, the Lebesgue inequality is understood to be the relation

$$\max_x|R_n(f,x)|\leq(L_n+1)E_n(f),\quad n=1,2,\ldots,$$

where $R_n(f,x)$ is the $n$-th remainder of the (trigonometric) Fourier series of a continuous $2\pi$-periodic function $f$, $L_n$ are the Lebesgue constants, and $E_n(f)$ is the best uniform approximation (error) (cf. Best approximation) by trigonometric polynomials of degree $n$. The Lebesgue inequality is a relation of general character: analogues of it hold for arbitrary orthonormal systems with suitable definitions of the Lebesgue constants and of the best approximation, and also for the comparison of the remainders of Fourier series with best approximations in norms of other spaces, for example, $L_p$, $1\leq p<\infty$. The Lebesgue inequality and relations similar to it are often used in approximation theory to obtain estimates of best approximations from below. The inequality was established by H. Lebesgue.

References

 [1] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)