Estimates of Best Approximations of Transformed Fourier Series in L p -Norm and p -Variational Norm
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ESTIMATES OF BEST APPROXIMATIONS OF TRANSFORMED FOURIER SERIES IN Lp-NORM AND p-VARIATIONAL NORM UDC 517.518.832
S. S. Volosivets and A. A. Tyuleneva
∞ Abstract. We consider functions F = F (λ, f ) with transformed Fourier series λn An (x), where ∞ n=1 An (x) is the Fourier series of a function f . Let Cp be the space of 2π-periodic p-absolutely conn=1
tinuous functions with p-variational norm. Estimates of best approximations of F in Lp in terms of best approximations of f in Cp are given. Also the dual problem for F in Cp and f in Lp is treated. In the important case of fractional derivative, the sharpness of estimates is established.
1. Introduction Let f be a 2π-periodic real bounded function, ξ = {x0 < x1 < · · · < xn = x0 + 2π} be a partition of the period, and 1/p n p p κξ (f ) : = |f (xi ) − f (xi−1 )| , 1 ≤ p < ∞. i=1
We set by definition for 1 < p < ∞
ω1−1/p (f, δ) = sup κξp (f ) : λ(ξ) : = max(xi − xi−1 ) ≤ δ i
and, for k ∈ N, k ≥ 2,
ωk−1/p (f, δ) = sup{ω1−1/p (Δk−1 h f (x), |h|) : |h| ≤ δ},
where Δkh (x)
=
k i=0
k−i
(−1)
k f (x + ih), i
k ∈ N,
is the kth difference of f with step h. It is known that ωk−1/p (f, nδ) ≤ nk−1/p ωk−1/p (f, δ) for n ∈ N and δ ∈ [0, 2π] (see [14, Lemma 1]), and this property explains the “fractional” notation ωk−1/p . For 1 < p < ∞, let us introduce the space Vp of all 2π-periodic bounded functions with the property f Vp : = max f ∞ , ω1−1/p (f, 2π) < ∞ Cp = f ∈ Vp : lim ω1−1/p (f, δ) = 0 .
and
δ→0
Here f ∞ = sup |f (x)|. x∈[0,2π]
The space Vp of functions of bounded p-variation was introduced in the case p = 2 by N. Wiener [17], while the space Cp of p-absolutely continuous functions with another but equivalent form was considered by L. C. Young [18] (see also the paper by Love [7]). Both Vp and Cp are Banach spaces with respect to ·Vp . Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 111–126, 2018. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2503–0463
463
If Tn is the space of trigonometric polynomials of order at most n, then nth best approximation in Vp is introduced by En (f )Vp : = inf f − tn Vp , n ∈ Z+ . tn ∈Tn
The quantities ωk−1/p (f, δ) and En (f )Vp are related via direct and inverse approximation theorems in Cp : for 1 < p < ∞, k ∈ N, and f ∈ Cp , we have 1 , n ∈ Z+ , (1.1) En (f )Vp ≤ C(k)ωk−1/p f, n+1 n 1 −k+1/p (k + 1)k−1/p−1 Ek (f )Vp , n ∈ N. (1.2) ωk−1/p f, ≤ C(k)n n k=0
Both inequalities (1.1) and (1.2) are due to A. P. Terekhin; for (1.1) we refer to [9], while (1.2) can be found in [14]. Let Lp , 1 ≤ p < ∞, be the space of 2π-periodic measurable functions with finite norm 2π
p
|f (x)| dx
f p =
1/p
0
and for k ∈ N, δ ∈ [0, 2π],
ωk (f, δ)p : = sup{Δkh f (x)p : |h| ≤ δ}.
The best approximation En (f )p in the space Lp is introduced similarly to En (f )Vp . The problems of approximation in Cp and Lp , 1 < p < ∞, are closely connected (see [9, 10] and [15]). We say that for f ∈ Lp (f ∈ Cp ) and r > 0 there exists the fractional derivativ
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