Two-Sided Estimates of the $$L^\infty$$ -Norm of the Sum of a Sine Series with Monotone Coefficients $$\{b_k\}$$ via t
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Sided Estimates of the L∞ -Norm of the Sum of a Sine Series with Monotone Coefficients {bk } via the ∞ -Norm of the Sequence {kbk } E. D. Alferova1, 2* and A. Yu. Popov1, 2* 1
2
Lomonosov Moscow State University, Moscow, 119991 Russia Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia Received December 13, 2019; in final form, April 23, 2020; accepted May 14, 2020
Abstract—We refine the classical boundedness criterion for sums of sine series with monotone coefficients bk : the sum of a series is bounded on R if and only if the sequence {kbk } is bounded. We derive a two-sided estimate of the Chebyshev norm of the sum of a series via a special norm of the sequence {kbk }. The resulting upper bound is sharp, and the constant in the lower bound differs from the exact value by at most 0.2. DOI: 10.1134/S0001434620090199 Keywords: two-sided estimate of a norm, sine series, monotone coefficients.
We consider sine series ∞
(1)
bk sin(kx) = g(b; x)
k=1
for which the sequences b = {bk }k∈N of coefficients are monotone: b1 > 0,
bk+1 ≤ bk
for all k ∈ N,
lim bk = 0.
k→∞
(2)
The set of all sequences b satisfying condition (2) is denoted by M. Series (1) with coefficients from M converge at each point x ∈ R, and their sums are continuous on the intervals (2πm, 2π(m + 1)), m ∈ Z. In view of the fact that the functions sin(kx), k ∈ N, are odd and 2π-periodic, it suffices to study the sum of the series (1) on the interval (0, π). While for sine series of general form (convergent everywhere or almost everywhere), it is hardly possible to find a boundedness criterion for their sums in terms of the sequence of coefficients, for the sums of series (1) with coefficients from M, such a criterion is fairly simple [1, Sec. 7, Theorem 7.27], [2, Chap. V, Theorem 1.3]. The sum of series (1) with coefficients from M is bounded on R (or, equivalently, is bounded on (0, π)) if and only if so is the sequence {kbk }k∈N . In connection with the given criterion, the problem arises of estimating the norms g(b; · )L∞ (R) = sup |g(b; x)| = sup |g(b; x)| = g(b; · )L∞ (0,π) x∈R
(3)
0 0. 3 Using (14) and taking f = ϕ and N = m + 1 in (13), we can write ˆ 1 m ϕ(0) k ϕ(1) π3 1 + + ≤ ϕ ϕ(t) dt + . m+1 2 m+1 2 36(m + 1)2 0 |ϕ (t)| ≤
(14)
(15)
k=1
Since ϕ(0) = π and ϕ(1) = 0, using (11) and (15), we obtain the inequality ˆ 1 π3 1 π − for all ϕ(t) dt + + Am ≤ 2 36(m + 1) 2(m + 1) 2(m + 1) 0 MATHEMATICAL NOTES
Vol. 108 No. 4 2020
m ∈ N.
(16)
474
ALFEROVA, POPOV
Further, we have π3 1 π 1 1 − < C0 (see Remark 2), and then the proof of the theorem will be complete. To prove inequality (23), we introduce the function ψ(t) = t−1 (1 − cos(At)). As above, we write the integral representation ˆ 1 − cos(Ax) A3 x 2 =− 3 y sin(Ay) dy x x 0 and use it to derive an upper bound: ˆ 1 − cos(At) A3 t 2 A3 y dy = ≤ |ψ (t)| = t t3 0 3
for all t > 0.
Using (24) and (13), we can write ˆ 1 n−1 1 ψ(0) k ψ(1) A3 + + + ψ(t) dt ≤ ψ n 2 n 2 36n2 0
(24)
for all n ∈ N.
k=1
We have ψ(0) = 0, Ther
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