On the Unimprovability of Some Theorems on the Convergence in the Mean of Trigonometric Series
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ON THE UNIMPROVABILITY OF SOME THEOREMS ON THE CONVERGENCE IN THE MEAN OF TRIGONOMETRIC SERIES A. S. Belov
UDC 517.518
Abstract. This paper puts forward a method for constructing trigonometric Fourier series with L2π -unbounded partial sums that have coefficients with some preassigned properties. In particular, examples of trigonometric Fourier series showing that some conditions of convergence in the mean of trigonometric series cannot be sharpened are constructed.
1. Introduction Let ∞
∞ 1 cn einx = a0 + an cos(nx) + bn sin(nx) 2 n=−∞
(1.1)
n=1
be an arbitrary trigonometric series written down in real or complex form, where an = cn + c−n and bn = (cn − c−n )i. As usual, let Sn (x) =
n
ck eikx and σn (x) =
k=−n
n
1 Sk (x) n+1 k=0
be, respectively, the partial sums and the Fej´er means of series (1.1), as considered for all n ≥ 0 and S˜n (x) =
n
i(−ck eikx + c−k e−ikx )
k=1
be the partial sums of the conjugate series. If f ∈ L2π (this means that f is 2π-periodic and Lebesgue-integrable function), then 1 f 1 = 2π
2π |f (x)| dx 0
is the norm. It is known (see [1, Chap. 1, Sec. 60]) that if series (1.1) is a Fourier series of some function f ∈ L2π , then σn − f 1 → 0 as n → ∞. Therefore, its partial sums Sn converge (respectively, are bounded) in the mean, i.e., in the metric of L2π , if and only if σn − Sn 1 = o(1) (respectively, σn − Sn 1 = O(1)). Of course, we assume that n tends to infinity in (1.2) and similar relations in the sequel. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 31–49, 2018.
404
c 2020 Springer Science+Business Media, LLC 1072–3374/20/2503–0404
(1.2)
It is also known (see [2, Lemma 3] or [8, Lemmas 1 and 2]) that condition (1.2) is equivalent to each of the following conditions: (respectively, ˜ σn − S˜n 1 = O(1)); ˜ σn − S˜n 1 = o(1) max
Sm − Sn−1 1 = o(1) (respectively,
max
S˜m − S˜n−1 1 = o(1) (respectively,
m=n,...,2n−1 m=n,...,2n−1
max
Sm − Sn−1 1 = O(1));
max
S˜m − S˜n−1 1 = O(1)).
m=n,...,2n−1 m=n,...,2n−1
A more detailed account on the condition (1.2) may be found in [6]. As usual, we shall make use of notation Δan = an − an+1 . The above theorem was proved in [2]. Theorem A. Suppose that there exists a positive constant C such that n
2
2n+1 −1
(|Δak |2 + |Δbk |2 ) ≤ Cn2 for all n ≥ 1.
(1.3)
k=2n
Then condition (1.2) is equivalent to the condition (|an | + |bn |) ln n = o(1)
(respectively, (|an | + |bn |) ln n = O(1)).
(1.4)
The following theorem is a consequence of Theorem A (see [2]). Theorem B. Suppose that there exists a positive constant C such that ln(n + 1) for all n ≥ 1. |Δan | + |Δbn | ≤ C n Then conditions (1.2) and (1.4) are equivalent.
(1.5)
In Theorems A and B, the trigonometric series (1.1) need not be a Fourier series. However, if series (1.1) is a Fourier series, then Theorem B states that under condition (1.5) the sequence of partial sums of series (1.1) converges (respectively, is bounded) in the mean if and only if condition (1.4) is fulfilled. The last assert
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