GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 1, 1999, 83-90

ON THE ABSOLUTE SUMMABILITY OF SERIES WITH RESPECT TO BLOCK-ORTHONORMAL SYSTEMS

G. NADIBAIDZE

Abstract. Theorems determining Weyl’s multipliers for the summability almost everywhere by the |c, 1| method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence {ω(n)} is the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all orthogonal series, then there exists a sequence {N_{k}} such that {ω(n)} will be the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all series with respect to the Δ_{k}-orthonormal systems.

The present paper deals with the summability almost everywhere (a.e.) by the |c, α| method of series with respect to block-orthonormal systems. Under the summability by the |c, α| method of the series

∑_{n=1}^{∞} a_{n}

is understood the convergence of the series

∑_{n=1}^{∞} |σ_{n+1}^{(α)} − σ_{n}^{(α)}|,

where

σ_{n}^{(α)} = \frac{1}{A_{n}^{α}} ∑_{k=1}^{∞} A_{n−k}^{α} a_{k}

are the Cesàro (c, α)-means.

The problem of the summability a.e. by the |c, α| method of orthogonal series was considered by P.L. Ul’yanov [1]. In particular, he proved that if

1991 Mathematics Subject Classification. 42C20. Key words and phrases. Block-orthonormal systems, Weyl multiplier, series with re-

spect to block-orthonormal systems, summability almost everywhere by the |c, 1| method.

83

1072-947X/99/0100-0083$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

the condition

∑_{n=1}^{∞} \frac{1}{n ω(n)} < ∞,  (1)

is fulfilled for a positive nondecreasing sequence {ω(n)}, then the convergence of the series

∑_{n=1}^{∞} a_{n}^{2} ω(n)

guarantees the summability a.e. on (0, 1) by the |c, α| method (α > \frac{1}{2}) of the series

∑_{n=1}^{∞} a_{n} φ_{n}(x)  (2)

for every orthonormal system from L^{2}(0, 1).

If however

∑_{n=1}^{∞} \frac{1}{n ω(n)} = ∞,

then there exists an even function f(x) ∈ ∩_{p≥1}L^{p}[0, 2π] such that its Fourier

series

f(x) ∼ ∑_{n=1}^{∞} c_{n} cos nx

converges a.e. on [0, 2π] and for every fixed α > 0 is not |c, α| summable a.e. on [0, 2π] though

∑_{n=1}^{∞} c_{n}^{2} ω(n) < ∞.

Definition 1 (see [2]). Let {N_{k}} be an increasing sequence of natural numbers, Δ_{k} = (N_{k}, N_{k+1}], k = 1, 2, …, and {φ_{n}} be a system of functions from L^{2}(0, 1). The system {φ_{n}} will be called a Δ_{k}-orthonormal system (Δ_{k}-ONS) if:

(1) ∥φ_{n}∥_{2} = 1, n = 1, 2, …;

(2) (φ_{i}, φ_{j}) = 0 for i, j ∈ Δ_{k}, i ≠ j, k ≥ 1.

Definition 2 (see [1]). A positive nondecreasing sequence {ω(n)} will be called the Weyl multiplier for the summability a.e. of series with respect to the Δ_{k}-ONS {φ_{n}} if the condition

∑_{n=1}^{∞} a_{n}^{2} ω(n) < ∞  (3)

guarantees the summability a.e. by the |c, α| method of the corresponding series (2).

Below we shall quote the theorem showing that if the sequence {ω(n)} is the Weyl multiplier for the summability a.e. by the |c, 1| method of all orthogonal series (2), then it will be the Weyl multiplier for the summability a.e. by the |c, 1| method of all series (2) with respect to the Δ_{k}-ONS for the increasing sequence of natural numbers {N_{k}}.

Theorem 1. If a positive nondecreasing sequence {ω(n)} is the Weyl multiplier for the summability a.e. by the |c, 1| method of all orthonormal series (2), then there exists an increasing sequence of natural numbers {N_{k}} such that {ω(n)} is the Weyl multiplier for the summability a.e. by the |c, 1| method of all series (2) with respect to the Δ_{k} = (N_{k}, N_{k+1}]-ONS.

Proof. We prove this theorem by the Wang–Ul’yanov’s scheme (see [1]) modifying it accordingly. Let the positive nondecreasing sequence {ω(n)} be the Weyl multiplier for the summability a.e. by the |c, 1| method of all orthogonal series (2). Then condition (1) is fulfilled.

As is known (see [1]), for the positive nondecreasing on [n_{0}, +∞) function ω(x) the series

∑_{m=n_{0}}^{∞} \frac{1}{m ω(m)} and ∑_{m=n_{0}^{2}}^{∞} \frac{1}{m ω(\sqrt{m})}

converge or diverge simultaneously. Therefore, taking into account (1), we have

∑_{n=1}^{∞} \frac{1}{n ω(\sqrt[4]{n})} < ∞.

Then

∑_{n=1}^{∞} \frac{R(n)}{n ω(\sqrt[4]{n})} < ∞,  (4)

where

R(n) = \begin{matrix}( & ∑_{k=2}^{∞} & \frac{1}{k ω( \sqrt[4]{k})} & ) & \frac{1}{2} \\ ( & ∑_{k=n+1}^{∞} & \frac{1}{k ω( \sqrt[4]{k})} & ) & \frac{1}{2} \end{matrix} .

Obviously, R(1) = 1, R(n) < R(n + 1) and lim_{n→∞} R(n) = +∞.

Define the sequence k(n) by the recursion formula

k(1) = 0, k(n + 1) = \begin{cases}k(n) + 1 & if R(n + 1) ≥ k(n) + 1, \\ k(n) & if R(n + 1) < k(n) + 1, \end{cases} n ≥ 1.

Thus we obtain the nondecreasing sequence of nonnegative integers for which

k(n) ≤ R(n), n = 1, 2, ….  (5)

Note that for the sequence k(n) there exists an increasing sequence of natural numbers {N_{k}} (it is assumed that N_{0} = 0) which is defined by the formula

k(n) = max{k : N_{k} < n}.

Then, taking into account (4) and (5), we find that

∑_{n=1}^{∞} \frac{k(n)}{n ω(\sqrt[4]{n})} < ∞.  (6)

Let {φ_{n}} be a block-orthonormal system with Δ_{k} = (N_{k}, N_{k+1}] and condition (3) be fulfilled. Then for the corresponding series (2) we have

σ_{n}(x) − σ_{n−1}(x) = \frac{1}{n(n − 1)} ∑_{i=1}^{∞} a_{i}(i − 1) φ_{i}(x), n ≥ 2.

Denoting by c the absolute positive constants which, generally speaking, may have different values in different inequalities and using (6), we find that

∑_{n=2}^{∞} ∫_{0}^{1} |σ_{n}(x) − σ_{n−1}(x)|dx ≤ ∑_{n=2}^{∞} ( ∫_{0}^{1} |σ_{n}(x) − σ_{n−1}(x)|^{2}dx)^{\frac{1}{2}} ≤

≤ c ∑_{n=2}^{∞} \frac{1}{n^{2}} ( ∫_{0}^{1} | ∑_{i=1}^{n} a_{i}(i − 1)φ_{i}(x)|^{2}dx)^{\frac{1}{2}} ≤

≤c ∑_{n=1}^{∞} \frac{1}{n^{2}} ( ∫_{0}^{1} | ∑_{i=1}^{N_{k(n)}} a_{i}(i−1)φ_{i}(x)|^{2}+∫_{0}^{1} | ∑_{i=N_{k(n)}+1}^{n} a_{i}(i−1)φ_{i}(x)|^{2}dx)^{\frac{1}{2}} ≤

≤ c ∑_{n=1}^{∞} \frac{1}{n^{2}} (k(n) ∑_{i=1}^{N_{k(n)}} a_{i}^{2}i^{2} + ∑_{i=N_{k(n)}+1}^{n} a_{i}^{2}i^{2})^{\frac{1}{2}} ≤

≤ c ∑_{n=1}^{∞} \frac{1}{n^{2}} (k(n) ∑_{i=1}^{n} i^{2}a_{i}^{2})^{\frac{1}{2}} ≤ c ∑_{n=1}^{∞} \frac{\sqrt{k(n)}}{n^{2}}[( ∑_{i=1}^{[ \sqrt[4]{n}]} i^{2}a_{i}^{2})^{\frac{1}{2}} +

+( ∑_{i=[ \sqrt[4]{n}]+1}^{n} i^{2}a_{i}^{2})^{\frac{1}{2}} ] ≤ c( ∑_{n=1}^{∞} \frac{\sqrt{k(n)}(\sqrt{n}\sqrt[4]{n})^{\frac{1}{2}}}{n^{2}} +

+ ∑_{n=1}^{∞} \frac{\sqrt{k(n)}}{n^{2}} \frac{nω(\sqrt[4]{n}))^{\frac{1}{2}}}{nω(\sqrt[4]{n}))^{\frac{1}{2}}} ( ∑_{i=[ \sqrt[4]{n}]+1}^{n} i^{2}a_{i}^{2})^{\frac{1}{2}} ) ≤

≤ ( ∑_{n=1}^{∞} \frac{1}{n^{\frac{9}{8}}} + ( ∑_{n=1}^{∞} \frac{ω(\sqrt[4]{n})}{n^{3}} ∑_{i=[ \sqrt[4]{n}]+1}^{n} i^{2}a_{i}^{2})^{\frac{1}{2}} ( ∑_{n=1}^{∞} \frac{k(n)}{kω(\sqrt[4]{n})})^{\frac{1}{2}} ) ≤

≤ c + c( ∑_{n=1}^{∞} \frac{1}{n^{3}} ∑_{i=1}^{∞} i^{2}a_{i}^{2}ω(i))^{\frac{1}{2}} ≤ c + c( ∑_{i=1}^{∞} a_{i}^{2}ω(i))^{\frac{1}{2}} < ∞,

whence by Levy’s theorem

∑_{n=2}^{∞} |σ_{n}(x) − σ_{n−1}(x)| < ∞ a.e. on (0, 1).

The theorem below makes it possible to determine the Weyl multipliers for the summability a.e. of the series (2) with respect to the Δ_{k}-ONS for regularly increasing sequences {N_{k}}.

Theorem 2. Let an increasing sequence of natural numbers {N_{k}} be given, for which the condition

∑_{k=n}^{∞} \frac{1}{N_{k}^{2}} = O(\frac{n}{N_{n}^{2}} ) (n → ∞)  (7)

is fulfilled, and let

k(n) = max{k : N_{k} < n}.

If for the positive nondecreasing sequence {ω(n)} condition (1) is fulfilled, then for every Δ_{k}-ONS {φ_{n}} the condition

∑_{n=1}^{∞} a_{n}^{2} ω(n) k(n) < ∞  (8)

guarantees the summability a.e. by the |c, 1| method of the corresponding series (2).

Proof. Let conditions (1), (7) and (8) be fulfilled. Then for the corresponding series (2) we have

∑_{n=2}^{∞} ∫_{0}^{1} |σ_{n}(x) − σ_{n−1}(x)|dx ≤ ∑_{n=2}^{∞} ( ∫_{0}^{1} |σ_{n}(x) − σ_{n−1}(x)|^{2}dx)^{\frac{1}{2}} ≤

≤c∑_{n=1}^{∞} \frac{1}{n^{2}}(∫_{0}^{1} | ∑_{i=1}^{N_{k(n)}} a_{i}(i−1)φ_{i}(x)|^{2}dx+∫_{0}^{1} | ∑_{i=N_{k(n)}+1}^{n} a_{i}(i−1)φ_{i}(x)|^{2}dx)^{\frac{1}{2}} ≤

≤ c ∑_{n=1}^{∞} \frac{1}{n^{2}} (k(n) ∑_{i=1}^{∞} i^{2}a_{i}^{2})^{\frac{1}{2}} ≤ c ∑_{n=1}^{∞} \frac{\sqrt{k(n)}}{n^{2}}[( ∑_{i=n}^{[ \sqrt[4]{n}]} i^{2}a_{i}^{2})^{\frac{1}{2}} +

+( ∑_{i=[ \sqrt[4]{n}]+1}^{n} i^{2}a_{i}^{2})^{\frac{1}{2}} ] ≤ c( ∑_{n=1}^{∞} \frac{\sqrt{k(n)}n^{\frac{3}{8}}}{n^{2}} +

+ ∑_{n=1}^{∞} \frac{\sqrt{k(n)}}{n^{2}} \frac{(nω(\sqrt[4]{n}))^{\frac{1}{2}}}{(nω(\sqrt[4]{n}))^{\frac{1}{2}}} ( ∑_{i=[ \sqrt[4]{n}]+1}^{n} i^{2}a_{i}^{2})^{\frac{1}{2}} ) ≤

≤ c + c( ∑_{n=1}^{∞} \frac{k(n)}{n^{3}} ∑_{i=1}^{∞} i^{2}a_{i}^{2}ω(i))^{\frac{1}{2}} ( ∑_{n=1}^{∞} \frac{1}{nω(\sqrt[4]{n})})^{\frac{1}{2}} ≤

≤ c + c( ∑_{i=1}^{∞} i^{2}a_{i}^{2}ω(i) ∑_{i=1}^{∞} \frac{k(n)}{n^{3}})^{\frac{1}{2}} =

= c + c( ∑_{i=1}^{∞} i^{2}a_{i}^{2}ω(i)( ∑_{n=i}^{N_{k(i)+1}} \frac{k(i)}{n^{3}} + ∑_{j=k(i)+1}^{∞} j ∑_{n=N_{j}+1}^{N_{j+1}} \frac{1}{n^{3}} ))^{\frac{1}{2}} ≤

≤ c + c( ∑_{i=1}^{∞} i^{2}a_{i}^{2}ω(i)(\frac{k(i)}{i^{2}} + (k(i) + 1) ∑_{n=N_{k(i)+1}+1}^{∞} \frac{1}{n^{3}} +

+ ∑_{j=k(i)+1}^{∞} \frac{1}{N_{j}^{2}} ))^{\frac{1}{2}} ≤ c + c( ∑_{i=1}^{∞} i^{2}a_{i}^{2}ω(i)k(i))^{\frac{1}{2}} < ∞,

whence by Levy’s theorem

∑_{n=2}^{∞} |σ_{n}(x) − σ_{n−1}(x)| < ∞ a.e. on (0, 1).

Remark 1. In Theorem 2, the Weyl multipliers defined by conditions (1) and (8) can be assumed to be exact on the set of sequences {N_{k}} with condition (7) in the sense that if condition (1) is violated, then one can construct a sequence {N_{k}} for which condition (7) is fulfilled and also there exists a trigonometric series

∑_{n=1}^{∞} b_{n} cos nx,

which is nonsummable by the |c, α| method for almost all x ∈ [0, 2π] (for every fixed α > 0) though

∑_{n=1}^{∞} b_{n}^{2} ω(n) k(n) < ∞.

Indeed, let the condition

∑_{n=1}^{∞} \frac{1}{n ω(n)} = ∞

be fulfilled for the sequence {ω(n)}.

We construct an increasing sequence of natural numbers {N_{k}} in such a way that the condition

k = O( ∑_{n=1}^{N_{k}} \frac{1}{n ω(n)})^{β}, 0 < β ≤ \frac{1}{2},

be fulfilled and the sequence \frac{N_{k}}{k} be increasing.

Clearly, condition (7) is fulfilled (see [3], Remark 2).

Take

s_{k} = ∑_{n=1}^{k} \frac{1}{n ω(n)}, k = 1, 2, …,

and

c_{m} = \frac{1}{\sqrt{m} ω(m) (s_{m})^{β+ \frac{1}{2}}} m = 1, 2, ….

Then for arbitrary ε_{m} = ±1 we have

∑_{m=1}^{∞}(ε_{m}c_{m})^{2}ω(m)k(m) = ∑_{m=1}^{∞} \frac{k(m)}{m ω(m)(s_{m})^{2β+1}} =

= ∑_{k=0}^{∞} ∑_{m=N_{k}+1}^{N_{k+1}} \frac{k(m)}{m ω(m)(s_{m})^{2β+1}} ≤ c ∑_{k=0}^{∞} ∑_{m=N_{k}+1}^{N_{k+1}} \frac{(s_{N_{k}})^{β}}{m ω(m)(s_{m})^{2β+1}} ≤

≤ c ∑_{k=0}^{∞} ∑_{m=N_{k}+1}^{N_{k+1}} \frac{1}{m ω(m)(s_{m})^{1+β}} ≤ c ∑_{m=1}^{∞} \frac{1}{m ω(m)(s_{m})^{1+β}} < ∞.

On the other hand,

∑_{n=0}^{∞} \{ ∑_{m=2^{n}+1}^{2^{n+1}} c_{m}^{2}\}^{\frac{1}{2}} ≥ ∑_{n=0}^{∞} \{ ∑_{m=2^{n}+1}^{2^{n+1}} \frac{1}{m(ω(m))^{2}(s_{m})^{1+2β}} \}^{\frac{1}{2}} ≥

≥ \frac{1}{2} ∑_{n=1}^{∞} \frac{1}{ω(2^{n})(s_{2^{n}})^{\frac{1}{2} +β}} = ∞.

Therefore by Billard’s theorem [1], for almost all choices of ε_{k} = ±1 the series

∑_{m=1}^{∞} ε_{m} c_{m} cos mx

is |c, α|-nonsummable (α > 0) at almost every point x ∈ [0, 2π] though

∑_{n=1}^{∞} b_{n}^{2} ω(n) k(n) < ∞,

where b_{n} = ε_{n}c_{n}.

Remark 2. The above theorems remain also valid for |c, α| methods with cα > \frac{1}{2}.

References

1. P. L. Ul’yanov, Solved and unsolved problems of the theory of trigonometric and orthogonal series. (Russian) Uspekhi Mat. Nauk 19(1964), No. 1, 3–69.

2. V. F. Gaposhkin, Series of block-orthogonal and block-independent systems. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 1990, No. 5, 12–18; English translation: Soviet Math. (Iz. VUZ), 34(1990), No. 5, 13–20.

3. G. G. Nadibaidze, On some problems connected with series with respect to Δ_{k}-ONS. Bull. Acad. Sci. Georgia 144(1991), No. 2, 233–236.

(Received 15.05.1997)

Author’s address:

J. Javakhishvili Tbilisi State University

Faculty of Mechanics and Mathematics

2, University St., Tbilisi 380043

Georgia