GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 1995, 517-534

ON THE CONVERGENCE AND SUMMABILITY OF SERIES WITH RESPECT TO BLOCK-ORTHONORMAL

SYSTEMS

G. NADIBAIDZE

Abstract. Statements connected with the so-called block-orthonor-malized systems are given. The convergence and summability almost everywhere by the (c, 1) method with respect to such systems are considered. In particular, the well-known theorems of Menshov-Rademacher and Kacmarz on the convergence and (c,1)-summability almost everywhere of orthogonal series are generalized.

1. The so-called block-orthonormal systems were introduced by V. F. Gaposhkin who obtained the first results [1] for series with respect to such systems. In particular, he generalized the well-known Menshov– Rademacher theorem. This paper presents the results on the convergence and (c,1)-summability almost everywhere of series with respect to blockorthonormal systems. These results were announced in [2] and [3] but here some of them are formulated in a slightly different form.

Let {N_{k}} be a strictly increasing sequence of natural numbers and Δ_{k} = (N_{k}, N_{k+1}], k = 1, 2, . . ..

Definition 1 ([1]). Let {φ_{n}} be a system of functions from L^{2}(0, 1). {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. A positive nondecreasing sequence {ω(n)} will be called the Weyl multiplier for the convergence ((c, 1)-summability) a.e. of series with respect to the Δ_{k}-ONS {φ_{n}(x)} if the convergence of the series

∑_{n=1}^{∞} a_{n}^{2}ω(n)

517

1072-947X/95/0900-0517$7.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

1991 Mathematics Subject Classification. 42C20.

Key words and phrases. Block-orthonormal systems, Weyl multiplier, convergence and (c,1)-summability almost everywhere of block-normal systems.

guarantees the convergence ((c, 1)-summability) a.e. of the series

∑_{n=1}^{∞} a_{n}φ(x).  (1)

2. Let the sequence {N_{k}} be fixed and Δ_{k} = (N_{k}, N_{k+1}]. Without loss of generality it can be assumed that

N_{0} = 0, N_{1} = 1, ω(0) = 1.

We have

Theorem 1. In order that a positive nondecreasing sequence {ω(n)} be the Weyl multiplier for the convergence a.e. of series with respect to any Δ_{k}-ONS, it is necessary and sufficient that the following two conditions be fulfilled:

(a) ∑_{k=1}^{∞} \frac{1}{ω(N_{k})} < ∞;

(b) log_{2}^{2} n = O(ω(n)) for n → ∞.

Proof. Sufficiency. Let the conditions of the theorem be fulfilled and for the sequence {a_{n}}

∑_{n=1}^{∞} a_{n}^{2}ω(n) < ∞.

We introduce

ψ_{k}(x) = ∑_{n=N_{k}+1}^{N_{k+1}} a_{n}φ_{n}(x), k = 0, 1, 2, ….

Then

∑_{k=0}^{∞} ∥ψ_{k}(x)∥_{1} ≤ ∑_{k=0}^{∞} ∥ψ_{k}(x)∥_{2} = ∑_{k=0}^{∞} ∥ψ_{k}(x)∥_{2}(ω(N_{k}))^{\frac{1}{2}} (ω(N_{k}))^{− \frac{1}{2}} ≤

≤∑_{k=0}^{∞} ∥ψ_{k}(x)∥_{2}^{2}ω(N_{k}) ∑_{k=0}^{∞} \frac{1}{ω(N_{k})} =∑_{k=0}^{∞} ( ∑_{n=N_{k}+1}^{N_{k+1}} a_{n}^{2}) ω(N_{k}) ∑_{k=0}^{∞} \frac{1}{ω(N_{k})} ≤

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

which by the Levy theorem implies that

∑_{k=0}^{∞} |ψ_{k}(x)| < ∞ a.e.

Therefore the sequence S_{N_{k}}(x), where

S_{k}(x) = ∑_{n=1}^{k} a_{n}φ_{n}(x),

converges a.e.

Let

δ_{k}(x) = max_{N_{k}<j≤N_{k+1}} | ∑_{n=N_{k}+1}^{j} a_{n}φ(x)|, k ≥ 1.

Using the Kantorovich inequality, we obtain

∥δ_{k}(x)∥_{2}^{2} ≤ c ∑_{n=N_{k}+1}^{N_{k+1}} a_{n}^{2} log_{2}^{2} n, k ≥ 1.

Now

∑_{k=0}^{∞} ∥δ_{k}(x)∥_{2}^{2} ≤ c ∑_{k=0}^{∞} ∑_{n=N_{k}+1}^{N_{k+1}} a_{n}^{2} log_{2}^{2} n ≤ c ∑_{n=1}^{∞} a_{n}^{2}ω(n) < ∞,^{1}

from which it follows that lim_{k→∞} δ_{k}(x) = 0 for a.e. x ∈ (0, 1). This together with the proven convergence almost everywhere of the series S_{N_{k}}(x) guarantees the convergence of series (1) a.e. on (0, 1).

Necessity.

(1) Let

∑_{k=1}^{∞} \frac{1}{ω(N_{k})} = ∞.

Then there exist numbers c_{k} > 0 such that

∑_{k=1}^{∞} c_{k}^{2}ω(N_{k}) < ∞ and ∑_{k=1}^{∞} c_{k} = ∞.

Let Φ_{N_{k}}(x) = 1 (k = 1, 2, …; x ∈ (0, 1)) and choose as other functions Φ_{n}(x) (n ∈ N, n ≠ N_{k}, k = 1, 2, …) an arbitrary ONS orthogonal to 1. The system {Φ_{n}(x)} is an Δ_{k}-ONS. Take b_{n} = 0 (n ≠ N_{1}, N_{2}, …), b_{N_{k}} = c_{k} (k = 1, 2, …). Then

∑_{n=1}^{∞} b_{n}Φ_{n}(x) = ∑_{k=1}^{∞} c_{k} = ∞, x ∈ (0, 1),

though

∑_{n=1}^{∞} b_{n}ω(n) = ∑_{k=1}^{∞} c_{k}^{2}ω(N_{k}) < ∞.

1In what follows c will denote, generally speaking , various absolute constants.

The necessity of condition (1) is proved.

(2) If equality (b) is not fulfilled, then

\frac{log_{2}^{2} 2^{k}}{ω(2^{k})} ≥ \frac{1}{4} \frac{log_{2}^{2} 2^{k+1}}{ω(2^{k})} ≥ \frac{1}{4} \frac{log_{2}^{2} n}{ω(n)}, n ∈ (2^{k}, 2^{k+1}] k = 1, 2, …,

which implies that the equality

log_{2}^{2} 2^{k} = O(ω(2^{k})) for k → ∞

is not fulfilled either. Therefore we can find an increasing sequence of natural numbers q_{j}, j = 1, 2, …, such that

1 ≤ \sqrt{ω(2^{q_{j}+1})} < \frac{q_{j}}{j^{3}} , j = 1, 2, ….  (2)

Inequality (2) makes it possible to construct an orthonormal system {Φ_{n}(x)} (which simultaneously will also be a Δ_{k}-ONS) and a sequence {b_{n}} (see [4], p. 298, the proof of Menshov’s theorem) such that

∑_{n=1}^{∞} b_{n}^{2}ω(n) < ∞,

but the series

∑_{n=1}^{∞} b_{n}Φ_{n}(x)

diverges a.e. on (0, 1).

Remark 1. The application of the proven theorem to orthonormal systems allows us to formulate the Menshov-Rademacher theorem as follows:

In order that a positive nondecreasing sequence {ω(n)} be the Weyl multiplier for the convergence a.e. of series with respect to any orthonormal systems, it is necessary and sufficient that the equality

log_{2}^{2} n = O(ω(n)) as n → ∞

be fulfilled.

Remark 2. If

ω(n) = log_{2}^{2} n,

then condition (b) of Theorem 1 is fulfilled and we obtain Gaposhkin’s theorem [1, Proposition 1].

Remark 3. If

N_{k} = [2^{k^{α}}], 0 < α ≤ \frac{1}{2},^{1}

then log_{2}^{2} n will be the Weyl multiplier for the convergence a.e. not for each Δ_{k}-ONS. From Theorem 1 it follows that in that case

ω(n) = log_{2}^{\frac{1}{α} +ε} n, ε > 0,

is the Weyl multiplier.

Analogously, if

N_{k} = [k^{α}], α ≥ 1,

then

ω(n) = n^{\frac{1}{α}} log_{2}^{1+ε} n, ε > 0.

Also note that in both cases one should not take ε = 0.

3. Here a necessary and sufficient condition is established to be imposed on the sequence {N_{k}} so that the well-known Kacmarz theorem on the (c, 1)- summability a.e. of series with respect to orthonormal systems (see [5], p. 223, theorem [5.8.6]) remains valid also with respect to block-orthonormal systems. Moreover, a generalization of the Kacmarz theorem is given for a Δ_{k}-ONS.

In what follows we shall use the notation

σ_{n}(x) = \frac{1}{n} ∑_{i=1}^{n} S_{i}(x), k(n) = max \{k : N_{k} < n\}.

Lemma 1. Let the sequence {N_{k}} be fixed, {φ_{n}} be an arbitrary Δ_{k}- ONS and for a positive nondecreasing sequence {ω(n)} let there be given

min \{k : N_{k} ≥ n\} + n^{2} ∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} = O(ω(n)) for n → ∞.  (3)

Then the condition

∑_{n=1}^{∞} a_{n}^{2}ω(n) < ∞  (4)

implies the convergence a.e. of the series

∑_{n=2}^{∞} n(σ_{n}(x) − σ_{n−1}(x))^{2},

1[p] denotes the integer part of the number p.

Proof. Let conditions (3) and (4) be fulfilled. Then

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

≤ \frac{4}{n^{3}} ∫_{0}^{1} ( ∑_{i=1}^{N_{k(n)}} a_{i}(i − 1)φ_{i}(x) + ∑_{i=N_{k(n)}+1}^{n} a_{i}(i − 1)φ(x))^{2} dx ≤

≤ \frac{8}{n^{3}} [∫_{0}^{1} ( ∑_{j=0}^{k(n)−1} ∑_{i=N_{j}+1}^{N_{j+1}} 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}] ≤

≤ \frac{8}{n^{3}} [k(n) ∑_{j=0}^{k(n)−1} ∫_{0}^{1}( ∑_{i=N_{j}+1}^{N_{j+1}} a_{i}(i − 1)φ_{i}(x))^{2}dx+ ∑_{i=N_{k(n)}+1}^{n} a_{i}^{2}(i − 1)^{2}]=

= \frac{8}{n^{3}} [k(n) ∑_{i=1}^{N_{k(n)}} a_{i}^{2}(i − 1)^{2} + ∑_{i=N_{k(n)}+1}^{n} a_{i}^{2}(i − 1)^{2}] ≤

≤ \frac{8}{n^{3}} [k(n) ∑_{i=1}^{N_{k(n)}} a_{i}^{2}i^{2} + ∑_{i=N_{k(n)}+1}^{n} a_{i}^{2}i^{2}] , n ≥ 2.

Therefore

∑_{n=2}^{∞} ∫_{0}^{1} n(σ_{n}(x) − σ_{n−1}(x))^{2}dx ≤ 8 ∑_{k=0}^{∞} ∑_{n=N_{k}+1}^{N_{k+1}} \frac{1}{n^{3}} (k(n) ∑_{i=1}^{N_{k(n)}} a_{i}^{2}i^{2}+

+ ∑_{i=N_{k(n)}+1}^{n} a_{i}^{2}i^{2})=8 ∑_{k=0}^{∞} ( ∑_{n=N_{k}+1}^{N_{k+1}} ∑_{i=1}^{N_{k}} \frac{k}{n^{3}} a_{i}^{2}i^{2}+ ∑_{n=N_{k}+1}^{N_{k+1}} ∑_{i=N_{k}+1}^{n} \frac{1}{n^{3}} a_{i}^{2}i^{2})=

= 8 ∑_{i=1}^{∞} a_{i}^{2}i^{2} ∑_{k:N_{k}≥i} k ∑_{n=N_{k}+1}^{N_{k+1}} \frac{1}{n^{3}} + 8 ∑_{k=0}^{∞} ∑_{i=N_{k}+1}^{N_{k+1}} a_{i}^{2}i^{2} ∑_{n=i}^{N_{k+1}} \frac{1}{n^{3}} ≤

≤ 8 ∑_{i=1}^{∞} a_{i}^{2}i^{2} ∑_{k=k(i)+1} k(\frac{1}{N_{k}^{2}} − \frac{1}{N_{k+1}^{2}} ) + c ∑_{k=0}^{∞} ∑_{i=N_{k}+1}^{N_{k+1}} a_{i}^{2} =

= 8 ∑_{i=1}^{∞} a_{i}^{2}i^{2} [(k(i) + 1)\frac{1}{N_{k(i)+1}^{2}} + ∑_{k=k(i)+2}^{∞} \frac{1}{N_{k}^{2}} ] + c ∑_{i=1}^{∞} a_{i}^{2} <

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

from which by the Levy theorem we obtain

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

Lemma 2. Let {N_{k}} be a given sequence, {φ_{n}(x)} be an arbitrary Δ_{k}- ONS, and conditions (3), (4) be fulfilled. Then for the corresponding series (1) the convegence a.e. of the sequence {S_{2^{n}}(x)} is equivalent to the convergence a.e. of the sequence {σ_{2^{n}}(x)}.

Proof. Let conditions (3) and (4) be fulfilled. We have

S_{n}(x) − σ_{n}(x) = \frac{1}{n} ∑_{i=1}^{n} a_{i}(i − 1)φ_{i}(x).

Then

∫_{0}^{1} (S_{2^{n}}(x) − σ_{2^{n}}(x))^{2}dx = ∫_{0}^{1} \frac{1}{4^{n}} ( ∑_{i=1}^{N_{k(2^{n})}} a_{i}(i − 1)φ_{i}(x)+

+ ∑_{i=N_{k(2^{n})}+1}^{2^{n}} a_{i}(i − 1)φ_{i}(x))^{2} dx ≤ \frac{2}{4^{n}} [k(2^{n}) ∑_{i=1}^{N_{k(2^{n})}} a_{i}^{2}(i − 1)^{2}+

+ ∑_{i=N_{k(2^{n})}+1}^{2^{n}} a_{i}^{2}(i − 1)^{2}] ≤ \frac{2}{4^{n}} [k(2^{n}) ∑_{i=1}^{N_{k(2^{n})}} a_{i}^{2}i^{2} + ∑_{i=N_{k(2^{n})}+1}^{2^{n}} a_{i}^{2}i^{2}] .

Therefore

∑_{n=1}^{∞} ∫_{0}^{1} (S_{2^{n}}(x) − σ_{2^{n}}(x))^{2}dx ≤ 2 (∑_{n=1}^{∞} \frac{k(2^{n})}{4^{n}} ∑_{i=1}^{N_{k(2^{n})}} a_{i}^{2}i^{2}+

+ ∑_{n=1}^{∞} \frac{1}{4^{n}} ∑_{i=N_{k(2^{n})}+1}^{2^{n}} a_{i}^{2}i^{2}) = 2(J_{1} + J_{2}).

524 G. NADIBAIDZE

We have

J_{1} = ∑_{n=1}^{∞} \frac{k(2^{n})}{4^{n}} ∑_{i=1}^{N_{k(2^{n})}} a_{i}^{2}i^{2} = ∑_{k=1}^{∞} ∑_{log_{2} N_{k}<n≤log_{2} N_{k+1}} \frac{k(2^{n})}{4^{n}} ∑_{i=1}^{N_{k(2^{n})}} a_{i}^{2}i^{2} =

= ∑_{k=1}^{∞} ∑_{log_{2} N_{k}<n≤log_{2} N_{k+1}} \frac{k}{4^{n}} ∑_{i=1}^{N_{k}} a_{i}^{2}i^{2} =

= ∑_{k=1}^{∞} ( ∑_{log_{2} N_{k}<n≤log_{2} N_{k+1}} \frac{k}{4^{n}} ) ∑_{i=1}^{N_{k}} a_{i}^{2}i^{2} =

= ∑_{i=1}^{∞} a_{i}^{2}i^{2} ∑_{k=k(i)+1}^{∞} ( ∑_{log_{2} N_{k}<n≤log_{2} N_{k+1}} \frac{k}{4^{n}} ) =

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

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

≤ \frac{4}{3} ∑_{i=1}^{∞} a_{i}^{2}i^{2} [\frac{1}{i^{2}} min \{k : N_{k} ≥ i\} + ∑_{k:N_{k}≥i} \frac{1}{N_{k}^{2}} ] ≤ c ∑_{i=1}^{∞} a_{i}^{2}ω(i) < ∞

and

J_{2} = ∑_{n=1}^{∞} \frac{1}{4^{n}} ∑_{i=N_{k(2^{n})}+1} a_{i}^{2}i^{2} ≤ ∑_{n=1}^{∞} \frac{1}{4^{n}} ∑_{i=1}^{2^{n}} a_{i}^{2}i^{2} =

= ∑_{i=1}^{∞} a_{i}^{2}i^{2} ∑_{2^{n}≥i} \frac{1}{4^{n}} ≤ c ∑_{i=1}^{∞} a_{i}^{2} < ∞.

Therefore

∑_{n=1}^{∞} ∫_{0}^{1} (S_{2^{n}}(x) − σ_{2^{n}}(x))^{2} < ∞

from which it follows that

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

and therefore

lim_{n→∞} ∫_{0}^{1} (S_{2^{n}}(x) − σ_{2^{n}}(x))^{2} = 0 a.e.

Theorem 2. Let {N_{k}} be a given sequence, {φ_{n}(x)} be an arbitrary Δ_{k}- ONS, and conditions (3), (4) be fulfilled. Then for series (1) to be (c, 1)- convergent a.e. it is necessary and sufficient that the subsequence of partial sums {S_{2^{n}}(x)} of (1) be convergent a.e.

Proof. Sufficiency. Let conditions (3), (4) be fulfilled and the subsequence {S_{2^{n}}(x)} of the corresponding series (1) converge a.e. Then by Lemma 3 the subsequence {σ_{2^{n}}(x)} also converges a.e. and we have

sup_{k∈(2^{n},2^{n+1}]} (σ_{k}(x) − σ_{2^{n}}(x))^{2} = ( sup_{k∈(2^{n},2^{n+1}]} ∑_{i=2^{n}+1}^{k} (σ_{i}(x) − σ_{i−1}(x)))^{2} ≤

≤ ∑_{i=2^{n}+1}^{2^{n+1}} i(σ_{i}(x) − σ_{i−1}(x))^{2},

which by Lemma 1 implies that {σ_{n}(x)} converges a.e., i.e., series (1) is (c, 1)-summable a.e.

Necessity. Let conditions (3), (4) be fulfilled and series (1) be (c, 1)- summable a.e. Then {σ_{2^{n}}(x)} converges almost everywhere and by Lemma 2 {S_{2^{n}}(x)}, too, converges almost everywhere.

Lemma 3. If

∑_{k=3}^{∞} \frac{1}{(log_{2} log_{2} N_{k})^{2}} < ∞,

then

min \{k : N_{k} ≥ n\} + n^{2} ∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} = O((log_{2} log_{2} n)^{2}) for n → ∞.

Proof. Let

∑_{k=2}^{∞} \frac{1}{(log_{2} log_{2} N_{k})^{2}} < ∞.

Then

lim_{k→∞} \frac{k}{(log_{2} log_{2} N_{k})^{2}} = 0

and therefore for sufficiently large k’s we have

2^{2^{\sqrt{k}}} < N_{k}.

By definition, n ∈ (N_{k(n)}, N_{k(n)+1}]. Putting

q(n) = \begin{cases}k(n) + 1, & if 2^{2^{\sqrt{k(n)+1}}} ≥ n, \\ m, & if 2^{2^{\sqrt{k(n)+1}}} < n and 2^{2^{\sqrt{m−1}}} ≤ n < 2^{2^{\sqrt{m}}}, \end{cases}

for sufficiently large n’s we have

∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} = ∑_{k=k(n)+1} \frac{1}{N_{k}^{2}} = ∑_{k=k(n)+1}^{q(n)−1} \frac{1}{N_{k}^{2}} + ∑_{k=q(n)}^{∞} \frac{1}{N_{k}^{2}} ≤

≤ \frac{q(n) − k(n) − 1}{N_{k(n)+1}} + ∑_{k=q(n)}^{∞} \frac{1}{(2^{2^{\sqrt{k}}})^{2}} ≤ \frac{q(n) − k(n) − 1}{n^{2}} +

+\frac{c}{(2^{2^{\sqrt{q(n)}}})^{2}} ≤ \frac{q(n) − k(n) − 1}{n^{2}} + \frac{c}{n^{2}} ≤ c \frac{(log_{2} log_{2} n)^{2}}{n^{2}} .

Therefore for sufficiently large n’s

min \{k : N_{k} ≥ n\} + n^{2} ∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} ≤ k(n) + 1 + n^{2}c \frac{(log_{2} log_{2} n)^{2}}{n^{2}} ≤

≤ c( log_{2} log_{2} n)^{2}.

Theorem 3. Let the sequence {N_{k}} be fixed. In order that the condition

∑_{n=2}^{∞} a_{n}^{2}( log_{2} log_{2} n)^{2} < ∞  (5)

guarantee the convergence a.e. of the sequence {S_{2^{k}}(x)} for series (1) with respect to any Δ_{k}-ONS {φ_{n}(x)}, it is necessary and sufficient that the condition

∑_{k=3}^{∞} \frac{1}{(log_{2} log_{2} N_{k})^{2}} < ∞  (6)

be fulfilled.

Proof. Sufficiency. Let conditions (5) and (6) be fulfilled. Define the sequence of natural numbers {M_{i}} by the recurrent formula

M_{1} =N_{1} = 1,
M_{i} = min { min{N_{k} : M_{k} > M_{i−1}, k ∈ N},
min{2^{m} : 2^{m} > M_{i−1}, m ∈ N}}, i ≥ 2,

(7)

i.e., {M_{i}} is the increasing sequence whose terms have the form N_{k} or 2^{m}, k ≥ 1, m ≥ 1.

Assume that N_{i} = M_{k_{i}}, i ≥ 1, and k_{0} = 0. Clearly,

M_{i} < 2^{i}, i ≥ 1,  (8)

and

log_{2} M_{p} + i + 1 ≥ p for p ∈ (k_{i}, k_{i+1}], i ≥ 0.  (9)

Now, applying condition (6) and inequality (9), for sufficiently large i’s and p ∈ (k_{i}, k_{i+1}] we have

p ≤ log_{2} M_{p} + i + 1 ≤ log_{2} M_{p} + log_{2} 2^{2^{\sqrt{i}}} ≤ log_{2} M_{p} + log_{2} N_{i} =
= log_{2} M_{p} + log_{2} M_{k_{i}} ≤ 2 log_{2} M_{p}.  (10)

Set

b_{n} =( ∑_{j=M_{n}+1}^{M_{n+1}} a_{j}^{2})^{\frac{1}{2}} , ψ_{n}(x)=\begin{cases}\frac{1}{b_{n}} ∑_{j=M_{n}+1}^{M_{n+1}} a_{j}φ_{j}(x), & for b_{n} ≠ 0, \\ φ_{Mn+1}(x), & for b_{n} = 0, \end{cases} n≥1.

Clearly, {ψ_{n}(x)} is a (k_{i}, k_{i+1}]-ONS. Moreover, by condition (6) and inequality (8) we have

∑_{i=3}^{∞} \frac{1}{log_{2}^{2} k_{i}} ≤ ∑_{i=3}^{∞} \frac{1}{(log_{2} log_{2} M_{k_{i}})^{2}} = ∑_{i=3}^{∞} \frac{1}{(log_{2} log_{2} N_{i})^{2}} < ∞

and by (5) and (10)

∑_{n=1}^{∞} b_{n}^{2} log_{2}^{2} n = ∑_{n=1} ( ∑_{j=M_{n}+1}^{M_{n+1}} a_{j}^{2}) log_{2}^{2} n ≤ c ∑_{n=1}^{∞} ( ∑_{j=M_{n}+1}^{M_{n+1}} a_{j}^{2}) ×

×( log_{2} log_{2} M_{n})^{2} ≤ c ∑_{n=1}^{∞} ∑_{j=M_{n}+1}^{M_{n+1}} a_{j}^{2}( log_{2} log_{2} j)^{2} < ∞.

Thus the conditions of V. Gaposhkin’s theorem (see [1], Proposition 1) are fulfilled for (k_{i}, k_{i+1}]-ONS {ψ_{n}(x)} and the sequence {b_{n}}. Therefore the series

∑_{n=1}^{∞} b_{n}ψ_{n}(x)

converges almost everywhere, which, in particular, guarantees the convergence a.e. of the sequence {S_{2^{k}}(x)} for the corresponding series (1).

Necessity. Let

∑_{k=3}^{∞} \frac{1}{(log_{2} log_{2} N_{k})^{2}} = ∞.

Then there exist numbers c_{k} > 0 such that

∑_{k=2}^{∞} c_{k}^{2}( log_{2} log_{2} N_{k})^{2} < ∞, ∑_{k=1}^{∞} c_{k} = ∞.

Take Φ_{N_{k}}(x) ≡ 1 (k ≥ 1) and as other functions Φ_{n}(x) (n ≠ N_{1}, N_{2}, …) choose an arbitrary ONS orthogonal to 1. The system {Φ_{n}(x)} is a Δ_{k}-ONS. Let b_{N_{k}} = c_{k} (k ≥ 1) and b_{n} = 0 (n ≠ N_{1}, N_{2}, …). Then

∑_{n=2}^{∞} b_{n}^{2}( log_{2} log_{2} n)^{2} = ∑_{k=2}^{∞} c_{k}^{2}( log_{2} log_{2} N_{k})^{2} < ∞,

but

∑_{n=1}^{∞} b_{n}Φ_{n}(x) = ∑_{k=1}^{∞} b_{N_{k}} = ∑_{k=1}^{∞} c_{k} = ∞, x ∈ (0, 1),

i.e., for the series

∑_{n=1}^{∞} b_{n}Φ_{n}(x)

the sequence {S_{2^{k}}(x)} diverges everywhere.

Theorem 4. Let the sequence {N_{k}} be fixed. In order that the sequence {(log_{2} log_{2} n)^{2}} be the Weyl multiplier for the (c, 1)-summability a.e. of series with respect to any Δ_{k}-ONS, it is necessary and sufficient that condition (6) be fulfilled.

Proof. Sufficiency. Let conditions (5) and (6) be fulfilled. Then by Theorem 3 the sequence {S_{2^{k}}(x)} converges a.e. for series (1), while by Lemma 3

min \{k : N_{k} ≥ n\} + n^{2} ∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} = O((log_{2} log_{2} n)^{2}), n → ∞,

holds and therefore series (1) is (c, 1)-summable by Theorem 2.

Necessity. Let

∑_{k=3}^{∞} \frac{1}{(log_{2} log_{2} N_{k})^{2}} = ∞.

Construct the Δ_{k}-ONS {Φ_{n}(x)} and {b_{n}} as we did when proving the necessity in Theorem 3. Then the series

∑_{n=1}^{∞} b_{n}Φ_{n}(x)

will not be (c, 1)-summable anywhere.

Remark 4. If

N_{k} = [2^{2^{k^{α}}} ] , α > \frac{1}{2},

then the above-mentioned Kacmarz theorem will hold for all Δ_{k}-ONS {φ_{n}(x)}.

Theorem 5. Let the sequence {N_{k}} be fixed. In order that the condition

∑_{n=1}^{∞} a_{n}^{2}ω(n) < ∞  (11)

guarantee the convergence almost everywhere of the subsequence of partial sums {S_{2^{k}}(x)} of series (1) with respect to any Δ_{k}-ONS {φ_{n}(x)}, it is necessary and sufficient that the following two conditions be fulfilled:

(a) ∑_{k=1}^{∞} \frac{1}{ω(N_{k})} < ∞; (12)

(b) log_{2}^{2} k = O(ω(M_{k})) for k → ∞, (13) where the sequence {M_{k}} is defined by the recurrent formula (7).

Proof. Sufficiency. Let conditions (11), (12), (13) be fulfilled. Construct the system {ψ_{n}(x)} and the sequence {b_{n}} as we did when proving the sufficiency in Theorem 3. Set

v(k) = ω(M_{k}), k ≥ 1.

Then we obtain

∑_{k=1}^{∞} b_{k}^{2}v(k) = ∑_{k=1}^{∞} ( ∑_{j=M_{k}+1}^{M_{k+1}} a_{j}^{2}) v(k) = ∑_{k=1}^{∞} ( ∑_{j=M_{k}+1}^{M_{k+1}} a_{j}^{2}) ω(M_{k}) ≤

≤ ∑_{k=1}^{∞} ∑_{j=M_{k}+1}^{M_{k+1}} a_{j}^{2}ω(j) < ∞,

∑_{i=1}^{∞} \frac{1}{v(k_{i})} = ∑_{i=1}^{∞} \frac{1}{ω(M_{k_{i}})} = ∑_{i=1}^{∞} \frac{1}{ω(N_{i})} < ∞.

By condition (b) of Theorem 5 we have

log_{2}^{2} k = O(ω(M_{k})) = O(v(k)) for k → ∞.

Hence we conclude that {ψ_{n}(x)} is an (k_{i}, k_{i+1}]-ONS and

∑_{i=1}^{∞} \frac{1}{v(k_{i})} < ∞, ∑_{k=1}^{∞} b_{k}^{2}v(k) < ∞, log_{2}^{2} k = O(v(k)) for k → ∞.

Now by Theorem 1 the series

∑_{n=1}^{∞} b_{n}ψ_{n}(x)

converges a.e. and therefore, in particular, it follows that the subsequence of partial sums {S_{2^{k}}(x)} of the corresponding series (1) converges a.e.

Necessity.

(1) Let

∑_{k=1}^{∞} \frac{1}{ω(N_{k})} = ∞.

Construct {Φ_{n}(x)} and {b_{n}} as we did in proving the necessity of condition (a) of Theorem 1. Then the sequence {S_{2^{k}}(x)} diverges a.e. for series (1).

(2) Let

∑_{k=1}^{∞} \frac{1}{ω(N_{k})} < ∞

but

log_{2}^{2} k = c_{k}ω(M_{k}), k ≥ 1,

where

\overline{lim}_{k→∞}c_{k} = ∞.

Let v(k) = ω(M_{k}). Then

log_{2}^{2} k = c_{k}v(k) and \overline{lim}_{k→∞}c_{k} = ∞.

Therefore there exist a {Φ_{n}(x)}-ONS and a sequence {b_{k}} (see Remark 1) such that

∑_{k=1}^{∞} b_{k}^{2}v(k) < ∞

but the series

∑_{k=1}^{∞} b_{k}Φ_{k}(x)

diverges a.e.

Construct the system {ψ_{n}(x)} and the sequence {a_{n}}. Namely, let

a_{M_{i}} = b_{i}, ψ_{M_{i}} (x) = Φ_{i}(x), i = 1, 2, ….

For the rest of n ∈ (N_{i}, N_{i+1}] assume that a_{n} = 0 and as ψ_{n}(x) take anyone of the functions Φ_{k}(x), k ∉ (k_{i}, k_{i+1}], so that ψ_{i}(x) ≠ ψ_{j}(x) for i ≠ j and i, j ∈ Δ_{k}. In that case we obtain an Δ_{k}-ONS {ψ_{n}(x)} for which

∑_{n=1}^{∞} a_{n}^{2}ω(n) = ∑_{i=1}^{∞} a2_{M_{i}} ω(M_{i}) = ∑_{i=1}^{∞} b_{i}^{2}v(i) < ∞

but the series

∑_{n=1}^{∞} a_{n}ψ_{n}(x)

diverges a.e. Then, following the construction of the terms of this series, the subsequence of partial sums {S_{M_{k}}(x)}, where {M_{k}} is defined by (7), diverges a.e. But since

∑_{k=1}^{∞} \frac{1}{ω(N_{k})} < ∞,

the subsequence of partial sums {S_{N_{k}}(x)} of the series

∑_{n=1}^{∞} a_{n}ψ_{n}(x)

converges alsmost everywhere. Let the {S_{2^{n}}(x)} converge on a set E ⊂ (0, 1), m(E) > 0.

It is clear that from the sequences {N_{m}} and {2^{n}} we must obtain subsequences {N_{m_{k}}} and {2^{n_{k}}} such that

S_{2^{n_{k}}} (x) − S_{N_{m_{k}}} (x) = a_{2^{n_{k}}} ψ_{2^{n_{k}}} (x), k ≥ 1.

Then

∑_{k=1}^{∞} ∫_{0}^{1} (S_{2^{n_{k}}} (x) − S_{N_{m_{k}}} (x))^{2}dx ≤ ∑_{k=1}^{∞} a_{2^{n_{k}}}^{2} < ∞

and therefore

lim_{k→∞} (S_{2^{n_{k}}} (x) − S_{N_{m_{k}}} (x)) = 0 a.e.,

i.e.,

lim_{n→∞} S_{2^{n}}(x) = lim_{k→∞} S_{2^{n_{k}}} (x) = lim_{k→∞} S_{N_{m_{k}}} (x) = lim_{m→∞} S_{N_{m}}(x)

almost every x ∈ E,

which contradicts the divergence a.e. of the sequence {S_{N_{k}}(x)}.

Theorem 6. Let the sequence {N_{k}} be given and the equality

∑_{k=n}^{∞} \frac{1}{N_{k}^{2}} = O(\frac{n}{N_{n}^{2}} ) for n → ∞  (14)

be fulfilled.

In order that the positive nondecreasing sequence {ω(n)} be the Weyl multiplier for the (c, 1)-summability a.e. of series with respect to any Δ_{k}- ONS, it is necessary and sufficient that conditions (12), and (13) be fulfilled.

Proof. Let condition (14) be fulfilled.

Sufficiency. Let conditions (11), (12) and (13) be fulfilled. Then for sufficiently large k’s we have

k < ω(N_{k})

and therefore for sufficiently large n’s

min \{k : N_{k} ≥ n\} + n^{2} ∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} = k(n) + 1 + n^{2} ∑_{k=k(n)+1}^{∞} \frac{1}{N_{k}^{2}} ≤

≤ 2k(n) + n^{2} \frac{c · k(n)}{N_{k(n)+1}^{2}} ≤ ck(n) ≤ cω(N_{k(n)}) < cω(n)

which yields

min \{k : N_{k} ≥ n\} + n^{2} ∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} = O(ω(n)) for n → ∞. (15)

Then by Theorem 5 the sequence {S_{2^{k}}(x)} converges a.e. for series (1), while by Theorem 2 series (1) is (c, 1)-summable slmost everywhere.

Necessity.

(a) Let

∑_{k=1}^{∞} \frac{1}{ω(N_{k})} = ∞.

Construct {Φ_{n}(x)} and {b_{n}} as we did when proving the necessity of condition (a) of Theorem 1. Then we have

∑_{n=1}^{∞} b_{n}^{2}ω(n) < ∞

and

∑_{n=1}^{∞} b_{n}Φ_{n}(x) = ∑_{k=1}^{∞} b_{N_{k}} = ∞, x ∈ (0, 1),

which imply that the series

∑_{n=1}^{∞} b_{n}Φ_{n}(x)

is nowhere (c, 1)-summable.

(b) Let

∑_{k=1}^{∞} \frac{1}{ω(N_{k})} < ∞

but condition (13) be not fulfilled. Then by Theorem 5 there exist a Δ_{k}- ONS {ψ_{n}(x)} and a sequence {a_{n}} such that

∑_{n=1}^{∞} a_{n}^{2}ω(n) < ∞

but the corresponding subsequence of partial sums {S_{2^{k}}(x)} diverges a.e. Moreover, if equality (15) is fulfilled, then by Theorem 2 the series

∑_{n=1}^{∞} a_{n}ψ_{n}(x)

is not (c, 1)-summable almost everywhere.

Remark 5. From the proof of Theorem 6 it is clear that condition (14) in this theorem can be replaced by condition (15). Then, assuming that ω(n) = (log_{2} log_{2} n)^{2} and condition (12) is fulfilled, by inequality (10) we have

log_{2}^{2} k = O((log_{2} log_{2} M_{k})^{2}) for k → ∞,

and by Lemma 3

min \{k : N_{k} ≥ n\} + n^{2} ∑_{k:N_{k}≥n} \frac{1}{N_{k}^{2}} = O((log_{2} log_{2} n)^{2}) for n → ∞,

and we obtain Theorem 4 as a corollary.

Remark 6. Theorem 6 implies that in the typical cases given below the Weyl multipliers for the (c, 1)-summability a.e. of series with respect to any Δ_{k}-ONS are:

(a) if

N_{k} = [2^{2^{k^{α}}} ] , 0 < α ≤ \frac{1}{2},

then

ω(n) = ( log_{2} log_{2} n)^{\frac{1}{α} +ε}, ε > 0;

(b) if

N_{k} = [2^{k^{α}}] , α > 0,

then

ω(n) = ( log_{2} n)^{\frac{1}{α} +ε}, ε > 0;

(c) if

N_{k} = [k^{α}], α ≥ 1,

then

ω(n) = n^{\frac{1}{α}} ( log_{2} n)^{1+ε}, ε > 0.

Note that if ε = 0, then in cases (a), (b) and (c) {ω(n)} will be the Weyl multiplier not for each Δ_{k}-ONS.

Remark 7. Condition (14) is fulfulled, in particular, if

N_{k} = kΦ(k),

where Φ(k) does not decrease.

References

1. V. F. Gaposhkin, On the series by block-orthogonal and block-inde-pendent systems. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 5(1990), 12–18.

2. G. G. Nadibaidze, On some problems connected with series with respect to Δ_{k}-ONS. Bull. Acad. Sci. Georgia 143(1991), No. 1, 16–19.

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.

4. B. S. Kashin and A. A. Saakyan, Orthogonal series. (Russian) Nauka, Moscow, 1984;Engl. transl.: Translations of Mathematical Monographs, 75, Amer. Math. Soc., Providnce, RI, 1989.

5. S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen. Warszawa-Lwow, 1935.

(Received 16.02.1994)

Author’s address: Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia