Georgian Mathematical Journal 1(1994), No. 3, 235-242

ON CONVERGENCE SUBSYSTEMS OF ORTHONORMAL SYSTEMS

G. BARELADZE

Abstract. It is proved that for any sequence {R_{k}}_{k=1}^{∞} of real numbers satisfying R_{k} ≥ k (k ≥ 1) and R_{k} = o(k log_{2} k), k → ∞, there exists a orthonormal system {φ_{n}(x)}_{n=1}^{∞}, x ∈ (0; 1), such that none of its subsystems {φ_{n_{k}} (x)}_{k=1}^{∞} with n_{k} ≤ R_{k} (k ≥ 1) is a convergence subsystem.

Let {φ_{n}(x)} be an orthonormal system (ONS) on (0; 1). It is called a convergence system if the series ∑ c_{n}φ_{n}(x) is convergent almost everywhere whenever the sequence {c_{n}} of real numbers satisfies ∑ c_{n}^{2} < ∞.

It is well-known [1] that not every ONS {φ_{n}(x)} is a convergence system. However [2], [3], each of them contains some convergence subsystem {φ_{n_{k}}(x)}. A question was formulated later [4] whether there exists a common estimate of growth rate of numbers n_{k} in the class of all ONS. B.S.Kashin [5] answered this question in the affirmative: one can determine a sequence of positive numbers {R_{k}} such that from any ONS it is possible to choose a convergence subsystem {φ_{n_{k}}} with n_{k} ≤ R_{k}, 1 ≤ k < ∞. In the same paper [5] the problem of finding {R_{k}} with a minimal admissible growth order is formulated and the hypothesis R_{k} = k^{1+ε} (ε > 0) is conjectured. G.A.Karagulyan [6] proved that one can take R_{k} = λ^{k}, λ > 1. However, this upper estimate is rougher than the one expected in [5].

In this paper we shall give the proof of the theorem providing the lower estimate for {R_{k}}.

Theorem. For any sequence {R_{k}}_{k=1}^{∞} of real numbers satisfying

R_{k} ≥ k (k ≥ 1) and R_{k} = o(k log_{2} k), k → ∞,  (1)

there exists an ONS {φ_{n}(x)}_{n=1}^{∞}, x ∈ (0; 1), such that none of its subsystems {φ_{n_{k}}(x)}_{k=1}^{∞} with n_{k} ≤ R_{k} (k ≥ 1) is a convergence subsystem.

Several lemmas are needed to prove this theorem.

1991 Mathematics Subject Classification. 42C20.

Lemma 1 (H.Rademacher [7]). For any ONS {ψ_{n}(x)}_{n=1}^{N}, x∈(0; 1), and any collection of real numbers {c_{n}}_{n=1}^{N}

∫_{0}^{1} ( max_{1≤j≤N} | ∑_{n=1}^{j} c_{n}ψ_{n}(x)|)^{2}dx ≤ c log_{2}^{2}(N + 1) ∑_{n=1}^{N} c_{n}^{2}, 1 ≤ N < ∞.^{1}

Lemma 2. For any N ≥ 1 there exists an ONS

ψ(N) := {ψ_{n}^{N}(x)}_{n=1}^{N}, x ∈ (0; 1),

satisfying, for any collection of natural numbers 1 ≤ n_{1} < n_{2} < ⋯ < n_{m} ≤ N (1 ≤ m ≤ N), the inequality

∫_{0}^{1} max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(x)|dx ≥ c\frac{log_{2} N}{\sqrt{N}} m.  (2)

Proof. We shall assure that the requirements of this lemma are satisfied by the ONS usually used in the proof of the Menshov-Rademacher theorem. The functions ψ_{n}^{N}(x), 1 ≤ n ≤ N, belonging to this ONS (see [8], p.295) have, in particular, the following properties:

(i)

ψ_{n}^{N}(x) := \begin{cases}\frac{c\sqrt{N}}{s−n} , & x ∈ (\frac{s−1}{2N}; \frac{2s−1}{4N}), \\ \frac{c\sqrt{N}}{n−s} , & x ∈ (\frac{2s−1}{4N}; \frac{s}{2N} ), \\ 0, & x ∈ (\frac{n−1}{2N}; \frac{n}{2N} ); \end{cases} 1 ≤ s ≤ N, s ≠ n;

(ii) ψ_{n}^{N}(x) is constant on each of the intervals

(\frac{s − 1}{4N}; \frac{s}{4N} ), 2N + 1 ≤ s ≤ 4N;

(iii)

∫_{0}^{1} ψ_{n}^{N}(x) dx = 0;

(iv) ψ_{n}^{N}(x) is extended from (0; 1) onto (−∞; ∞) with period 1.

Denote δ_{s} := (\frac{s−1}{2N}; \frac{2s−1}{4N}), 1 ≤ s ≤ N. When x ∈ δ_{s}, because of (i) we have ψ_{p}^{N}(x) ≥ 0 for 1 ≤ p ≤ s and ψ_{p}^{N}(x) ≤ 0 for s ≤ p ≤ N (1 ≤ s ≤ N).

1Here and in what follows c denotes positive absolute constants which, in general, may differ from one equality (inequality) to another.

Therefore for fixed numbers 1 ≤ n_{1} < n_{2} < ⋯ < n_{m} ≤ N (1 ≤ m ≤ N) and for each x ∈ δ_{s} (1 ≤ s ≤ N) we obtain

∑_{k=1}^{m} |ψ_{n_{k}}^{N}(x)| = ∑_{k:n_{k}≤s} + ∑_{k:n_{k}>s} = | ∑_{k:n_{k}≤s} ψ_{n_{k}}^{N}(x)| +

+| ∑_{k:n_{k}>s} ψ_{n_{k}}^{N}(x)| ≤ 3 max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(x)|.

Hence

∫_{δ_{s}} max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(x)|dx ≥ \frac{1}{3} ∫_{δ_{s}} ∑_{k=1}^{m} |ψ_{n_{k}}^{N}(x)|dx =

= \frac{c}{\sqrt{N}} ∑_{\begin{matrix}k:1≤k≤m, \\ n_{k}≠s \end{matrix}} \frac{1}{|n_{k} − s|}, 1 ≤ s ≤ N;

∫_{0}^{1} max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(x)|dx ≥ ∑_{s=1}^{N} ∫_{δ_{s}} max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(x)|dx ≥

≥ \frac{c}{\sqrt{N}} ∑_{s=1}^{N} ∑_{\begin{matrix}k:1≤k≤m, \\ n_{k}≠s \end{matrix}} \frac{1}{|n_{k} − s|} =

= \frac{c}{\sqrt{N}} ∑_{k=1}^{m} ∑_{\begin{matrix}s:1≤s≤N, \\ s≠n_{k} \end{matrix}} \frac{1}{|n_{k} − s|} ≥ c\frac{log_{2} N}{\sqrt{N}} m. ■

Remark 1. For any positive integer Q, {ψ_{n}^{N}(Qx)}_{n=1}^{N} is an ONS on (0; 1) also satisfying the inequality (2).

Remark 2. Let N_{0}, N_{1}, Q_{0}, Q_{1}, p be positive integers. If Q_{1} = 4pN_{0}Q_{0}, then functions belonging to different collections {ψ_{n}^{N_{0}}(Q_{0}x)}_{n=1}^{N_{0}} and {ψ_{n}^{N_{1}}(Q_{1}x)}_{n=1}^{N_{1}} are mutually orthogonal and pairwise stochastically independent on (0; 1).

Both conclusions follow readily from (i)–(iv).

Lemma 3. Suppose a function f ∈ L_{(0;1)}^{1} is not equivalent to zero and

A := \{x ∈ (0; 1) : |f(x)| > \frac{1}{2}∥f∥_{L_{(0;1)}^{1}}\}.

Then

mes A ≥ ∥f∥_{L_{(0;1)}^{1}}^{2}/4∥f∥_{L_{(0;1)}^{2}}^{2}.  (3)

Proof. Indeed, we obtain (using Hölder’s inequality)

∥f∥_{L_{(0;1)}^{1}} = ∫_{A} |f(x)|dx + ∫_{(0;1)\A} |f(x)|dx ≤
≤(mes A)^{1/2} · ∥f∥_{L_{(0;1)}^{2}} + \frac{1}{2}∥f∥_{L_{(0;1)}^{1}} ,

which immediately implies (3).

Lemma 4. Let N, Q, m (1 ≤ m ≤ N) be positive integers. Then for any collection of natural numbers 1 ≤ n_{1} < n_{2} < ⋯ < n_{m} ≤ N

mes \{x ∈ (0; 1) : max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(Qx)| > \frac{J}{2} \} ≥ c\frac{m}{N} ,

where ψ_{n}^{N}(x) ∈ ψ(N), 1 ≤ n ≤ N (see Lemma 2), and

J := ∫_{0}^{1} max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(Qx)| dx.

Proof. By Lemmas 1 and 2 (see also Remark 1) we have

∫_{0}^{1} ( max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(Qx)|)^{2}dx ≤ cm log_{2}^{2}(m + 1),

J^{2} ≥ c\frac{m^{2}}{N} log_{2}^{2} N; 1 ≤ m ≤ N < ∞.

Thus, applying Lemma 3 , we obtain

mes \{x ∈ (0; 1) : max_{1≤j≤m} | ∑_{k=1}^{j} ψ_{n_{k}}^{N}(Qx)| > \frac{J}{2} \} ≥ c\frac{m}{N} ·\frac{log_{2}^{2} N}{log_{2}^{2}(m + 1)} ≥ c\frac{m}{N} . ■

Proof of the theorem. If the conditions (1) are fulfilled, then R_{k} =k log_{2}k/ε(k), 2 ≤ k < ∞, where ε(k) tends to infinity. Moreover, without loss of generality, it can be assumed that

a) R_{1} = 1;

b) ε(k) is a nondecreasing sequence of positive integers;

c) the sets Δ_{m} := {k : ε(k) = m} have the form (ν_{m−1}; ν_{m}] ∩ ℕ (m = 1, 2, …), where ν_{0} = 0, log_{2} log_{2} log_{2} ν_{m} = p_{m} (m ≥ 1) and {p_{m}}_{m=1}^{∞} is some increasing sequence of positive integers.

In particular, for k ∈ Δ_{m} (m ≥ 2) we have

log_{2} k > log_{2} ν_{m−1} = 2^{2^{p_{m−1}}} ≥ 2^{2^{m−1}} > m = ε(k),

ε(k log_{2} k) ≤ ε(ν_{m} log_{2} ν_{m}) < ε(ν_{m+1}) = m + 1 ≤ 2ε(k). (4)

Denote

T_{1} := 0, T_{k} := 2^{[k log_{2} k log_{2} log_{2} k]}, 2 ≤ k < ∞,  (5)

and E_{m} := {k : R_{k} ≤ T_{m}}, 2 ≤ m < ∞ (here [x] is the integer part of the number x).

Since the function φ(x) = xε(x)/ log_{2} x increases on (e; ∞), taking into account (4) and (5), we have, for m > ν_{1},

E_{m} = \{1; 2\} ∪ \{k ≥ 3 : R_{k} ≤ T_{m}\} = \{1; 2\} ∪ \{k ≥ 3 : \frac{k log_{2} k}{ε(k)} ≤ T_{m}\} =

= \{1; 2\} ∪ \{k ≥ 3 : φ(\frac{k log_{2} k}{ε(k)} ) ≤ φ(T_{m})\} ⊃

⊃ \{1; 2\} ∪ \{k ≥ 3 : 2k ≤ φ(T_{m})\} = \{k ≥ 1 : k ≤ \frac{1}{2}φ(T_{m})\}.

Therefore for a large m (m ≥ m_{1} > ν_{1})

|E_{m}| > \frac{1}{2}φ(T_{m}) − 1 ≥ \frac{1}{3} \frac{T_{m}ε(T_{m})}{log_{2} T_{m}},  (6)
|E_{m+1}| ≥ \frac{1}{3} \frac{T_{m+1}ε(T_{m+1})}{log_{2} T_{m+1}} ≥ \frac{1}{3} \frac{T_{m+1}}{T_{m} log_{2} T_{m+1}} · T_{m} ≥ 2T_{m};  (7)

Because of

∑_{k∈Δ_{m}} \frac{1}{log_{2} T_{k}} = ∑_{k∈Δ_{m}} \frac{1}{[k log_{2} k log_{2} log_{2} k]} > \frac{1}{8}(p_{m} − p_{m−1}) ≥ \frac{1}{8} (m ≥ 2)

we can select a subsequence T_{q_{k}} ≡ T̃_{k} (1 ≤ k < ∞; T̃_{1} = 0, T̃_{2} = ν_{1}) such that

\frac{1}{2m^{2}} < ∑_{k:q_{k}∈Δ_{m}} \frac{1}{log_{2} T̃_{k}} ≡ ∑_{k:q_{k}∈Δ_{m}} \frac{1}{log_{2} T_{q_{k}}} ≤ \frac{1}{m^{2}} , m ≥ 2,  (8)

and hence

∑_{k:q_{k}∈Δ_{m}} \frac{ε(T̃_{k})}{log_{2} T̃_{k}} = ∑_{k:q_{k}∈Δ_{m}} \frac{ε(T_{q_{k}})}{log_{2} T_{q_{k}}} ≥

≥ ∑_{k:q_{k}∈Δ_{m}} \frac{ε(q_{k})}{log_{2} T_{q_{k}}} = m ∑_{k:q_{k}∈Δ_{m}} \frac{1}{log_{2} T_{q_{k}}} > \frac{1}{2m}, m ≥ 2.  (9)

Let

\begin{matrix}N_{m} := T̃_{m} − T̃_{m−1}, \\ Q_{2} := 1, Q_{m+1} := 4N_{m}Q_{m}, 2 ≤ m < ∞. \end{matrix}  (10)

Consider the orthonormal collections

{ψ_{n}^{N_{m}}(Q_{m}x)}_{n=1}^{N_{m}}, 2 ≤ m < ∞,

and construct with their aid the desired ONS {φ_{n}(x)}_{n=1}^{∞} as

φ_{n}(x) := ψ_{k}^{N_{m}}(Q_{m}x),  (11)

where n ∈ (T̃_{m−1}; T̃_{m}], k = n − T̃_{m−1}, 2 ≤ m < ∞, x ∈ (0; 1) (the orthonormality follows from (10) and Remark 2).

Let {n_{k}}_{k=1}^{∞} be a sequence of positive integers with k ≤ n_{k} ≤ R_{k}, 1 ≤ k < ∞. We set

G_{m} := {k : T̃_{m−1} < n_{k} ≤ T̃_{m}},
M_{m} := |G_{m}|,
a_{n} := ((1 + M_{m}) log_{2} T̃_{m})^{−1/2} for n ∈ (T̃_{m−1}; T̃_{m}]; m ≥ 2.

On account of (8)

∑_{k=1}^{∞} a_{n_{k}}^{2} = ∑_{m=2}^{∞} ∑_{k:_{T_{m−1}}<n_{k}≤_{T_{m}}} a_{n_{k}}^{2} = ∑_{m=2}^{∞} \frac{M_{m}}{(1 + M_{m}) log_{2} T̃_{m}} <

< ∑_{k=2}^{∞} \frac{1}{log_{2} T_{q_{k}}} = ∑_{m=2}^{∞} ∑_{k:q_{k}∈Δ_{m}} \frac{1}{log_{2} T_{q_{k}}} < ∞.

It is thus sufficient to show that the series

∑_{k=1}^{∞} a_{n_{k}}φ_{n_{k}}(x)  (12)

diverges on some set of positive measure.

Note that

G_{m} = {k : n_{k} ≤ T̃_{m}}\{k : n_{k} ≤ T̃_{m−1}} ⊃
⊃ {k : R_{k} ≤ T̃_{m}}\{k : k ≤ T̃_{m−1}}, m ≥ 2.

Consequently in view of (6) and (7)

M_{m} ≥ |{k : R_{k} ≤ T̃_{m}}| − |{k : k ≤ T̃_{m−1}}| = |{k : R_{k} ≤ T_{q_{m}}}| − T_{q_{m−1}} >
> |E_{q_{m}}| − \frac{1}{2}|E_{q_{m−1}+1}| ≥ \frac{1}{2}|E_{q_{m}}| ≥ \frac{1}{6} \frac{T̃_{m}ε(T̃_{m})}{log_{2} T̃_{m}}, m ≥ m_{1}.  (13)

Hence by (11) and Lemma 2

J̃_{m} : = ∫_{0}^{1} max_{_{T_{m−1}}<j≤_{T_{m}}} | ∑_{k:_{T_{m−1}}<n_{k}≤j} a_{n_{k}}φ_{n_{k}}(x)|dx ≥

≥ c · \frac{1}{\sqrt{(1 + M_{m}) log_{2} T̃_{m}}} · \frac{log_{2} N_{m}}{\sqrt{N_{m}}} · M_{m} ≥

≥ c\sqrt{\frac{M_{m} log_{2} T̃_{m}}{T̃_{m}}} ≥ c\sqrt{ε(T̃_{m})}, m ≥ m_{1},

and therefore

lim_{m→∞} J̃_{m} = ∞.  (14)

If A_{m} denotes the set

\{x ∈ (0; 1) : max_{_{T_{m−1}}<j≤_{T_{m}}} | ∑_{k:_{T_{m−1}}<n_{k}≤j} a_{n_{k}}φ_{n_{k}}(x)| > \frac{1}{2}J̃_{m}\}, m ≥ 2,

then by (11), Lemma 4, (13) and (9)

mes A_{m} ≥ \frac{M_{m}}{N_{m}} > \frac{M_{m}}{T̃_{m}} > \frac{1}{6} \frac{ε(T̃_{m})}{log_{2} T̃_{m}} , m ≥ m_{1};

∑_{m=2}^{∞} mes A_{m} ≥ \frac{1}{6} ∑_{k=m_{1}}^{∞} \frac{ε(T̃_{k})}{log_{2} T̃_{k}} = \frac{1}{6} ∑_{k:q_{k}≥q_{m_{1}}} \frac{ε(T_{q_{k}})}{log_{2} T_{q_{k}}} >

> \frac{1}{6} ∑_{k:q_{k}>ν_{m_{1}}} \frac{ε(T_{q_{k}})}{log_{2} T_{q_{k}}} = \frac{1}{6} ∑_{m=1+m_{1}} ∑_{k:q_{k}∈Δ_{m}} \frac{ε(T_{q_{k}})}{log_{2} T_{q_{k}}} = ∞.  (15)

It is easy to verify (see (10),(11), Remark 2) that {A_{m}}_{m=2}^{∞} is a sequence of stochastically independent sets. Therefore by (15) and the Borel-Cantelli lemma

mes ( lim_{m→∞} sup A_{m}) = 1.

Hence we conclude because of (14) and the definition of sets A_{m} that the series (12) diverges almost everywhere on (0;1). ■

References

1. D.E. Menshov, Sur les séries des fonctions orthogonales I. Fund. Math. 4(1923), 82-105.

2. J. Marcinkiewicz, Sur la convergence des séries orthogonales. Studia Math. 6(1936), 39-45.

3. D.E. Menshov, Sur la convergence et la sommation des séries des fonctions orthogonales. Bull. Soc. Math. de France 64(1936), 147-170.

4. G. Bennet, Lectures on matrix transformation of l_{p} spaces. Notes in Banach spaces, pp. 39-80, Univ. Texas Press, Austin, Tex., 1980.

5. B.S. Kashin, The choice of convergence subsystem from the given orthonormal system. (Russian) Uspehi Mat. Nauk 40(1985), No.2(242), 181-182. English translation: Russ. Math. Surv. 40(1985), 215-216.

6. G.A. Karagulyan, The choice of convergence subsystem with logarithmic density from an arbitrary orthonormal system. (Russian) Mat. Sb. (N.S.) 136(178)(1988), No.1, 41-45.

7. H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen. Math. Ann. 87(1922), 112-138.

8. B.S. Kashin, A.A. Saakyan, Orthogonal series. (Russian) ”Nauka”, Moscow, 1984. English translation: Translations of Mathematical Monographs, 75, American Math. Society, Providence, RI, 1989.

(Received 25.02.1993)

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