Georgian Mathematical Journal Volume 10 (2003), Number 2, 363–379

ESTIMATES OF FOURIER COEFFICIENTS

V. TSAGAREISHVILI

Abstract. Some well-known properties of the trigonometric system as well as of the Haar and Welsh systems are generalized to general orthonormal systems.

2000 Mathematics Subject Classification: 42A16, 42C10.

Key words and phrases: Orthonormal system, Fourier coefficients, modulus of continuity, bounded variation.

1. Introduction

In the theory of functions an important place is occupied by generalization of properties of specific orthonormal series to general orthonormal systems.

Here we note only some of the authors who have significant results concerning the mentioned problems: Marcinkiewicz [1], Stechkin [2], Olevskii [3], Bochkarev [4], [5], Mitiagin [6], Kashin [7], McLaughlin [8].

It was proved that in many cases some properties of the well-known orthogonal systems are typical for general orthogonal systems (see, e.g., [2], [3], [4], [5]). However, not all properties of the well-known orthogonal systems are extending on general orthogonal systems. Therefore, in order to obtain well-known results for general orthogonal systems, we need to impose specific conditions on the given system.

2. Auxiliary Notation and Results

Let δ ∈ (0, 1]. If a function f ∈ C([0, 1]), then its modulus of continuity is defined as follows

ω(δ, f) = sup_{|h|≤δ} max_{0≤x≤1−h} |f(x) − f(x + h)|,

where 0 < h ≤ 1.

If a function f ∈ L_{2}([0, 1]), then the integral modulus of continuity has the form

ω_{2}(δ, f) = sup_{|h|≤δ} (∫_{0}^{1−h} |f(x) − f(x + h)|^{2}dx)^{\frac{1}{2}} .

We say that a function f ∈ Lip α if ω(δ, f) = O(δ^{α}) as δ → 0.

Let (φ_{n}) be an orthogonal system on [0, 1]. Then the Fourier coefficients with respect to (φ_{n}) for the function f ∈ L([0, 1]) are defined as follows

c_{n}(f) = ∫_{0}^{1} f(x)φ_{n}(x) dx, n = 1, 2, ….

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

The best approximation with respect to the system (φ_{n}) in the sense of L_{2}([0, 1]) is defined by the following equality:

E_{n}^{(2)}(f) = inf_{{a_{m}}} ∥f(x) − ∑_{m=1}^{n} a_{m}φ_{m}(x)∥_{2} .

If (φ_{n}) is a complete system on [0, 1], then

E_{n}^{(2)}(f) = (∑_{k=n}^{∞} c_{k}^{2}(f))^{\frac{1}{2}} .

In the sequel we will denote by (ψ_{n}) either one of the orthonormal systems of Haar or Walsh (see, e.g., [9, p. 53, 54]), or the trigonometric system. For these systems the following results are valid. They are important for our purpose.

Theorem A. If

c_{n}(f) = ∫_{0}^{1} f(x)ψ_{n}(x) dx,

then the following relations are valid:

a) c_{n}(f) = O (ω(\frac{1}{n} , f)) for every f ∈ C([0, 1]);

b) c_{n}(f) = O (ω_{2}(\frac{1}{n} , f)) for every f ∈ L_{2}([0, 1]).

Theorem B. For every function f ∈ L_{2}([0, 1]) the relation

E_{n}^{(2)} = O (ω_{2} (\frac{1}{n} , f))

holds.

The relation a) of Theorem A for the Haar system was proved by B. Golubov [10] and the relation b) was proved by P. Ul’yanov [11], for the Walsh system it was proved by Fine [12].

Theorem B for the Haar system was proved by Ul’yanov [11]. As to the trigonometric system, Theorems A and B are well-known (see, e.g., [13, p. 79]).

It is well-known that Theorems A and B are not valid for general orthogonal systems. In fact, the following theorem holds.

Theorem C. Let f be a function from L_{2}([0, 1]) and (c_{n}) be an arbitrary sequence from ℓ_{2} satisfying the condition

∑_{n=1}^{∞} c_{n}^{2} = ∫_{0}^{1} f^{2}(x) dx.

Then there exists an orthonormal system (φ_{n}) on [0, 1] such that

c_{n} = ∫_{0}^{1} f(x)φ_{n}(x) dx, n = 1, 2, ….

Theorem C is proved by A. Olevskii [3]. From this theorem follows that if f ∈ L_{2}([0, 1]) and the numbers c_{n} satisfy the conditions

a) ∑_{n=1}^{∞} c_{n}^{2} = ∫_{0}^{1} f^{2}(x) dx and b) lim_{n→∞} \frac{|c_{n}|}{ω_{2}(\frac{1}{n} , f)} = +∞,

then there exists an orthonormal system of functions (φ_{n}) such that

c_{n} = ∫_{0}^{1} f(x)φ_{n}(x) dx, n = 1, 2, ….

Therefore there exists a function f ∈ L_{2}([0, 1]) such that

lim_{n→∞} \frac{|c_{n}(f)|}{ω_{2}(\frac{1}{n} , f)} = +∞.

The following propositions are valid (see [15]).

Lemma 1. Let the function f ∈ L_{2}([0, 1]) take finite values at every point of the interval [0, 1] and Φ ∈ L_{2}([0, 1]). Then the following equality is valid:

∫_{0}^{1} f(x)Φ(x) dx = ∑_{i=1}^{n−1} (f (\frac{i}{n}) − f (\frac{i + 1}{n})) ∫0^{\frac{i}{n}} Φ(x) dx

+ ∑_{i=1}^{n} ∫ \frac{i}{n} ; \frac{i−1}{n} (f(x) − f (\frac{i}{n})) Φ(x) dx + f(1) ∫_{0}^{1} Φ(x) dx. (1)

Lemma 2. Let (φ_{n}) be an orthonormal system of functions on [0, 1]. Let E_{n} denote the set of the natural numbers i (i = 1, 2, …, n) for which there exists a point t ∈ (\frac{i−1}{n}, \frac{i}{n}) such that

sign ∫_{0}^{t} φ_{n}(x) dx ≠ sign ∫0^{\frac{i−1}{n}} φ_{n}(x) dx.

Then the inequality

∑_{i∈E_{n}} |∫0^{\frac{i}{n}} φ_{n}(x) dx| ≤ (∫_{0}^{1} φ_{n}^{2}(x) dx)^{\frac{1}{2}}

holds.

366 V. TSAGAREISHVILI

3. Main Results

Let (a_{n}) be a sequence of real numbers from ℓ_{2}, Introduce the following notation:

P_{N}^{(m)}(x) ≡ ∑_{k=N}^{N+m} a_{k}φ_{k}(x), m = 1, 2, …,

e_{N} ≡ (∑_{k=N}^{∞} c_{k}^{2}(f))^{\frac{1}{2}} ,

H_{N} ≡ sup_{m} ∑_{k=1}^{N−1} |∫0^{\frac{k}{N}} P_{N}^{(m)}(x) dx| ,

h_{N} ≡ (∑_{k=N}^{∞} a_{k}^{2})^{\frac{1}{2}} .

Theorem 1. Let f ∈ C([0, 1]) and (φ_{n}) be an orthonormal system of functions on [0, 1] satisfying ∫_{0}^{1} φ_{n}(x) dx = 0, n = 1, 2, . . ..

Then the relation

e_{N} = O (ω (\frac{1}{N} , f))

holds if and only if the condition

H_{N} = O(h_{N}) for every sequence (a_{k}) ∈ ℓ_{2}

is fulfilled.

Proof. Sufficiency. If c_{n}(f) = ∫_{0}^{1} f(x)φ_{n}(x)dx, then

∑_{k=N}^{N+m} c_{k}^{2}(f) = ∑_{k=N}^{N+m} c_{k}(f) ∫_{0}^{1} f(x)φ_{k}(x) dx
= ∫_{0}^{1} f(x) ∑_{k=N}^{N+m} c_{k}(f)φ_{k}(x) dx = ∫_{0}^{1} f(x)P_{N}^{(m)}(x) dx.  (2)

Using the assertion of Lemma 1, we obtain

∫_{0}^{1} f(x)P_{N}^{(m)}(x) dx = ∑_{i=1}^{N−1} (f (\frac{i}{N} ) − f (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m)}(x) dx

+ ∑_{i=1}^{N} ∫ \frac{i}{N} ; \frac{i−1}{N} (f(x) − f (\frac{i}{N} )) P_{N}^{(m)}(x) dx + f(1) ∫_{0}^{1} P_{N}^{(m)}(x) dx.  (3)

ESTIMATES OF FOURIER COEFFICIENTS 367

Estimate the right-hand side of equality (3). We have

|∑_{i=1}^{N−1} (f (\frac{i}{N} ) − f (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m)}(x) dx|

≤ ∑_{i=1}^{N−1} |f (\frac{i}{N} ) − f (\frac{i + 1}{N})| |∫0^{\frac{i}{N}} P_{N}^{(m)}(x) dx|

≤ ω (\frac{1}{N} , f) ∑_{i=1}^{N−1} |∫0^{\frac{i}{N}} P_{N}^{(m)}(x) dx| ≤ O(1) ω (\frac{1}{N} , f) h_{N}.  (4)

Applying the Hölder inequality, we obtain

|∑_{i=1}^{N−1} ∫ \frac{i}{N} ; \frac{i−1}{N} (f(x) − f (\frac{i}{N} )) P_{N}^{(m)}(x) dx|

≤ ω (\frac{1}{N} , f) ∑_{i=1}^{N−1} ∫ \frac{i}{N} ; \frac{i−1}{N} |P_{N}^{(m)}(x)| dx ≤ ω (\frac{1}{N} , f) (∫_{0}^{1} (P_{N}^{(m)}(x))^{2} dx)^{\frac{1}{2}}

= O(1)ω (\frac{1}{N} , f) ( ∑_{k=N}^{N+m} c_{k}^{2}(f))^{\frac{1}{2}} ≤ O(1)ω (\frac{1}{N} , f) e_{N}.  (5)

Finally, since ∫_{0}^{1} φ_{n}(x) dx = 0, n = 1, 2, …, from (3), (4) and (5) we have

|∫_{0}^{1} f(x)P_{N}^{(m)}(x) dx| ≤ O (ω (\frac{1}{N} , f) e_{N}) + O (ω (\frac{1}{N} , f) e_{N})

= O (ω (\frac{1}{N} , f) e_{N}) .

Hence, applying (2), we get

∑_{k=N}^{N+m} c_{k}^{2}(f) = O (ω (\frac{1}{N} , f) e_{N}) ,

i.e.,

∑_{k=N}^{∞} c_{k}^{2}(f) = O (ω (\frac{1}{N} , f) e_{N}) ,  (6)

From (6) we have

(∑_{k=N}^{∞} c_{k}^{2}(f))^{\frac{1}{2}} = O (ω (\frac{1}{N} , f)) .

Sufficiency of the theorem is proved.

Necessity. Let H_{N} ≠ O(h_{N}), i.e.,

\overline{lim}_{N} \frac{H_{N}}{h_{N}} = +∞.

368 V. TSAGAREISHVILI

Therefore there exist a sequence (a_{n}) ∈ ℓ_{2} and natural numbers m_{N} ↑ ∞ such that

\overline{lim}_{N} \frac{H_{N}^{(m_{N})}}{h_{N}} = +∞,  (7)

where

H_{N}^{(m_{N})} = ∑_{k=1}^{N−1} |∫0^{\frac{k}{N}} P_{N}^{(m_{N})}(x) dx|

and

P_{N}^{(m_{N})}(x) = ∑_{k=N}^{N+m_{N}} a_{k}φ_{k}(x).

Consider the sequence of functions

f_{N}(x) = ∫_{0}^{x} (sign ∫_{0}^{u} P_{N}^{(m_{N})}(t) dt) du, N = 1, 2, ….  (8)

Taking into account the condition ∫_{0}^{1} φ_{n}(x) dx = 0, n = 1, 2, …, from (1) we obtain

∫_{0}^{1} f_{N}(x)P_{N}^{(m_{N})}(x) dx = ∑_{i=1}^{N−1} (f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx
+ ∑_{i=1}^{N} ∫ \frac{i}{N} ; \frac{i−1}{N} (f_{N}(x) − f_{N} (\frac{1}{N} )) P_{N}^{(m_{N})}(x) dx.  (9)

Applying (8) we have

|∑_{i=1}^{N} ∫ \frac{i}{N} ; \frac{i−1}{N} (f_{N}(x) − f_{N} (\frac{i}{N} )) P_{N}^{(m_{N})}(x) dx|

≤ \frac{1}{N} ∑_{i=1}^{N} ∫ \frac{i}{N} ; \frac{i−1}{N} |P_{N}^{(m_{N})}(x)| dx

≤ \frac{1}{N} (∫_{0}^{1} (P_{N}^{(m_{N})}(x))^{2} dx)^{\frac{1}{2}} = \frac{1}{N} ( ∑_{k=N}^{N+m_{N}} a_{k}^{2})^{\frac{1}{2}} .  (10)

Taking into account the assertions of Lemma 2 and (8), we get

∑_{i=1}^{N−1} (f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx

= ∑_{i∈F_{N} \E_{N}} (f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx

+ ∑_{i∈E_{N}} (f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx,  (11)

where F_{N} = {1, 2, …, N}.

If i ∈ F_{N} \ E_{N}, then

f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N}) = − ∫ \frac{i+1}{N} ; \frac{i}{N} sign ∫_{0}^{u} P_{N}^{(m_{N})}(t) dt du

= −\frac{1}{N} sign ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(t)dt.

Therefore

| ∑_{i∈F_{N} \E_{N}} (f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx|

= \frac{1}{N} ∑_{i∈F_{N} \E_{N}} |∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx| .  (12)

At last, as far as

|f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})| < \frac{1}{N}  (13)

and, by virtue of Lemma 2, we have

∑_{i∈E_{N}} |∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx| ≤ (∫_{0}^{1} (P_{N}^{(m_{N})}(x))^{2} dx)^{\frac{1}{2}}

= ( ∑_{k=N}^{N+m_{N}} a_{k}^{2})^{\frac{1}{2}} ≤ (∑_{k=N}^{∞} a_{k}^{2})^{\frac{1}{2}} ≡ h_{N}.  (14)

Then, taking into account (13) and (14), we obtain

| ∑_{i∈E_{N}} (f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx| ≤ \frac{1}{N} h_{N},  (15)
N = 1, 2, ….

Using (12) and (15), we get

|∑_{i=1}^{N−1} (f_{N} (\frac{i}{N} ) − f_{N} (\frac{i + 1}{N})) ∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx|

≥ \frac{1}{N} ∑_{i∈F_{N} \E_{N}} |∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx| − \frac{1}{N} ∑_{i∈E_{N}} |∫0^{\frac{i}{N}} P_{N}^{(m_{N})}(x) dx|

≥ \frac{1}{N} H_{N}^{(m_{N})} − \frac{h_{N}}{N} .

Hence due to (7) we have

\frac{N |∫_{0}^{1} f_{N}(x)P_{N}^{(m_{N})}(x) dx|}{h_{N}} ≥ \frac{H_{N}^{(m_{N})}}{h_{N}} − \frac{h_{N}}{h_{N}} ,

and consequently

\overline{lim}_{n→∞} \frac{N |∫_{0}^{1} f_{N}(x)P_{N}^{(m_{N})}(x) dx|}{h_{N}} = +∞.  (16)

Further, since

∥f_{N}∥_{Lip 1} = ∥f_{N}∥_{C} + sup_{x,y} \frac{|f_{N}(x) − f_{N}(y)|}{|x − y|} ,

then from (8) it follows that

∥f_{N}∥_{Lip 1} ≤ 2.  (17)

Finally, by virtue of the Banach–Steinhaus theorem (see (16) and (17)) there exists a function f_{0} ∈ Lip 1 such that

\overline{lim}_{n→∞} \frac{N |∫_{0}^{1} f_{N}(x)P_{N}^{(m_{N})}(x) dx|}{h_{N}} = +∞.  (18)

As far as

|∫_{0}^{1} f_{0}(x)P_{N}^{(m_{N})}(x) dx| = |∫_{0}^{1} f_{0}(x) ∑_{k=N}^{N+m_{N}} a_{k}φ_{k}(x) dx|

= | ∑_{k=N}^{N+m_{N}} a_{k} ∫_{0}^{1} f_{0}(x)φ_{k}(x) dx| = | ∑_{k=N}^{N+m_{N}} a_{k}c_{k}(f_{0})|

≤ ( ∑_{k=N}^{N+m_{N}} a_{k}^{2})^{\frac{1}{2}} ( ∑_{k=N}^{N+m_{N}} c_{k}^{2}(f_{0}))^{\frac{1}{2}} < h_{N} · e_{N},

then taking into account that ω (\frac{1}{N} , f_{0}) = O (\frac{1}{N} ), from (18) we have

lim_{n→∞} \frac{h_{N} · e_{N}}{ω (\frac{1}{N} , f_{0}) · h_{N}} = +∞,

i.e.,

lim_{n→∞} \frac{e_{N}}{ω (\frac{1}{N} , f_{0})} = +∞. □

Theorem 2. Let the functions f and Φ be from L_{2}([0, 1]). Then

|∫_{0}^{1} f(x)Φ(x) dx| ≤ ω_{2} (\frac{1}{n} , f) (V_{n} + 2∥Φ∥_{2})
+ n ∫_{1− \frac{1}{n}}^{1} |f(x)| dx |∫_{0}^{1} Φ(x) dx| ,  (19)

where V_{n} = ∑_{k=1}^{n−1} | ∫_{0}^{k/n} Φ(x) dx|.

Proof. Applying the Abel transformation, we get

n ∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} f(x) dx ∫ \frac{k}{n} ; \frac{k−1}{n} Φ(x) dx

= n ∑_{k=1}^{n−1} (∫ \frac{k}{n} ; \frac{k−1}{n} f(x) dx − ∫ \frac{k+1}{n} ; \frac{k}{n} f(x) dx) ∫0^{\frac{k}{n}} Φ(x) dx

+ n ∫_{1− \frac{1}{n}}^{1} f(x) dx ∫_{0}^{1} Φ(x) dx

= n ∑_{k=1}^{n−1} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx ∫0^{\frac{k}{n}} Φ(x) dx

+ n ∫_{1− \frac{1}{n}}^{1} f(x) dx ∫_{0}^{1} Φ(x) dx.  (20)

Since

∫ \frac{k}{n} ; \frac{k−1}{n} f(x)Φ(x) dx − n ∫ \frac{k}{n} ; \frac{k−1}{n} f(t) dt ∫ \frac{k}{n} ; \frac{k−1}{n} Φ(x) dx

= n ∫ \frac{k}{n} ; \frac{k−1}{n} dt ∫ \frac{k}{n} ; \frac{k−1}{n} f(x)Φ(x) dx − n ∫ \frac{k}{n} ; \frac{k−1}{n} f(t) dt ∫ \frac{k}{n} ; \frac{k−1}{n} Φ(x) dx

= n ∫ \frac{k}{n} ; \frac{k−1}{n} (∫ \frac{k}{n} ; \frac{k−1}{n} f(x) dt − ∫ \frac{k}{n} ; \frac{k−1}{n} f(t) dt) Φ(x) dx

= n ∫ \frac{k}{n} ; \frac{k−1}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t)) Φ(x) dx dt  (21)

and

∫_{0}^{1} f(x)Φ(x) dx = ∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} f(x)Φ(x) dx,  (22)

372 V. TSAGAREISHVILI

then, taking into account (20), (21) and (22), one has

∫_{0}^{1} f(x)Φ(x) dx = ∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} f(x)Φ(x) dx − n ∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} f(t) dt ∫ \frac{k}{n} ; \frac{k−1}{n} Φ(x) dx

+n ∑_{k=1}^{n−1} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx ∫0^{\frac{k}{n}} Φ(x) dx + n ∫_{1− \frac{1}{n}}^{1} f(x) dx ∫_{0}^{1} Φ(x) dx

= ∑_{k=1}^{n} (∫ \frac{k}{n} ; \frac{k−1}{n} f(x)Φ(x) dx − n ∫ \frac{k}{n} ; \frac{k−1}{n} f(t) dt ∫ \frac{k}{n} ; \frac{k−1}{n} Φ(x) dx)

+n ∑_{k=1}^{n−1} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx ∫0^{\frac{k}{n}} f(x)Φ(x) dx

+n ∫_{1− \frac{1}{n}}^{1} f(x) dx ∫_{0}^{1} Φ(x) dx = n ∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t)) Φ(x) dx dt

+n ∑_{k=1}^{n−1} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx ∫0^{\frac{k}{n}} Φ(x) dx+n ∫_{1− \frac{1}{n}}^{1} f(x) dx ∫_{0}^{1} Φ(x) dx.

Consequently,

∫_{0}^{1} f(x)Φ(x) dx = n ∑_{k=1}^{n−1} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx ∫0^{\frac{k}{n}} Φ(x) dx

+n ∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t)) Φ(x) dx dt + n ∫_{1− \frac{1}{n}}^{1} f(x) dx ∫_{0}^{1} Φ(x) dx.  (23)

Assuming ξ_{n}^{(k)}(t) = \frac{1}{n} t + \frac{k−1}{n}, we get

∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx = \frac{1}{n} ∫_{0}^{1} (f (\frac{1}{n} t + \frac{k − 1}{n}) − f (\frac{1}{n} t + \frac{k}{n})) dt

= \frac{1}{n} ∫_{0}^{1} (f (ξ_{n}^{(k)}(t)) − f (ξ_{n}^{(k)}(t) + \frac{1}{n})) dt.

Hence we have

|∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx| ≤ \frac{1}{n} ∫_{0}^{1} |f (ξ_{n}^{(k)}(t)) − f (ξ_{n}^{(k)}(t) + \frac{1}{n})| dt

≤ \frac{1}{n} ∫_{0}^{1} |f(t) − f (t + \frac{1}{n})| dt ≤ \frac{1}{n} sup_{|h|≤ \frac{1}{n}} (∫_{0}^{1} |f(t) − f(t + h)|^{2} dt)^{\frac{1}{2}}

= \frac{1}{n} ω_{2} (\frac{1}{n} , f) .  (24)

ESTIMATES OF FOURIER COEFFICIENTS 373

By virtue of the Hölder inequality we obtain

|∫ \frac{k}{n} ; \frac{k−1}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t)) Φ(x) dx dt|

≤ ∫ \frac{k}{n} ; \frac{k−1}{n} (∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t))^{2} dx)^{\frac{1}{2}} (∫ \frac{k}{n} ; \frac{k−1}{n} Φ^{2}(x) dx)^{\frac{1}{2}} dt

≤ \frac{1}{\sqrt{n}} (∫ \frac{k}{n} ; \frac{k−1}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t))^{2} dx dt)^{\frac{1}{2}} (∫ \frac{k}{n} ; \frac{k−1}{n} Φ^{2}(x) dx)^{\frac{1}{2}} .  (25)

Applying the equality (see [11])

∫_{a}^{b} ∫_{a}^{b} |f(x) − f(t)|^{p}dx dt = 2 ∫_{0}^{b−a} (∫_{a}^{b−ξ} |f(y + ξ) − f(y)|^{p}dy) dξ, we get

∫ \frac{k}{n} ; \frac{k−1}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t))^{2} dx dt = 2 ∫0^{\frac{1}{n}} (∫ \frac{k}{n} −ξ_{\frac{k−1}{n}} |f(y + ξ) − f(y)|^{2}dy) dξ. (26)

Therefore, by virtue of (25) and (26) we have

|∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f(t)) Φ(x)dx dt|

≤ \frac{2}{\sqrt{n}} ∑_{k=1}^{n} (∫0^{\frac{1}{n}} ∫ \frac{k}{n} −ξ_{\frac{k−1}{n}} (f(y + ξ) − f(y))^{2} dy dξ)^{\frac{1}{2}} · (∫ \frac{k}{n} ; \frac{k−1}{n} Φ^{2}(x) dx)^{\frac{1}{2}}

≤ \frac{2}{\sqrt{n}} (∫0^{\frac{1}{n}} ∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(y + ξ) − f(y))^{2} dy dξ)^{\frac{1}{2}} · (∑_{k=1}^{n} ∫ \frac{k}{n} ; \frac{k−1}{n} Φ^{2}(x) dx)^{\frac{1}{2}}

≤ \frac{2}{\sqrt{n}} (∫0^{\frac{1}{n}} ∫_{0}^{1} (f(y + ξ) − f(y))^{2} dy dξ)^{\frac{1}{2}} · (∫_{0}^{1} Φ^{2}(x) dx)^{\frac{1}{2}}

≤ \frac{2}{\sqrt{n}} · \frac{1}{\sqrt{n}} ω_{2} (\frac{1}{n} , f) · ∥Φ∥_{2} = \frac{2∥Φ∥_{2}}{n} ω_{2} (\frac{1}{n} , f) .  (27)

At last, taking into account (24), we have

|∑_{k=1}^{n−1} ∫ \frac{k}{n} ; \frac{k−1}{n} (f(x) − f (x + \frac{1}{n})) dx ∫0^{\frac{k}{n}} Φ(x) dx|

≤ \frac{1}{n} ω_{2} (\frac{1}{n} , f) · ∑_{k=1}^{n−1} |∫0^{\frac{k}{n}} Φ(x) dx| = \frac{ω_{2} (\frac{1}{n} , f)}{n} V_{n}.

374 V. TSAGAREISHVILI

Hence from (23) and (27) we get

|∫_{0}^{1} f(x)Φ(x) dx| ≤ n · \frac{ω_{2} (\frac{1}{n} , f)}{n} V_{n} + n \frac{2∥Φ∥_{2}}{n} ω_{2} (\frac{1}{n} , f)

+n ∫_{1− \frac{1}{n}}^{1} |f(x)|dx |∫_{0}^{1} Φ(x) dx| .

Theorem 2 is proved completely. □

Theorem 3. Let (φ_{n}) be an orthonormal system of functions on [0, 1] satisfying the condition ∫_{0}^{1} φ_{n}(x) dx = 0, n = 1, 2, . . .. Then

c_{n}(f) = O (ω_{2} (\frac{1}{n} , f)) for every f ∈ L_{2}([0, 1])

if and only if

V_{n} = O(1),

where V_{n} = ∑_{k=1}^{n−1} | ∫0^{\frac{k}{n}} φ_{n}(x) dx|.

Proof. Sufficiency. Assuming in (19) that Φ(x) = φ_{n}(x), we obtain

|∫_{0}^{1} f(x)φ_{n}(x) dx| ≤ ω_{2} (\frac{1}{n} , f) (V_{n} + 2∥φ_{n}∥_{2})+n ∫_{1− \frac{1}{n}}^{1} |f(x)| dx |∫_{0}^{1} φ_{n}(x) dx| .

Since V_{n} = ∑_{k=1}^{n−1} | ∫_{0}^{\frac{k}{n}} φ_{n}(x) dx| = O(1) and ∫_{0}^{1} φ_{n}(x) dx = 0, n = 1, 2, …, we have

|c_{n}(f)| = O (ω_{2} (\frac{1}{n} , f)) .

Necessity. If V_{n} ≠ O(1), then as it is known (see [15]) there exists f_{0} ∈ Lip 1 such that

\overline{lim}_{n→∞} n|c_{n}(f_{0})| = +∞.  (28)

As far as ω_{2} (\frac{1}{n} , f_{0}) = O (\frac{1}{n}), from (28) we have

lim_{n→∞} \frac{c_{n}(f_{0})}{ω_{2} (\frac{1}{n} , f_{0})} = +∞. □

Theorem 4. Let (φ_{n}) be an orthonormal system on [0, 1] satisfying

∫_{0}^{1} φ_{n}(x)dx = 0, n=1, 2, ….

Then the relation

e_{N} = O (ω_{2} (\frac{1}{N} , f)) for every f ∈ L_{2}([0, 1])

holds if and only if

H_{N} = O(h_{N})

for every sequence (a_{n}) ∈ ℓ_{2}.

Proof. Sufficiency. Let f ∈ L_{2}([0, 1]). As it was shown above (see (2)), the equality

∑_{m=N}^{N+k} c_{m}^{2}(f) = ∫_{0}^{1} f(x) ∑_{m=N}^{N+k} c_{m}φ_{m}(x) dx = ∫_{0}^{1} f(x)P_{N}^{(k)}(x) dx  (29)

holds. By virtue of Theorem 2, we get

|∫_{0}^{1} f(x)P_{N}^{(k)}(x) dx| ≤ ω_{2} (\frac{1}{N} , f) (H_{N} + 2∥P_{N}^{(k)}∥_{2}) .  (30)

Since

∥P_{N}^{(k)}∥_{2} = (∫_{0}^{1} (P_{N}^{(k)}(x))^{2})^{\frac{1}{2}} = (∫_{0}^{1} ( ∑_{m=N}^{N+k} c_{m}(f)φ_{m}(x))^{2} dx)^{\frac{1}{2}}

= ( ∑_{m=N}^{N+k} c_{m}^{2}(f))^{\frac{1}{2}} ≤ e_{N},

and by virtue of the condition of Theorem 4 H_{N} = O(e_{N}), from (30) we have

|∫_{0}^{1} f(x)P_{N}^{(k)}(x) dx| ≤ c · ω_{2} (\frac{1}{N} , f) e_{N},

where c does not depend on N.

Finally, from (29) we get

∑_{m=N}^{N+k} c_{m}^{2}(f) ≤ c · ω_{2} (\frac{1}{N} , f) e_{N}, k = 1, 2, …,

hence

∑_{m=N}^{∞} c_{m}^{2}(f) ≤ c · ω_{2} (\frac{1}{N} , f) e_{N}.  (31)

From (31) we have

e_{N} = O (ω_{2} (\frac{1}{N} , f)) .

Thus the sufficiency is proved.

Necessity. Let H_{N} ≠ O(h_{N}). Then by virtue of Theorem 1 there exists a function f_{0} ∈ Lip 1 such that

\overline{lim}_{n→∞} Ne_{N} = +∞.

As far as for the function f_{0} the relation ω_{2} (\frac{1}{N} , f_{0}) = O (\frac{1}{N} ) holds, from (31) it follows

lim_{n→∞} \frac{e_{N}}{ω_{2} (\frac{1}{n} , f_{0})} = +∞.

376 V. TSAGAREISHVILI

Theorem is proved completely. □

Theorem 5. From every orthonormal system φ_{n} on [0, 1] satisfying the condition ∫_{0}^{1} φ_{n}(x) dx = 0, n = 1, 2, …, one can choose a subsystem ψ_{k} = φ_{n_{k}} for which the following conditions are fulfilled:

1) c_{n}(f) = O (ω_{2} (\frac{1}{n} , f)) ,

2) e_{N} = O (ω (\frac{1}{N} , f)) ,

3) e_{N} = O (ω_{2} (\frac{1}{N} , f)) . Proof. Let

ε_{i}(k) = ∑_{s=1}^{∞} (∫0^{\frac{i}{k}} φ_{s}(x) dx)^{2} , i = 1, 2, …, k.

By virtue of the Bessel inequality, ε_{i}(k) ≤ 1 for every i = 1, 2, …, k. Therefore for each fixed k we can choose a number s_{i}(k) such that the inequality

∑_{s=s_{i}(k)}^{∞} (∫0^{\frac{i}{k}} φ_{s}(x) dx)^{2} < \frac{1}{k^{3}}

holds. Now if s(k) = max_{1≤i≤k} s_{i}(k), then

∑_{s=s(k)}^{∞} (∫0^{\frac{i}{k}} φ_{s}(x) dx)^{2} < \frac{1}{k^{3}} .

Hence for every i = 1, 2, …, k we have

|∫0^{\frac{i}{k}} φ_{s}(x) dx| < \frac{1}{k^{3/2}}

if s ≥ s(k). Assuming φ_{s(k)} = ψ_{k}, we get

|∫0^{\frac{i}{k}} ψ_{k}(x) dx| < \frac{1}{k^{3/2}} .

This inequality implies the inequality

V_{k} = ∑_{i=1}^{k−1} |∫0^{\frac{i}{k}} ψ_{k}(x) dx| < 1, n = 1, 2, …,  (32)

and also the estimate

∑_{i=1}^{k−1} |∫0^{\frac{i}{k}} ∑_{m=k}^{k+l} a_{m}ψ_{m}(x) dx| ≤ (∑_{m=k}^{k+l} a_{m}^{2})^{\frac{1}{2}} ∑_{i=1}^{k−1} (∑_{m=k}^{k+l} (∫0^{\frac{i}{k}} ψ_{m}(x) dx)^{2})^{\frac{1}{2}}

≤ (∑_{m=k}^{k+l} a_{m}^{2})^{\frac{1}{2}} · ∑_{i=1}^{k−1} \frac{1}{k} < (∑_{m=k}^{k+l} a_{m}^{2})^{\frac{1}{2}} .

ESTIMATES OF FOURIER COEFFICIENTS 377

is valid. Therefore

H_{k} = O(h_{k}).  (33)

Hence the validity of Theorem 5 follows from Theorems 1,2,3,4 and from the relations (32) and (33). □

Remark 1. It follows from the proof of Theorem 1 that the relation

e_{N} = O (ω_{2} (\frac{1}{N} , f))  (34)

holds for every f ∈ C([0, 1]) if and only if relation (34) holds for every f ∈ Lip 1. Hence it follows Corollary 1. Relation (34) holds for every f ∈ L_{2}([0, 1]) if and only if the

condition

e_{N} = O (\frac{1}{N} )

holds for every function f ∈ Lip 1.

Corollary 2. The relation

c_{n}(f) = O (ω_{2} (\frac{1}{n} , f))

holds for every f ∈ L_{2}([0, 1]) if and only if the condition

c_{n}(f) = O (\frac{1}{n})

holds for every f ∈ Lip 1 (see [15]).

Remark 2. a) If (φ_{n}) is a trigonometric system, then the inequality

V_{n} = ∑_{k=1}^{n−1} |∫0^{\frac{k}{n}} sin πnx dx| ≤ ∑_{k=1}^{n−1} \frac{1}{πn} < \frac{1}{π}

is valid.

b) Let (χ_{n}) be the Haar system (see [11]). Then if n = 2^{m} + l (1 ≤ l ≤ 2^{m}), we have

|∫0^{\frac{k}{n}} χ_{n}(x) dx| ≤ 2^{− \frac{m}{2}} < \frac{2}{\sqrt{n}} .

On the other hand, note that only one integral ∫_{0}^{\frac{k}{n}} χ_{n}(x) dx is different from zero when k = 1, 2, …, n. Thus

V_{n} = ∑_{k=1}^{n−1} |∫0^{\frac{k}{n}} χ_{n}(x) dx| ≤ \frac{2}{\sqrt{n}} .

Further on, it is easy to verify that if n = 2^{s} + l, in case l ≤ 2^{s} the following estimate is valid:

∑_{k=1}^{n−1} |∫0^{\frac{k}{n}} P_{n}^{(q)}(x) dx| ≡ ∑_{k=1}^{n−1} |∫0^{\frac{k}{n}} ∑_{m=n}^{n+q} a_{m}χ_{m}(x) dx|

≤ ∑_{k=1}^{n} |∫0^{\frac{k}{2^{s+1}}} ∑_{m=n}^{n+q} a_{m}χ_{m}(x) dx| + ∑_{k=1}^{n} |∫ \frac{k}{n} ; \frac{k}{2^{s+1}} P_{n}^{(q)}(x) dx| .  (36)

Applying the Hölder inequality, we get

∑_{k=1}^{n} |∫ \frac{k}{n} ; \frac{k}{2^{s+1}} P_{n}^{(q)}(x) dx| ≤ ∫_{0}^{1} |P_{n}^{(q)}(x)| dx ≤ (∑_{m=n}^{n+q} a_{m}^{2})^{\frac{1}{2}} ≤ h_{n}.  (37)

Then, if n + q > 2^{s+1}, then

∑_{k=1}^{n} |∫0^{\frac{k}{2^{s+1}}} ∑_{m=n}^{n+q} a_{m}χ_{m}(x) dx| ≤ ∑_{k=1}^{n} |∫0^{\frac{k}{2^{s+1}}} ∑_{m=n}^{2^{s+1}} a_{m}χ_{m}(x) dx|

+ ∑_{k=1}^{n} |∫0^{\frac{k}{2^{s+1}}} ∑_{m=2^{s+1}+1}^{n+q} a_{m}χ_{m}(x) dx| .  (38)

Since m ≥ 2^{s+1}, we have

∫0^{\frac{k}{2^{s+1}}} χ_{m}(x) dx = 0

and the second summand in (38) is equal to zero.

On the other hand, as far as 2^{s+1} − n ≤ 2^{s}, we get

∑_{k=1}^{n} |∫0^{\frac{k}{2^{s+1}}} ∑_{m=n}^{2^{s+1}} a_{m}χ_{m}(x) dx| ≤ ∑_{m=n}^{2^{s+1}} |a_{m}| ∑_{k=1}^{n} |∫0^{\frac{k}{2^{s+1}}} χ_{m}(x) dx|

≤ ∑_{m=n}^{2^{s+1}} |a_{m}| · \frac{1}{\sqrt{m}} ≤ (∑_{m=n}^{2^{s+1}} a_{m}^{2})^{\frac{1}{2}} (∑_{m=n}^{2^{s+1}} \frac{1}{m})^{\frac{1}{2}} ≤ h_{n}.  (39)

Finally, taking into account in (36) the inequalities (37), (38) and (39), we obtain

∑_{k=1}^{n} |∫0^{\frac{k}{n}} P_{n}^{(q)}(x) dx| = O(h_{n}).

Consequently,

H_{n} = O(h_{n}).

c) Let now (φ_{n}) be the Walsh system (see [12]). Then since

|∫0^{\frac{k}{n}} φ_{n}(x) dx| < \frac{1}{n},

we get

V_{n} = ∑_{k=1}^{n−1} |∫0^{\frac{k}{n}} φ_{n}(x) dx| < 1.

In this case the inequality

H_{n} = O(h_{n})

is analogously proved as in case of the Haar system.

In conclusion, we can say that the efficiency of the conditions of Theorems 1, 2, and 3 is evident.

It should be noted that the above results were partially announced in [14].

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(Received 8.04.2002)

Author’s address:

Faculty of Mechanics and Mathematics I. Javakhishvili Tbilisi State University 2, University St., Tbilisi 0143, Georgia