GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 3, 1995, 313-322

SOME NEGATIVE RESULTS CONCERNING THE PROCESS OF FEJÉR TYPE TRIGONOMETRIC

CONVOLUTION

G. ONIANI

Abstract. The process of Fejér type trigonometric convolution and its discrete analogue have equivalent uniform residues. The situation changes under pointwise comparison. In this direction negative results have been obtained by different authors. One result of such a type is given in the present paper. In particular, a counter-example is constructed for which both comparisons diverge on the set of complete measure. The smoothness of the counter-example as well as some other problems are investigated.

1. Let C_{2π} denote a Banach space of 2π-periodic functions continuous on ℝ, with sup-norm ∥ · ∥_{c}.

Let for every n ∈ ℕ

χ_{n}(x) := ∑_{k=−n}^{n} ρ_{k,n}e^{ikx},  (1.1)

where ρ_{−k,n} = ρ_{k,n}, ρ_{0,n} = 1.

F_{n} denotes an operator of trigonometric convolution

F_{n}f(x) := \frac{1}{2π} ∫_{0}^{2π}f(u)χ_{n}(x − u)du  (1.2)

and J_{n} is its discrete analogue

J_{n}f(x) := \frac{1}{2n + 1} ∑_{j=0}^{2n} f(u_{j,n})χ_{n}(x − u_{j,n}),  (1.3)

1991 Mathematics Subject Classification. 41A35,42A24. Key words and phrases. Sequence of convolution operators, sequence of discrete ana-

logues of convolution operators, Fejér type kernels, pointwise comparison.

313

1072-947X/95/0500-0313$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

where u_{j,n} = 2πj / (2n + 1), 0 ≤ j ≤ 2n.

The following theorem on the operators {F_{n}} and {J_{n}} is valid (see [2]- [5]).

Theorem 1.1. Let {χ_{n}} be a sequence of even polynomial kernels (1.1) satisfying the conditions

∥χ_{n}∥_{1} = O(1) (n → ∞),  (1.4)
1 − ρ_{k,n} = o_{k}(1) (k ∈ ℕ, n → ∞).  (1.5)

Then for every f ∈ C_{2π} we have

∥F_{n}f − f∥_{c} = o(1) and ∥J_{n}f − f∥_{c} = o(1) (n → ∞),  (1.6)
\begin{matrix}∥J_{n}f − f∥_{c} = O(∥F_{n}f − f∥_{c}), \\ ∥F_{n}f − f∥_{c} = O(∥J_{n}f − f∥_{c}) (n ∈ ℕ). \end{matrix}  (1.7)

Note also that (1.6) is equivalent to the pair of conditions (1.4) and (1.5), i.e., the sequences (processes) of the operators {F_{n}} and {J_{n}} ensure uniform approximation of continuous functions (such sequences of operators are called Fejér) and their residues are equivalent with respect to the norm in C_{2π}.

Pointwise comparison changes the situation. In this case residues of the operators {F_{n}} and {J_{n}} may turn out to be nonequivalent on the set of complete measure. In particular, the following theorems are valid (see resp.

[2] and [1]):

Theorem 1.2. Let {χ_{n}} be a sequence of even polynomial kernels satisfying the conditions (1.4) and

1 − ρ_{k,n} = O_{k}(\frac{1}{n}) (k ∈ ℕ, n → ∞).  (1.8)

Let Ω denote a class of functions continuous on [0, ∞) with the following properties:

0 = ω(0) < ω(s) ≤ ω(s + t) ≤ ω(s) + ω(t) (s, t > 0)

and lim_{t→0} ω(t) / t = ∞

and let f ∈ Lip_{1} ω denote that |f(x) − f(y)| ≤ c[ω(|x − y|)], x, y ∈ [0, 2π] (c > 0 is a constant).

Then for every ω ∈ Ω there is a counter-example f_{ω} ∈ Lip_{1} ω such that for n → ∞

|J_{n}f_{ω}(x) − f_{ω}(x)| ≠ O(|F_{n}f_{ω}(x) − f_{ω}(x)|) for a.e. x ∈ ℝ. (1.9)

Theorem 1.3. Let {χ_{n}} be a sequence of even polynomial kernels (1.1) satisfying the conditions (1.4) and (1.8). Then there is a counter-example f ∈ C_{2π} such that as n → ∞

|F_{n}f(x) − f(x)| ≠ O(|J_{n}f(x) − f(x)|) for a.e. x ∈ ℝ. (1.10)

Comparing Theorems 1.2 and 1.3 there naturally arises the question: Is there a function f ∈ C_{2π} with properties (1.9) and (1.10) when the conditions (1.4) and (1.8) are fulfilled, i.e., is there a continuous function with two-sided divergence?

2. The theorem below gives a positive answer to the above question.

Theorem 2.1. Let {χ_{n}} be a sequence of even polynomial kernels (1.1) satisfying the conditions (1.4) and (1.8). Then there is a counter-example f ∈ C_{2π} such that as n → ∞

\begin{matrix}|F_{n}f(x) − f(x)| ≠ O(|J_{n}f(x) − f(x)|), \\ |J_{n}f(x) − f(x)| ≠ O(|F_{n}f(x) − f(x)|) \end{matrix}  (2.1)

simultaneously almost for all x ∈ ℝ.

Proof. Let {s_{k}} ⊂ ℕ be an arbitrary sequence and

n_{1} = 4, n_{k+1} = \frac{1}{2}[(4s_{k} + 1)(2n_{k} + 1) − 1], k = 2, 3, …. (2.2)

For natural numbers n and k we introduce the notation

\begin{matrix}g_{n}(x) := cos(2n + 1)x, \\ H_{k} := ⋃_{j∈ℤ} \frac{π}{2(2n_{k} + 1)}[2j + 1 − \frac{1}{k + 1}, 2j + 1 + \frac{1}{k + 1}], \\ D_{k} := [0, 2π] ∩ ⋃_{j∈ℤ} \frac{π}{2(2n_{k} + 1)}[2j − \frac{1}{2}, 2j + \frac{1}{2}]. \end{matrix}  (2.3)

The following lemma is valid.

Lemma 2.1. (See [2].) Let x ∈ H_{k} − y_{k}, y_{k} ∈ D_{k}. Then for every k ∈ ℕ

\begin{matrix}|J_{n_{k}}g_{n_{k}}(x + y_{k}) − g_{n_{k}}(x + y_{k})| ≥ C_{1} − ε_{k} \\ (C_{1} = 1/\sqrt{2}, ε_{k} = π/2(k + 1)), \\ |F_{n_{k}}g_{n_{k}}(x + y_{k}) − g_{n_{k}}(x + y_{k})| ≤ ε_{k}. \end{matrix}  (2.4)

There is also a sequence {y_{k}} (y_{k} ∈ D_{k}, k ∈ ℕ) such that

lim sup_{k→∞}(H_{k} − y_{k})

is a set of full measure.

316 G. ONIANI

Let

\begin{matrix}M_{n} := ∥χ_{n}∥_{c} (n ∈ ℕ), \\ E_{n} := ⋃_{j=0}^{2n} (\frac{2πj}{2n + 1} + \frac{1}{2n\sqrt{M_{n}}} , \frac{2π(j + 1)}{2n + 1}) (n ∈ ℕ). \end{matrix}  (2.5)

The following assertions are valid (see [1]).

Lemma 2.2. Let the sequence {χ_{n}} satisfy the conditions (1.4) and (1.5). Then ∥χ_{n}∥_{c} → ∞ (n → ∞).

Lemma 2.3. Let the sequence {χ_{n}} satisfy the conditions (1.4) and (1.8). Then for every n ≥ n_{0} there is a trigonometric polynomial P_{n}(x) and a set A_{n} ⊂ [0, 2π] such that for n ∈ ℕ

∥P_{n}∥_{c} ≤ 2,  (2.6)
A_{n} ⊂ E_{n},  (2.7)
μA_{n} ≥ C_{2},  (2.8)

where μ is the Lebesgue measure,

|F_{n}P_{n}(x) − P_{n}(x)| ≥ C_{3} \frac{\sqrt{M_{n}}}{n} (x ∈ A_{n}),  (2.9)
|P_{n}(x)| ≤ \frac{1}{nM_{n}} (x ∈ E_{n}),  (2.10)

where n_{0}, C_{2}, C_{3} are positive constants.

Consider the series

f(x) = ∑_{k=1}^{∞} [\frac{1}{\sqrt{n_{k}}} g_{n_{k}}(x + y_{k}) + \frac{1}{\sqrt[4]{M_{m_{k}}}} P_{m_{k}}(x + t_{k})]

(here ∑_{k=1}^{∞} [\frac{1}{\sqrt{n_{k}}} + _{\frac{1}{\sqrt[4]{M_{m_{k}}}}} ] < ∞, whence f ∈ C_{2π}). The sequences n_{k} ↑ ∞, m_{k} ↑ ∞, {y_{k}}, and {t_{k}} will be constructed later on.

Let for k ≥ ℕ

\begin{matrix}α_{k}(x) := ∑_{j=1}^{k−1} \frac{1}{\sqrt{n_{j}}} g_{n_{j}}(x + y_{j}) + ∑_{j=1}^{k−1} \frac{1}{\sqrt[4]{M_{m_{j}}}} P_{m_{j}}(x + t_{j}), \\ γ_{k}(x) := ∑_{j=k+1}^{∞} \frac{1}{\sqrt{n_{j}}} g_{n_{j}}(x + y_{j}) + ∑_{j=k}^{∞} \frac{1}{\sqrt[4]{M_{m_{j}}}} P_{m_{j}}(x + t_{j}), \\ α′_{k}(x) := ∑_{j=1}^{k} \frac{1}{\sqrt{n_{j}}} g_{n_{j}}(x + y_{j}) + ∑_{j=1}^{k−1} \frac{1}{\sqrt[4]{M_{m_{j}}}} P_{m_{j}}(x + t_{j}), \\ γ′_{k}(x) := ∑_{j=k+1}^{∞} \frac{1}{\sqrt{n_{j}}} g_{n_{j}}(x + y_{j}) + ∑_{j=k+1}^{∞} \frac{1}{\sqrt[4]{M_{m_{j}}}} P_{m_{j}}(x + t_{j}). \end{matrix} (2.11)

We can easily conclude from (1.8) that as n → ∞

\begin{matrix}|J_{n}α_{k}(x) − α_{k}(x)| ≤ S_{k−1}/n, |F_{n}α_{k}(x) − α_{k}(x)| ≤ S_{k−1}/n \\ |J_{n}α′_{k}(x) − α′_{k}(x)| ≤ S_{k−1}/n, |F_{n}α′_{k}(x) − α′_{k}(x)| ≤ S_{k−1}/n, \end{matrix} (2.12)

where S_{k−1} depends on k − 1 only. It is not difficult to see that for k ∈ ℕ

\begin{matrix}|J_{n_{k}}f(x) − f(x)| ≥ \frac{1}{\sqrt{n_{k}}} |J_{n_{k}}g_{n_{k}}(x + y_{k}) − g_{n_{k}}(x + y_{k})| − \\ −|J_{n_{k}}α_{k}(x) − α_{k}(x)| − |J_{n}γ_{k}(x) − γ_{k}(x)|, \\ |F_{n_{k}}f(x) − f(x)| ≤ \frac{1}{\sqrt{n_{k}}} |F_{n_{k}}g_{n_{k}}(x + y_{k}) − g_{n_{k}}(x + y_{k})| + \\ +|F_{n_{k}}α_{k}(x) − α_{k}(x)| − |F_{n_{k}}γ_{k}(x) − γ_{k}(x)|, \end{matrix} (2.13)

\begin{matrix}|F_{m_{k}}f(x) − f(x)| ≥ \frac{1}{\sqrt[4]{M_{m_{k}}}} |F_{m_{k}}P_{m_{k}}(x + t_{k}) − P_{m_{k}}(x + t_{k})| − \\ −|F_{m_{k}}α′_{k}(x) − α′_{k}(x)| − |F_{m_{k}}γ′_{k}(x) − γ′_{k}(x)|, \\ |J_{m_{k}}f(x) − f(x)| ≤ \frac{1}{\sqrt[4]{M_{m_{k}}}} |J_{m_{k}}P_{m_{k}}(x + t_{k}) − \\ −P_{m_{k}}(x + t_{k})| + |J_{m_{k}}α′_{k}(x) − α′_{k}(x)| + |J_{m_{k}}γ′_{k}(x) − γ′_{k}(x)| \end{matrix}(2.14)

and for C_{4} = const and n ∈ ℕ

|F_{n}f(x) − f(x)| ≤ C_{4}∥f∥_{c}, |J_{n}f(x) − f(x)| ≤ C_{4}∥f∥_{c}. (2.15)

Let us construct the sequences {n_{k}} and {m_{k}} such that for k ∈ ℕ

\begin{matrix}n_{0} < n_{1} < m_{1} < ⋯ < n_{k} < m_{k} < n_{k+1} < m_{k+1} < …, \\ \sqrt{n_{k}} > kS_{k−1}, \\ \sqrt[4]{M_{m_{k}}} > kS_{k−1}, \\ \frac{1}{\sqrt[4]{M_{m_{k}}}} < \frac{1}{2n_{k}} , \\ \frac{1}{\sqrt{n_{k+1}}} < \frac{1}{2m_{k}} . \end{matrix} (2.16)

Put

T_{k} := ⋃_{j=0}^{2m_{k}} [\frac{2πj}{2m_{k} + 1} + \frac{1}{2m_{k}\sqrt{M_{m_{k}}}} , \frac{2π(j + 1)}{2m_{k} + 1} − \frac{1}{2m_{k}\sqrt{M_{m_{k}}}} ].

Let A_{m_{k}}^{∗} and E_{m_{k}}^{∗} be 2π-periodic extensions of the sets A_{m_{k}} and E_{m_{k}}, respectively.

We can easily show that

\frac{2πj}{2m_{k} + 1} + t_{k} ∈ E_{m_{k}}^{∗} (k ∈ ℕ, t_{k} ∈ T_{k}),

whence by Lemma 2.3 we have

|P_{m_{k}}(\frac{2πj}{2m_{k} + 1} + t_{k})| ≤ \frac{1}{m_{k}M_{m_{k}}} (k ∈ ℕ, t_{k} ∈ T_{k}, j = 0, 1, …, 2n). (2.17)

From (2.17) we obtain that if x ∈ A_{m_{k}}^{∗} −t_{k}, t_{k} ∈ T_{k}, then for every k ∈ ℕ

|J_{m_{k}}P_{m_{k}}(x + t_{k}) − P_{m_{k}}(x + t_{k})| ≤
≤ |J_{m_{k}}P_{m_{k}}(x + t_{k})| + |P_{m_{k}}(x + t_{k})| ≤ \frac{1}{m_{k}} + \frac{1}{m_{k}M_{m_{k}}} . (2.18)

We can prove that (see [1]) there is a sequence {t_{k}} (t_{k} ∈ T_{k}, k ∈ ℕ) such that

lim sup_{k→∞}(A_{m_{k}}^{∗} − t_{k})  (2.19)

is the set of full measure.

From Lemma 2.3 and relations (2.13)–(2.16) and (2.18) we get that if x ∈ A_{m_{k}}^{∗} − t_{k}, t_{k} ∈ T_{k}, then for every k ∈ ℕ

\begin{matrix}|F_{m_{k}}f(x) − f(x)| ≥ \frac{1}{\sqrt[4]{M_{m_{k}}}} C_{3} \frac{\sqrt{M_{m_{k}}}}{m_{k}} − \frac{S_{k−1}}{m_{k}} − \\ −C_{4}∥γ′_{k}∥_{c} = \frac{\sqrt[4]{M_{m_{k}}}}{m_{k}} (C_{3} − o(1)), \\ |J_{m_{k}}f(x) − f(x)| ≤ \frac{1}{\sqrt[4]{M_{m_{k}}}} (\frac{1}{m_{k}} + \frac{1}{m_{k}M_{m_{k}}} ) + \\ +\frac{S_{k−1}}{m_{k}} + C_{4}∥γ′_{k}∥_{c} = \frac{\sqrt[4]{M_{m_{k}}}}{m_{k}} o(1). \end{matrix}  (2.20)

From (2.19) and (2.20) we obtain that as n → ∞

|F_{n}f(x) − f(x)| ≠ O(|J_{n}f(x) − f(x)|) for a.e. x ∈ ℝ.  (2.21)

On the other hand, from Lemma 2.1 and conditions (2.13)–(2.15) we easily see that if x ∈ H_{k} − y_{k}, y_{k} ∈ D_{k}, then for k ∈ ℕ

\begin{matrix}|J_{n_{k}}f(x)−f(x)| ≥ \frac{1}{\sqrt{n_{k}}} C_{1} − \frac{S_{k−1}}{n_{k}} − C_{4}∥γ_{k}∥_{c} = \frac{1}{\sqrt{n_{k}}} (C_{1} − o(1)), \\ |F_{n_{k}}f(x)−f(x)| ≤ \frac{1}{\sqrt{n_{k}}} ε_{k} + \frac{S_{k−1}}{n_{k}} + C_{4}∥γ_{k}∥_{c} = \frac{1}{\sqrt{n_{k}}} o(1)). \end{matrix} (2.22)

It follows from (2.22) that as n → ∞

|J_{n}f(x) − f(x)| ≠ O(|F_{n}f(x) − f(x)|) for a.e. x ∈ ℝ.  (2.23)

With the help of (2.21) and (2.23) we can conclude that f is the sought for counter-example.

Here without proof we shall give the result concerning the smoothness of the counter-example constructed in Theorem 2.1.

Theorem 2.2. Let the sequence {χ_{n}} satisfy the conditions (1.4) and

1 − ρ_{k,n} = O_{k}(\frac{M_{n}}{n^{+}} ) (k ∈ ℕ, n → ∞).

Then for every ω ∈ Ω there is a counter-example f_{ω} ∈ Lip_{1} ω such that

|F_{n}f_{ω}(x) − f_{ω}(x)| ≠ O(|J_{n}f_{ω}(x) − f_{ω}(x)|) (n → ∞),
|J_{n}f_{ω}(x) − f_{ω}(x)| ≠ O(|F_{n}f_{ω}(x) − f_{ω}(x)|) (n → ∞)

simultaneously for a.e. x ∈ ℝ.

3. As we have mentioned, if the sequence (1.1) is of Fejér type (i.e., ∥χ_{n}∥_{1} = O(1) and 1 − ρ_{k,n} = o_{k}(1) (k ∈ ℕ, n → ∞)), then ∥χ_{n}∥_{c} → ∞ (n → ∞) (see Lemma 2.2).

In this section we shall investigate the rate of the above-mentioned convergence.

Does any condition of the type 1−ρ_{k,n} = O_{k}(ε_{n}) (k ∈ ℕ, ε_{n} ↓ 0, n → ∞) improve the rate of convergence of ∥χ_{n}∥_{c} to ∞? Theorem below gives a negative answer to this question.

Theorem 3.1. For every sequence α_{n} → ∞ there is a sequence of kernels {χ_{n}} such that as n → ∞

∥χ_{n}∥_{1} = O(1),
1 − ρ_{k,n} = 0 (k ∈ ℕ, n > n_{k}),  (3.1)
∥χ_{n}∥_{c} = O(α_{n})  (3.2)

(i.e., convergence to infinity can be arbitrarily slow).

Proof. Let

K_{n}(x) := \frac{1}{2} + \frac{n}{n + 1} cos x + ⋯ + \frac{n + 1 − k}{n + 1} cos kx +

+ ⋯ + \frac{1}{n + 1} cos nx (Fejér type kernel),  (3.3)

R_{k,n}(x) := \frac{1}{n + 1} cos x + \frac{2}{n + 1} cos 2x +

+ ⋯ + \frac{k}{n + 1} cos kx (k ≤ n).  (3.4)

Clearly,

|R_{k,n}(x)| ≤ \frac{k^{2}}{n + 1} (k, n ∈ ℕ).  (3.5)

Consider the sequence n_{k} ↑ ∞ with the following properties:

n_{k} ≥ 2k^{2} (k ∈ ℕ),  (3.6)
α_{n} ≥ 2k^{2} (n ≥ n_{k}).  (3.7)

Let χ_{n}(x) = 1 (n ∈ [1, n_{1}]), and

\frac{χ_{n}(x)}{2} := K_{k^{2}}(x) + R_{k,k^{2}}(x) (n ∈ [n_{k} + 1, n_{k+1}]),  (3.8)

i.e.,

χ_{n}(x)/2 = \frac{1}{2} + \frac{k^{2}}{k^{2} + 1} cos x + ⋯ + \frac{k^{2} + 1 − k}{k^{2} + 1} cos kx +

+ ⋯ + \frac{1}{k^{2} + 1} cos k^{2}x + (\frac{1}{k^{2} + 1} cos x + ⋯ + \frac{k}{k^{2} + 1} cos kx) =

\frac{1}{2} + cos x + ⋯ + cos kx + (\frac{k^{2} − k}{k^{2} + 1} cos(k + 1)x + ⋯ +

+\frac{1}{k^{2} + 1} cos k^{2}x) (n ∈ [n_{k} + 1, n_{k+1}]).  (3.9)

It is easily seen from (3.5) that

∥χ_{n}∥_{1} ≤ 2∥K_{k^{2}}∥_{1} + 2∥R_{k,k^{2}}∥_{1} < 2π + 4π =
= 6π (n ∈ [n_{k} + 1, n_{k+1}]).  (3.10)

From (3.9) we obtain that

1 − ρ_{k,n} = 0 (k ∈ ℕ, n > n_{k}).  (3.11)

The relations (3.5)–(3.8) show that

∥χ_{n}∥_{c} ≤ 2∥K_{k^{2}}∥_{c} + 2∥R_{k,k^{2}}∥_{c} ≤ 2\frac{k^{2} + 1}{2} + 2 <

< 2k^{2} < α_{n} (n ∈ [n_{k} + 1, n_{k+1}]).  (3.12)

From (3.9), (3.10), and (3.11) we can conclude that {χ_{n}} is the desired sequence.

It is not difficult to see that

∥χ_{n}∥_{c} = O(n) (n → ∞).

Acknowledgement. I thank Professor R. Getsadze for posing the question and for interest in this work.

References

1. R. D. Getsadze, Problem of N.Kirchhoff and R.Nessel. J. Approx. Theory (to appear).

2. N. Kirchhoff and R. J. Nessel, Divergence almost everywhere of a pointwise comparison of trigonometric convolution process with their discrete analogues. J. Approx. Theory 1(1992), 29-38.

3. S. M. Lozinski, On an analogy between the summation of Fourier series and that of interpolation trigonometric polynomials. C. R. Acad. Sci. URSS (N.S.) 39(1943), 83-97.

4. S. Lozinski, On convergence and summability of Fourier series and interpolation processes. Mat. Sb. 14(56)(1944), No. 3, 175-268.

5. N. K. Bari, A Treatise on Trigonometric Series, Vols 1 and 2. Pergamonm New York, 1964. Russian original: Fizmatgiz, Moscow, 1961.

(Received 16.12.1994)

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