INTEGRABLE MAJORANT OF THE PARTIAL

SUMS FOR MULTIPLE HAAR SERIES

Ondrej Kováčik

Let [a, b] be an interval in ℝ, let {g_{n}} be some orthonormal system of real functions defined on [a, b] and let

∑_{n=1}^{∞} c_{n}g_{n}(x)

be a Fourier series of some function f ∈ L^{2}(a, b) with respect to the system {g_{n}}, i.e., c_{n} = (f, g_{n}) is the n-th Fourier coefficient. In problems concerning the almost everywhere convergence of orthogonal Fourier series the majorant of the partial sums

(P f)(x) = sup_{m} | ∑_{n=1}^{m} c_{n}g_{n}(x)|

is frequently used.

Numerous propositions concerning the convergence of Fourier series are expressed in the form of estimates for P f. For example the classical Menshov-Rademacher theorem (see e.g. [1]) states the inequality

∥P f∥_{L^{2}}^{2} ≤ K ∑_{n=1}^{∞} c_{n}^{2} log^{2}(n + 1).

Also in the theory of convergence systems of Fourier series the majorant P f plays an important role.

Definition 1. Let {g_{n}} be an orthonormal system in L^{2}(a, b). We call it a convergence system if the condition

(A)  ∑_{n=1}^{∞} c_{n}^{2} < ∞

implies

(B)  ∑_{n=1}^{∞} c_{n}g_{n}(x) < ∞

almost everywhere on [a, b].

Definition 2. Let {g_{n}} be a convergence system defined on [a, b]. If the validity of (B) on some subinterval E of [a, b] implies (A), then we call this system a strong convergence system.

Theorem (see e.g. [2]). An orthonormal system {g_{n}} defined on [a, b] is the convergence system if and only if P f is finite a.e. for any f ∈ L^{2}(a, b).

Remark 1. P. L. Ulyanov studied convergence problems of Fourier series and he stated in [3] that under certain conditions on {c_{n}} a complete system becomes a strong convergence system.

In this paper we deal with the majorant P f for a Haar orthonormal system in ℝ^{N}. Namely, we shall deal with conditions on f ensuring P f ∈ L^{1}. Let us define the Haar system on [0, 1].

Definition 3. Given a natural number n we can write n uniquely as n = 2^{k} + i with k = 0, 1, 2, … and i = 0, 1, …, 2^{k} − 1. Denote I = [0, 1] and I_{n} = (i2^{−k}, (i + 1)2^{−k}). We introduce the Haar system {h_{n}} in I in the form:

h_{0}(t) = 1 for t ∈ I,

h_{n}(t) = \{ \begin{matrix}2^{k/2} & for t ∈ I_{n}^{+} = (i2^{−k}, (2i + 1)2^{−k−1}) \\ −2^{k/2} & for t ∈ I_{n}^{−} = ((2i + 1)2^{−k−1}, (i + 1)2^{−k}) \\ 0 & for t ∈ I \ \overline{I_{n}^{+} ∪ I_{n}^{−}} \\ h_{n}(0+) & for t = 0 \\ h_{n}(1−) & for t = 1 \\ \frac{1}{2}(h_{n}(t−) + h_{n}(t+)) & for all other t ∈ I, \end{matrix}

where h_{n}(t−) = lim_{s→t−} h_{n}(s) and h_{n}(t+) = lim_{s→t+} h_{n}(s).

Let N be a natural number and let ℕ be the set of all non-negative integers. We shall use the following notation: ℕ^{N} = ℕ × ⋯ × ℕ, I^{N} = I × ⋯ × I, x = (x_{1}, …, x_{N}) ∈ I^{N}, m = (m_{1}, …, m_{N}) ∈ ℕ^{N}, |m| = m_{1} + ⋯ + m_{N}, 0 = (0, …, 0), ∞ = (∞, …, ∞), I_{m} = I_{m_{1}} × ⋯ × I_{m_{N}}, D_{m} = (0, 2^{−m_{1}}) × ⋯ × (0, 2^{−m_{N}}), J_{m} = (2^{−m_{1}−1}, 2^{−m_{1}}) × ⋯ × (2^{−m_{N} −1}, 2^{−m_{N}}) and 2^{m} = (2^{m_{1}}, …, 2^{m_{N}}).

We shall say that the series

(1)  ∑_{k_{1}=0}^{∞} … ∑_{k_{N} =1}^{∞}b_{k_{1}...k_{N}} h_{k_{1}}(x_{1}) … h_{k_{N}} (x_{N})

= ∑_{k=0}^{∞} b_{k}h_{k_{1}}(x_{1}) … h_{k_{N}} (x_{N})

is a multiple Haar series of some measurable function f(x) = f(x_{1}, …, x_{N}) defined on I^{N}, i.e.,

b_{k} = ∫_{I^{N}} f(x)h_{k_{1}}(x_{1}) … h_{k_{N}} (x_{N}) dx,

and that the sum

(2)  S_{m}(f; x) = S_{m_{1}...m_{N}} (f; x_{1}, …, x_{N})

= ∑_{k_{1}=0}^{m_{1}} … ∑_{k_{N} =0}^{m_{N}} b_{k_{1}...k_{N}} h_{k_{1}}(x_{1}) … h_{k_{N}} (x_{N})

= ∑_{k=0}^{m} b_{k}h_{k_{1}}(x_{1}) … h_{k_{N}} (x_{N})

is a partial sum of the series (1) of order |m|. We introduce the function

(P f)(x) = sup_{m} |S_{m}(f; x)|

defined on I^{N}. We shall call this function a majorant of the partial sums for the multiple Haar series (1).

Using the Haar kernel on [0, 1] (see e.g. [1]),

(3) K_{n}(u, v) = ∑_{l=0}^{n} h_{l}(u)h_{l}(v) = \{ \begin{matrix}(meas I_{n})^{−1} & for (u, v) ∈ I_{n} × I_{n}, \\ 0 & for other (u, v), \end{matrix}

we obtain the following representation for the sum (2):

S_{m}(f; x) = ∑_{k=0}^{m} ∫_{I^{N}} f(t)h_{k_{1}}(t_{1}) … h_{k_{N}} (t_{N}) dt · h_{k_{1}}(x_{1}) … h_{k_{N}} (x_{N})

= ∫_{I^{N}} f(t)K_{m_{1}}(t_{1}, x_{1}) … K_{m_{N}} (t_{N}, x_{N}) dt,

i.e.,

S_{m}(f; x) = \{ \begin{matrix}(meas I_{m})^{−1} ∫_{I_{m}} f(t) dt & for x ∈ I_{m}, \\ 0 & otherwise. \end{matrix}

Theorem A. Let Φ(u) be a non-negative function increasing on [0, ∞) such that

Φ(u) = o( log^{N}(u + 1)) for u → +∞.

Then there exists a function f_{0} in the Orlicz class LΦ(L) of measurable functions defined on I^{N} such that the majorant P f_{0} of the partial sums of its multiple Haar series is not integrable (in the Lebesgue sense) over I^{N}.

Proof. Choose u = 2^{Nn} (n ∈ ℕ). Due to the properties of Φ we obtain

Φ(2^{Nn}) = o( log^{N}(2^{Nn} + 1)),

and thus

Φ(2^{Nn}) = o((Nn + 1)^{N}).

Hence there exists a sequence {n_{l}} of natural numbers such that

∑_{l=1}^{∞} \frac{Φ(2^{Nn_{l}})}{(Nn_{l} + 1)^{N}} < +∞,

and due to the monotonicity of Φ we get

(4)  ∑_{l=1}^{∞} \frac{Φ(2^{Nn_{l}}(Nn_{l} + 1)^{−N})}{(Nn_{l} + 1)^{N}} < +∞.

For m, |m| = 0, 1, …, we take a step function

f_{0}(x) = a_{m} ≥ 0 for x ∈ J_{m}.

Then

S_{2^{m}}(f_{0}; x) = 2^{|m|} ∫_{D_{m}} f_{0}(t) dt for x ∈ D_{m}

which yields (for x ∈ D_{m}) that

S_{2^{m}}(f_{0}; x) = 2^{|m|} ∑_{i=m}^{∞} ∫_{J_{i}} f_{0}(t) dt = 2^{|m|} ∑_{i=m}^{∞} a_{i}2^{−|i|−N}

INTEGRABLE MAJORANT FOR MULTIPLE HAAR SERIES 237

and

(P f_{0})(x) ≥ |S_{2^{m}}(f_{0}; x)| = 2^{|m|−N} ∑_{i=m}^{∞} a_{i}2^{−|i|}.

Therefore, we obtain

∫_{I^{N}} (P f_{0})(x) dx = ∑_{k=0}^{∞} ∫_{J_{k}} (P f_{0})(x) dx

≥ ∑_{k=0}^{∞} ∫_{J_{k}} 2^{|k|−N} ∑_{i=k}^{∞} a_{i}2^{−|i|} dx

= 2^{−2N} ∑_{k=0}^{∞} ∑_{i=k}^{∞} a_{i}2^{−|i|}

= 2^{−2N} ∑_{i=0}^{∞} ∑_{k=0}^{i} a_{i}2^{−|i|}

= 2^{−2N} ∑_{i=0}^{∞} a_{i}2^{−|i|} ∏_{j=1}^{N}(i_{j} + 1)

and

∫_{I^{N}} |f_{0}(x)|Φ(|f_{0}(x)|) dx = ∑_{i=0}^{∞} ∫_{J_{i}} |f_{0}(x)|Φ(|f_{0}(x)|) dx

= ∑_{i=0}^{∞} a_{i}Φ(a_{i})2^{−|i|−N}.

According to the selection of the sequence {n_{l}} above and to the property (4) we choose

a_{i} = \{ \begin{matrix}2^{Nn_{l}}(Nn_{l} + 1)^{−N} & for i_{1} = ⋯ = i_{N} = n_{l}, l = 1, 2, … \\ 0 & for other i. \end{matrix}

Then we conclude that

∫_{I^{N}} (P f_{0})(x) dx ≥ 2^{−2N} ∑_{l=1}^{∞} \frac{2^{Nn_{l}}}{(Nn_{l} + 1)^{N}} 2^{−Nn_{l}}(n_{l} + 1)^{N} = +∞

and

∫_{I^{N}} |f_{0}(x)|Φ(|f_{0}(x)|) dx = 2^{−N} ∑_{l=1}^{∞} \frac{2^{Nn_{l}}}{(Nn_{l} + 1)^{N}} Φ(\frac{2^{Nn_{l}}}{(Nn_{l} + 1)^{N}} )2^{−Nn_{l}}

= 2^{−N} ∑_{l=1}^{∞} \frac{1}{(Nn_{l} + 1)^{N}} Φ(\frac{2^{Nn_{l}}}{(Nn_{l} + 1)^{N}} ) < +∞,

which completes the proof of Theorem A. □

Remark 2. Using the properties of maximal function (see e.g. [4]) we obtain the following theorem.

Theorem B. Let Φ(u) = log^{N}(u + 1). Then P f ∈ L^{1}(I^{N}) for every function f ∈ LΦ(L) on I^{N}.

References

[1] G. Alexits, Convergence problems of orthogonal series, Pergamon Press, New York, 1961.

[2] A. M. Olevskii, A certain orthonormal systems and their applications, Mat. Sb. 71 (1966), 297–335. (Russian)

[3] P. L. Ulyanov, Series with respect to the Haar system with monotone coefficients, Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 925–950. (Russian)

[4] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, New Jersey, 1970.