[PPT] - Topics on N orlund logarithmic means Nacima Memi c University of PowerPoint Presentation

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Topics on N¨

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logarithmic means Nacima Memi´ c Topics on N¨

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Topics on N¨

rlund logarithmic means

Nacima Memi´ c

University of Sarajevo, Bosnia and Herzegovina

28.08.2017

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Content

Almost everywhere convergence of some subsequences (tmnf )n

f N¨
rlund logarithmic means of Walsh Fourier coefficients for

every integrable function f and divergence for other classes of subsequences. Convergence and divergence in norm of N¨

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means of generalized Walsh Fourier coefficients on some unbounded Vilenkin groups.

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Motivation for Walsh-Paley system

The Riesz logarithmic means of Walsh or trigonometric Fourier series

1 log n

n−1

k=1 Skf k

f any integrable function f converges

almost everywhere to the original function f . This is not true for the Walsh Fourier series which diverges everywhere for some integrable function f satisfying

ϕ(|f |) < ∞, where ϕ(u) = o(u√log u).

As G´ at and Goginava mentioned the following results show a similarity of N¨

rlund logarithmic means with Walsh Fourier

series rather than classical logarithmic means.

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Goginava’s results for Walsh-Paley system

(Goginava 2005)

Theorem

Let {(mn)n : n ≥ 1} be a sequence of positive integers for which

∞

n=1

log2(mn − 2⌊log mn⌋ + 1) log mn < ∞. Then, the operator t∗f := sup

n≥1

|tmnf | is of weak type (1, 1).

Corollary

Let {(mn)n : n ≥ 1} be the sequence defined in the theorem above and f an integrable function. Then, tmnf → f , a.e.

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Goginava’s results for Walsh-Paley system

Corollary

For all integrable function f , we have t2nf → f , a.e.

I. Blahota [1] proved the validity of the same results on the

2-adic group Bijection between 2-adic and dyadic integers Important difference between the two systems of characters

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Motivation for Walsh-Paley system

(G´ at-Goginava 2009):

Theorem

Let ϕ : [0, ∞) → [0, ∞) be a function such that ϕ(u)

u

is nondecreasing and ϕ(u) = o(u√log u). Then there exist a function f ∈ L and a measurable set E with positive measure for which

ϕ(|f |) < ∞

and lim sup tnf (x) = ∞, ∀x ∈ E.

Remark

Due to the corollary above it is impossible to replace lim sup tnf (x) = ∞ by lim tnf (x) = ∞, in this theorem.

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Notations and definitions

The N¨

rlund logarithmic means are defined by

tnf := 1 ln

n−1

k=1

Skf n − k , ln :=

n−1

k=1

1 k . Fn := 1 ln

n−1

k=1

Dk n − k , tnf = Fn ∗ f . Define the function ϕ : N \ {0} → N by ϕ(n) = n − 2[log2 n]. Set ϕ1(n) = ϕ(n), ϕ0(n) = n and ϕi(n) = ϕ ◦ ϕi−1(n) when i ≥ 2. For every n ∈ N \ {0}, i ≥ 0, such that ϕi(n) > 0, define the functions αi(n) = [log2(ϕi(n))] and βi(n) = lϕi(n).

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Main results

Theorem

Let (mn)n be an increasing sequence of positive integers. Suppose that

i:ϕi(mn)>0

βi(mn) lmn = O(1). Then, tmnf → f , a.e. The condition of [4, Theorem 1] from which Goginava proves that tmnf → f , a.e. formulated in our notations is

∞

n=1

α2

1(mn)

α0(mn) < +∞.

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Main results

Since ♯{i : ϕi(n) > 0} ≤ α1(n), it follows that

i:ϕi(mn)>0

αi(mn) α0(mn) < α2

1(mn)

α0(mn). If the sequence (mn)n satisfies the condition of [4, Theorem 1], then α2

1(mn) = o(α0(mn)),

which implies that

i:ϕi(mn)>0

αi(mn) = o(α0(mn)),

r equivalently,
i:ϕi(mn)>0

βi(mn) = o(β0(mn)) = o(lmn). Therefore, this theorem is a generalization of [4, Theorem 1].

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Main results

Theorem

Let (mn)n and (sn)n be increasing sequences of positive integers for which:

1

the sequence (ϕsn(mn))n is increasing,

2

lmn = o(βsn(mn)√sn), when n → ∞, then there exists an integrable function f such that tmnf f on a subset of positive measure.

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Convergence in norm of logarithmic means for Walsh-Fourier coefficients

(F. Schipp, W.R. Wade, P. Simon, and J. P´ al, 1990)

Theorem

Let f ∈ C and ω(δ, f )∞ = o

1

log( 1

δ )

, then Snf − f ∞→ 0.

(G´ at-Goginava 2006)

Theorem

Let f ∈ C and ω(δ, f )∞ = o

1

log( 1

δ )

, then tnf − f ∞→ 0.

Theorem

There exists a function g ∈ C such that ω(δ, g)∞ = O

1

log( 1

δ )

, and

tng(0) diverges.

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Convergence in norm of logarithmic means for Walsh-Fourier coefficients

(B.I. Golubov, A.V. Efimov, and V.A. Skvortsov, 1991)

Theorem

Let f ∈ L1 and ω(δ, f )L1 = o

1

log( 1

δ )

, then Snf − f 1→ 0.

(G´ at-Goginava 2006)

Theorem

Let f ∈ L1 and ω(δ, f )L1 = o

1

log( 1

δ )

, then tnf − f 1→ 0.

Theorem

There exists a function g ∈ L1 such that ω(δ, g)L1 = O

1

log( 1

δ )

, and

tng − g 1 0.

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Fourier series, Fej´ er means-Convergence in norm

For Vilenkin systems Snf − f p→ 0, 1 < p < ∞ (P. Simon 1976). SMnf − f 1→ 0, (G. H. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, A. I. Rubinstein, 1981). For bounded groups σnf − f p→ 0, 1 ≤ p < ∞ (Simon, P., P´ al, J., 1977). On arbitrary groups as a trivial consequence of the convergence

f the partial sums: σnf − f p→ 0, 1 < p < ∞.

On every unbounded group there exists an integrable function f such that σMnf − f 1 0 (Price, J., 1957). For every Vilenkin system and every integrable function σMnf → f a.e (G´ at 2003).

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Motivations for Part 2

In general, the Fej´ er (C, 1) means have better properties, than the logarithmic ones. In the case of some unbounded Vilenkin systems the situation may change. In their paper [2] the authors have proved a convergence result

f the subsequence (tMnf )n to the integrable function f in the

L1 norm for some unbounded Vilenkin groups. The main tool was the boundedness of the sequence (FMn1)n. Paradoxically, this is the reason for the divergence of the whole sequence (tnf )n. Therefore, in order to construct unbounded groups on which the sequence (tnf )n converges in the L1 norm, the property of uniform boundedness should be avoided.

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Blahota-G´ at’s results

Theorem

If f ∈ Lp(1 ≤ p < ∞) and lim sup

n

FMn1 < ∞, then tMnf − f p→ 0. If f is continuous then the convergence holds in the supremum norm.

Theorem

If log mn = O(nδ) for some 0 < δ < 1

2, then there exists an integrable

function such that tnf − f 1 0.

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Blahota-G´ at’s results

Example

Let mn =

⌊exp(n

1 4 )⌋,

if n = j2, j ∈ N; 2,

therwise.

In this case we have

1

lim supn mn = ∞,

2

log mn = O(n

1 4 ), 3

lim sup

n

FMn1 < ∞.

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Main result and examples

Theorem

If the sequence (mn)n is unbounded and if the sequence (FMn)n is bounded in L1, then there exists a function f ∈ L1 such that tnf f in L1.

Example

There exists an unbounded Vilenkin group represented by the sequence (mn)n such that

1

log mnk ∼ √nk, for some subsequence (mnk)k and

2

tnf f in L1.

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Example

Using the the previous theorem and [2, Lemma 4] it suffices to construct a sequence (mn)n such that sup

n FMn1 ≤ sup n

n−1

k=0(log mk)2

n−1

k=0 log mk

< +∞. Let mk = 2 if k = 4s for all positive integers s, and log mk = 2s = √ k if k = 4s. Hence we have

n−1

k=0

(log mk)2 =

n−1

s=[log √n−1]+1

(log 2)2 +

[log √n−1]

s=0

4s ≤ n(log 2)2 + C4log√n ∼ n,

n−1

k=0

log mk ∼ n log 2 + 2log√n ∼ n, from which we easily obtain the result.

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References

I. Blahota, Almost everywhere convergence of a subsequence of

logarithmic means of Fourier series on the group of 2-adic integers, Georgian Math. J. 19 (2012), no. 3, 417-425.

I. Blahota and G.G´

at, Norm summabitity of N¨

rlund logarithmic

means on unbounded Vilenkin groups, Anal. Theory Appl., Vol 24, 1(2008), 1-17.

G. G´

at, U. Goginava, On the divergence of N¨

rlund logarithmic

means of Walsh-Fourier series, Acta Math. Sinica , 25 (6) (2009), 903–916.

U. Goginava, Almost everywhere convergence of subsequence of

logarithmic means of Walsh-Fourier series, Acta Math. Acad.

Paedagog. Nyhzi. (N.S.) 21 (2005), no. 2, 169–175.

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