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Convergence of subsequences of partial sums of trigonometric Fourier series Gy orgy G at Institute of Mathematics, University of Debrecen, gat.gyorgy@science.unideb.hu 6th Workshop on Fourier Analysis and Related Fields, P ecs, Hungary,

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Convergence of subsequences of partial sums of trigonometric Fourier series

Gy¨

  • rgy G´

at

Institute of Mathematics, University of Debrecen, gat.gyorgy@science.unideb.hu

6th Workshop on Fourier Analysis and Related Fields, P´ ecs, Hungary, 24-31 August 2017 1 / 23 Gy¨

  • rgy G´

at Convergence of subsequences of partial sums of trigonometric Fourier

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The trigonometric system

The trigonometric system: (

1 √ 2πeınx

n = 0, ±1, ±2, . . . ) (x ∈ R, ı = √−1). Orthonormal over any interval of length 2π. Let T := [−π, π]. Let f ∈ L1(T). The kth Fourier coefficient of f : ˆ f (k) := 1 2π

  • T

f (x)e−ıktdt, k ∈ Z. The nth (n ∈ N) partial sum of the Fourier series of f : Snf (y) :=

n

  • k=−n

ˆ f (k)eıky.

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The trigonometric system

The nth (n ∈ N) Fej´ er or (C, 1) mean of function f : σnf (y) := 1 n + 1

n

  • k=0

Skf (y). σnf (y) = 1 π

  • T

f (x)Kn(y − x)dx, Kn is the nth Fej´ er kernel. Lebesgue (1905, Mathematische Annalen): For each integrable function a.e. convergence of Fej´ er means σnf =

1 n+1

n

k=0 Skf → f .

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Partial sums, first negative results, the trigonometric system

It is of main interest in the theory of trigonometric Fourier series that how to reconstruct the function from the partial sums of its Fourier series. Du Bois-Reymond (1876, Abhand. Akad. M¨ unchen) the Fourier series of a continuous function can unboundedly diverge at some point. Kolmogoroff (1923, Fund. Math.) constructed an example of a function f ∈ L1(T) such that the partial sums Smf (x) diverges unboundedly almost everywhere. Kolmogoroff (1926, C. R. Acad. Sci. Paris) there exists an integrable function with everywhere divergent Fourier series.

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Partial sums, positive results, the trigonometric system

Carleson (1966, Acta Math.) f ∈ L2(T), then Snf → f almost everywhere. Hunt (1968, University Press, Carbondale, Ill.) f ∈ Lp(p > 1)

  • impl. a.e. conv.

Antonov (1996, East J. Approx.) f ∈ L log+ L log+ log+ log+ L, (

  • |f | log+ |f | log+ log+ log+ |f | < ∞) then the partial sums

converge to the function almost everywhere again.

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The trigonometric system

What if we have only a subsequence of the partial sums? With respect to the partial sums and the Lebesgue space L1 bad news... Totik (1982, Publicationes Mathematicae-Debrecen): for each subsequence (nj) of the sequence of natural numbers there exists an integrable function f such that supj |Snjf | = +∞

  • everywhere. Moreover,

Konyagin (2005, Proc. Steklov Inst. Math. Suppl.): for any increasing sequence (nj) of positive integers and any nondecreasing function φ : [0, +∞) → [0, +∞) satisfying condition φ(u) = o(u log log u), there is a function f ∈ φ(L) such that supj |Snjf | = +∞ everywhere. Special subsequences?

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Lacunary partial sums, the endpoint theorems, trigonometric system

Di Plinio (2014, Collect. Math.) for (nj) lacunary and f ∈ L1 log+ log+ L log+ log+ log+ log+ L we have Snjf → f a.e. Victor Lie (2017, to appear in European Math. Journal) for any (nj) lacunary and φ(u) = o(u log+ log+ u log+ log+ log+ log+ u), there exists a f ∈ φ(L) such that supj |Snjf | = +∞ everywhere. What about the L1 case? Some summation method is needed.

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Zygmunt Zalcwasser’s problem, trigonometric system

In 1936 Zalcwasser (1936, Stud. Math.) asked how fast can the sequence of integers (nj) grow that it still holds: 1 N

N

  • j=1

Snjf → f a.e. for every function f ∈ L1. This problem with respect to the trigonometric system was completely solved for continuous functions and uniform convergence: Salem, (1955, Am. J. Math.): If the sequence (nj) is convex, then the condition supj j−1/2 log nj < +∞ is necessary and sufficient for the uniform convergence for every continuous function.

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The trigonometric system

With respect to convergence almost everywhere, and integrable functions the situation is more complicated. In 1936 Zalcwasser (1936, Stud. Math.) proved the a.e. relation 1

N

N

j=1 Sj2f → f for each integrable function f .

Salem, (1955, Am. J. Math.) writes that this theorem of Zalcwasser is extended to j3 and j4. Belinsky proved (1997, Proc. Am. Math. Soc.) the existence

  • f a sequence nj ∼ exp( 3

√j) such that the relation

1 N

N

j=1 Snjf → f holds a.e. for every integrable function.

Belinsky also conjectured that if the sequence (nj) is convex, then the condition supj j−1/2 log nj < +∞ is necessary and sufficient again. So, that would be the answer for the problem

  • f Zalcwasser (1936, Stud. Math.).

(in this point of view (trigonometric system, a.e. convergence and L1 functions.)) 9 / 23 Gy¨

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An example for kernel function, Fej´ er kernel

Figure: 1

4

3

k=0 Dk(x) Fej´

er kernel

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An example for kernel function, Zalcwasser’s kernel

Figure: 1

4

3

k=0 Dk2(x) Zalcwasser kernel

”Hopeless” to investigate a general Zalcwasser kernel.

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An example for kernel function

Figure: 1

4

3

k=0 D2k(x) kernel

KN ≥ 0 fails to hold and KN1 ≥ C log nN

N

if nN ր ∞ fast enough.

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The result, Zalcwasser’s problem

Theorem (G´ at, submitted) Let nj+1 ≥

  • 1 + 1

jδ

  • nj for some 0 < δ <

√ 5/2 − 1/2 ≈ 0.618, f ∈ L1 . Then a.e.: lim

N→∞

1 N

N

  • j=1

Snjf = f . Corollary Let (nj) be a lacunary sequence of natural numbers. Then it holds the almost everywhere relation: limN→∞ 1

N

N

j=1 Snjf = f for every f ∈ L1(T).

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The main tool, Zalcwasser’s problem

Main tool: (G´ at, submitted) Let f ∈ L1, λ > f 1 and (nj) lacunary: there exists: F1 ⊂ F2 ⊂ . . . , mes Fj ≤ Cf 1/λ and

  • 1

N

N

  • j=1
  • Snjf − Vnjf

σmj1Fj

  • 2

2

≤ 1 N C log5 Nf 1λ, where mj ∼ nj (Vnf is the nth de La Vall´

ee Poussin mean).

  • Remark. Of course, without σmj1Fj this inequlity does not hold for

all f ∈ L1. However, the mes. of Fj is ”small”, the mes. of Fj is ”big” and σmj1Fj is close to 1 on a ”big” set.

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The Walsh system

expansion with resp. the binary number system n = ∞

i=0 ni2i ∈ N, x = ∞ i=0 xi 2i+1 ∈ [0, 1)

ni, xi ∈ {0, 1} i = 0, 1, . . . , If x is a dyadic rational number (x ∈ { p

2n : p, n ∈ N}) we choose

the expansion which terminates in 0 ’s. Walsh function n-th Walsh-Paley function: ωn(x) := (−1)

∞

i=0 nixi Paley, A remarkable series of orthogonal functions, Proceedings of the London Mathematical Society (1932).

Can take +1 and −1 as a value.

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Dirichlet and Fej´ er kernel functions:

Dirichlet and Fej´ er kernel functions Dn :=

n−1

  • k=0

ωk, Kn := 1 n

n−1

  • k=0

Dk, Fourier coefficients, partial sums of Fourier series, Fej´ er means: ˆ f (n) := 1 f (x)ωn(x)dx (n ∈ N), Snf (y) :=

n−1

  • k=0

ˆ f (k)ωk(y) = 1 f (x ∔ y)Dn(x)dx σnf (y) := 1 n

n−1

  • k=0

Skf (y) = 1 f (x ∔ y)Kn(x)dx

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Fej´ er means:

trigonometric system:

  • H. Lebesgue σnf (x) → f (x) for a.e. x. ,,Reconstruction the

function” Walsh case Walsh-Paley system N.J. Fine, Trans. Am. Math. Soc., 1949. For the Walsh-Kaczmarz system

  • G. G´
  • at. On (C; 1) summability of integrable functions with respect

to the Walsh-Kaczmarz system. Stud. Math., 1998. What does this Walsh-Kaczmarz system mean?

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Fej´ er means:

trigonometric system:

  • H. Lebesgue σnf (x) → f (x) for a.e. x. ,,Reconstruction the

function” Walsh case Walsh-Paley system N.J. Fine, Trans. Am. Math. Soc., 1949. For the Walsh-Kaczmarz system

  • G. G´
  • at. On (C; 1) summability of integrable functions with respect

to the Walsh-Kaczmarz system. Stud. Math., 1998. What does this Walsh-Kaczmarz system mean?

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Walsh-Kaczmarz system:

Let the Walsh-Kaczmarz functions be defined by κ0 = 1 and for n ≥ 1, |n| = ⌊log2 n⌋ κn(x) := r|n|(x)(−1)

|n|−1

k=0

nkx|n|−1−k.

The Walsh-Paley system is ω := (ωn : n ∈ N) and the Walsh-Kaczmarz system is κ := (κn : n ∈ N). {κn : 2k ≤ n < 2k+1} = {ωn : 2k ≤ n < 2k+1} for all k ∈ N and κ0 = ω0. A dyadic blockwise ,,rearrangement”.

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Walsh-Paley Fej´ er kernel function

Figure: A K8 and K11 Walsh-Paley-Fej´ er kernels

Kn(x) → 0 (n → ∞) for every x = 0.

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Walsh-Kaczmarz Fej´ er kernel function

Figure: A K26 Walsh-Kaczmarz

|Kn(x)| → ∞ (n → ∞) at every dyadic rational.

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subsequences of Walsh-Fourier series

Theorem (G´ at, JAT, 2010) Let (nj) be a lacunary sequence of natural numbers. Then it holds the almost everywhere relation: limN→∞ 1

N

N

j=1 Snjf = f for every f ∈ L1(T).

Conjecture Let nj+1 ≥

  • 1 + 1

jδ

  • nj for some

0 < δ < √ 5/2 − 1/2 ≈ 0.618, f ∈ L1 . Then a.e.: lim

N→∞

1 N

N

  • j=1

Snjf = f . For the Riesz logarithmic means we have: Theorem (G´ at, JAT, 2010) Let (nj) be any convex sequence of natural numbers tending to +∞. Then it holds the almost everywhere relation: limN→∞

1 log N

N

j=1 Snj f j

= f for every f ∈ L1(T).

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subsequences of Walsh and other Fourier series

Conjecture: The theorem above concerning the Riesz log. means and trig. sys. holds. Problems: What about (C, 1) means of (Snjf ) with resp. the Walsh-Kaczmarz system. Nothing is known yet. Norm convergence of (C, 1) means of (Snjf ) with resp. the Walsh-Paley, Walsh-Kaczmarz system. Not complete characterization. Two or more dimensional situation?

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Thank you!

Thank you for your attention!

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