convergence of subsequences of partial sums of...

Convergence of subsequences of partial sums oftrigonometric Fourier series

Gyorgy Gat

Institute of Mathematics, University of Debrecen, [email protected]

6th Workshop on Fourier Analysis and Related Fields, Pecs, Hungary, 24-31 August 2017

1 / 23 Gyorgy Gat Convergence of subsequences of partial sums of trigonometric Fourier series

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 :

f (k) :=1

2π

∫Tf (x)e−ıktdt,

k ∈ Z.

The nth (n ∈ N) partial sum of the Fourier series of f :

Snf (y) :=n∑

k=−nf (k)eıky .



The nth (n ∈ N) Fejer or (C , 1) mean of function f :

σnf (y) :=1

n + 1

n∑k=0

Sk f (y).

σnf (y) =1

π

∫Tf (x)Kn(y − x)dx ,

Kn is the nth Fejer kernel.

Lebesgue (1905, Mathematische Annalen):

For each integrable function a.e. convergence of Fejer meansσnf = 1

n+1

∑nk=0 Sk f → f .


Partial sums, first negative results, the trigonometricsystem

It is of main interest in the theory of trigonometric Fourier seriesthat how to reconstruct the function from the partial sums of itsFourier series.

Du Bois-Reymond (1876, Abhand. Akad. Munchen) theFourier series of a continuous function can unboundedlydiverge at some point.

Kolmogoroff (1923, Fund. Math.) constructed an example ofa function f ∈ L1(T ) such that the partial sums Smf (x)diverges unboundedly almost everywhere.

Kolmogoroff (1926, C. R. Acad. Sci. Paris) there exists anintegrable function with everywhere divergent Fourier series.


Partial sums, positive results, the trigonometric system

Carleson (1966, Acta Math.) f ∈ L2(T ), then Snf → falmost 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.



What if we have only a subsequence of the partial sums? Withrespect to the partial sums and the Lebesgue space L1 bad news...

Totik (1982, Publicationes Mathematicae-Debrecen): for eachsubsequence (nj) of the sequence of natural numbers thereexists an integrable function f such that supj |Snj f | = +∞everywhere. Moreover,

Konyagin (2005, Proc. Steklov Inst. Math. Suppl.): for anyincreasing sequence (nj) of positive integers and anynondecreasing function φ : [0,+∞)→ [0,+∞) satisfyingcondition φ(u) = o(u log log u), there is a function f ∈ φ(L)such that supj |Snj f | = +∞ everywhere.

Special subsequences?


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+ Lwe have Snj f → f a.e.

Victor Lie (2017, to appear in European Math. Journal)for any (nj) lacunary andφ(u) = o(u log+ log+ u log+ log+ log+ log+ u), there exists af ∈ φ(L) such that supj |Snj f | = +∞ everywhere.

What about the L1 case? Some summation method is needed.


Zygmunt Zalcwasser’s problem, trigonometric system

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

1

N

N∑j=1

Snj f → f

a.e. for every function f ∈ L1.This problem with respect to the trigonometric system wascompletely solved for continuous functions and uniformconvergence:Salem, (1955, Am. J. Math.): If the sequence (nj) is convex, thenthe condition supj j

−1/2 log nj < +∞ is necessary and sufficient forthe uniform convergence for every continuous function.



With respect to convergence almost everywhere, and integrablefunctions the situation is more complicated.

In 1936 Zalcwasser (1936, Stud. Math.) proved the a.e.relation 1

N

∑Nj=1 Sj2f → f for each integrable function f .

Salem, (1955, Am. J. Math.) writes that this theorem ofZalcwasser is extended to j3 and j4.

Belinsky proved (1997, Proc. Am. Math. Soc.) the existenceof a sequence nj ∼ exp( 3

√j) such that the relation

1N

∑Nj=1 Snj f → 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 problemof Zalcwasser (1936, Stud. Math.). (in this point of view (trigonometric system,

a.e. convergence and L1 functions.))


An example for kernel function, Fejer kernel

Figure: 14

∑3k=0 Dk(x) Fejer kernel


An example for kernel function, Zalcwasser’s kernel

Figure: 14

∑3k=0 Dk2 (x) Zalcwasser kernel

”Hopeless” to investigate a general Zalcwasser kernel.11 / 23 Gyorgy Gat Convergence of subsequences of partial sums of trigonometric Fourier series

An example for kernel function

Figure: 14

∑3k=0 D2k (x) kernel

KN ≥ 0 fails to hold and ‖KN‖1 ≥ C log nNN if nN ↗∞ fast enough.


The result, Zalcwasser’s problem

Theorem (Gat, submitted)

Let nj+1 ≥(

1 + 1jδ

)nj for some 0 < δ <

√5/2− 1/2 ≈ 0.618,

f ∈ L1 . Then a.e.:

limN→∞

1

N

N∑j=1

Snj f = f .

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

1N

∑Nj=1 Snj f = f for every f ∈ L1(T ).


The main tool, Zalcwasser’s problem

Main tool: (Gat, submitted) Let f ∈ L1, λ > ‖f ‖1 and (nj)lacunary: there exists: F1 ⊂ F2 ⊂ . . . ,mes

⋃Fj ≤ C‖f ‖1/λ and∥∥∥∥∥∥ 1

N

N∑j=1

(Snj f − Vnj f

) (σmj 1Fj

)∥∥∥∥∥∥2

2

≤ 1

NC log5 N‖f ‖1λ,

where mj ∼ nj (Vnf is the nth de La Vallee Poussin mean).

Remark. Of course, without σmj 1Fjthis inequlity does not hold for

all f ∈ L1. However, the mes. of Fj is ”small”, the mes. of Fj is”big” and σmj 1Fj

is close to 1 on a ”big” set.


The Walsh system

expansion with resp. the binary number system

n =∑∞

i=0 ni2i ∈ N, x =

∑∞i=0

xi2i+1 ∈ [0, 1) ni , xi ∈ {0, 1}

i = 0, 1, . . . ,

If x is a dyadic rational number (x ∈ { p2n : 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.


Dirichlet and Fejer kernel functions:

Dirichlet and Fejer kernel functions

Dn :=n−1∑k=0

ωk , Kn :=1

n

n−1∑k=0

Dk ,

Fourier coefficients, partial sums of Fourier series, Fejer means:

f (n) :=

∫ 1

0f (x)ωn(x)dx (n ∈ N),

Snf (y) :=n−1∑k=0

f (k)ωk(y) =

∫ 1

0f (x u y)Dn(x)dx

σnf (y) :=1

n

n−1∑k=0

Sk f (y) =

∫ 1

0f (x u y)Kn(x)dx


Fejer means:

trigonometric system:

H. Lebesgue σnf (x)→ f (x) for a.e. x . ,,Reconstruction thefunction”

Walsh case

Walsh-Paley system

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

G. Gat. On (C; 1) summability of integrable functions with respectto the Walsh-Kaczmarz system. Stud. Math., 1998.

What does this Walsh-Kaczmarz system mean?


Walsh-Kaczmarz system:

Let the Walsh-Kaczmarz functions be defined by κ0 = 1 and forn ≥ 1, |n| = blog2 nc

κn(x) := r|n|(x)(−1)∑|n|−1

k=0 nkx|n|−1−k .

The Walsh-Paley system is ω := (ωn : n ∈ N) and theWalsh-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”.


Walsh-Paley Fejer kernel function

Figure: A K8 and K11 Walsh-Paley-Fejer kernels

Kn(x)→ 0 (n→∞) for every x 6= 0.


Walsh-Kaczmarz Fejer kernel function

Figure: A K26 Walsh-Kaczmarz

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


subsequences of Walsh-Fourier series

Theorem (Gat, JAT, 2010) Let (nj) be a lacunary sequence ofnatural numbers. Then it holds the almost everywhere relation:limN→∞

1N

∑Nj=1 Snj f = f for every f ∈ L1(T ).

Conjecture Let nj+1 ≥(

1 + 1jδ

)nj for some

0 < δ <√

5/2− 1/2 ≈ 0.618, f ∈ L1 . Then a.e.:

limN→∞

1

N

N∑j=1

Snj f = f .

For the Riesz logarithmic means we have:

Theorem (Gat, JAT, 2010) Let (nj) be any convex sequence ofnatural numbers tending to +∞. Then it holds the almost

everywhere relation: limN→∞1

log N

∑Nj=1

Snj f

j = f for every

f ∈ L1(T ).


subsequences of Walsh and other Fourier series

Conjecture: The theorem above concerning the Riesz log. meansand trig. sys. holds.

Problems:

What about (C , 1) means of (Snj f ) with resp. theWalsh-Kaczmarz system. Nothing is known yet.

Norm convergence of (C , 1) means of (Snj f ) with resp. theWalsh-Paley, Walsh-Kaczmarz system. Not completecharacterization.

Two or more dimensional situation?


Thank you!

Thank you for your attention!


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