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, 24-31 August 2017 1 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
The trigonometric system 1 2 π e ı nx The trigonometric system: ( n = 0 , ± 1 , ± 2 , . . . ) √ ( x ∈ R , ı = √− 1). Orthonormal over any interval of length 2 π . Let T := [ − π, π ]. Let f ∈ L 1 ( T ). The k th Fourier coefficient of f : f ( k ) := 1 � ˆ f ( x ) e − ı kt dt , 2 π T k ∈ Z . The n th ( n ∈ N ) partial sum of the Fourier series of f : n � ˆ f ( k ) e ı ky . S n f ( y ) := k = − n 2 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
The trigonometric system The n th ( n ∈ N ) Fej´ er or ( C , 1) mean of function f : n 1 � σ n f ( y ) := S k f ( y ) . n + 1 k =0 σ n f ( y ) = 1 � f ( x ) K n ( y − x ) dx , π T K n is the n th Fej´ er kernel. Lebesgue (1905, Mathematische Annalen): For each integrable function a.e. convergence of Fej´ er means � n 1 σ n f = k =0 S k f → f . n +1 3 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
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 ∈ L 1 ( T ) such that the partial sums S m f ( x ) diverges unboundedly almost everywhere. Kolmogoroff (1926, C. R. Acad. Sci. Paris) there exists an integrable function with everywhere divergent Fourier series. 4 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
Partial sums, positive results, the trigonometric system Carleson (1966, Acta Math.) f ∈ L 2 ( T ), then S n f → f almost everywhere. Hunt (1968, University Press, Carbondale, Ill.) f ∈ L p ( 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. 5 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
The trigonometric system What if we have only a subsequence of the partial sums? With respect to the partial sums and the Lebesgue space L 1 bad news... Totik (1982, Publicationes Mathematicae-Debrecen): for each subsequence ( n j ) of the sequence of natural numbers there exists an integrable function f such that sup j | S n j f | = + ∞ everywhere. Moreover, Konyagin (2005, Proc. Steklov Inst. Math. Suppl.): for any increasing sequence ( n j ) 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 sup j | S n j f | = + ∞ everywhere. Special subsequences? 6 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
Lacunary partial sums, the endpoint theorems, trigonometric system Di Plinio (2014, Collect. Math.) for ( n j ) lacunary and f ∈ L 1 log + log + L log + log + log + log + L we have S n j f → f a.e. Victor Lie (2017, to appear in European Math. Journal) for any ( n j ) lacunary and φ ( u ) = o ( u log + log + u log + log + log + log + u ), there exists a f ∈ φ ( L ) such that sup j | S n j f | = + ∞ everywhere. What about the L 1 case? Some summation method is needed. 7 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
Zygmunt Zalcwasser’s problem, trigonometric system In 1936 Zalcwasser (1936, Stud. Math.) asked how fast can the sequence of integers ( n j ) grow that it still holds: N 1 � S n j f → f N j =1 a.e. for every function f ∈ L 1 . 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 ( n j ) is convex, then the condition sup j j − 1 / 2 log n j < + ∞ is necessary and sufficient for the uniform convergence for every continuous function. 8 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
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. � N relation 1 j =1 S j 2 f → f for each integrable function f . N Salem, (1955, Am. J. Math.) writes that this theorem of Zalcwasser is extended to j 3 and j 4 . Belinsky proved (1997, Proc. Am. Math. Soc.) the existence √ j ) such that the relation of a sequence n j ∼ exp( 3 � N 1 j =1 S n j f → f holds a.e. for every integrable function. N Belinsky also conjectured that if the sequence ( n j ) is convex, then the condition sup j j − 1 / 2 log n j < + ∞ is necessary and sufficient again. So, that would be the answer for the problem of Zalcwasser (1936, Stud. Math.). (in this point of view (trigonometric system, a.e. convergence and L 1 functions.)) 9 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
An example for kernel function, Fej´ er kernel � 3 Figure: 1 k =0 D k ( x ) Fej´ er kernel 4 10 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
An example for kernel function, Zalcwasser’s kernel � 3 Figure: 1 k =0 D k 2 ( x ) Zalcwasser kernel 4 ”Hopeless” to investigate a general Zalcwasser kernel. 11 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
An example for kernel function � 3 Figure: 1 k =0 D 2 k ( x ) kernel 4 K N ≥ 0 fails to hold and � K N � 1 ≥ C log n N if n N ր ∞ fast enough. N 12 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
The result, Zalcwasser’s problem Theorem (G´ at, submitted) √ � � 1 + 1 Let n j +1 ≥ n j for some 0 < δ < 5 / 2 − 1 / 2 ≈ 0 . 618, j δ f ∈ L 1 . Then a.e.: N 1 � lim S n j f = f . N N →∞ j =1 Corollary Let ( n j ) be a lacunary sequence of natural numbers. Then it holds the almost everywhere relation: lim N →∞ 1 � N j =1 S n j f = f for every f ∈ L 1 ( T ). N 13 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
The main tool, Zalcwasser’s problem at, submitted) Let f ∈ L 1 , λ > � f � 1 and ( n j ) Main tool: (G´ lacunary: there exists: F 1 ⊂ F 2 ⊂ . . . , mes � F j ≤ C � f � 1 /λ and 2 � � N � � 1 ≤ 1 � � � N C log 5 N � f � 1 λ, � � � � S n j f − V n j f σ m j 1 F j � � N � � j =1 � � 2 where m j ∼ n j ( V n f is the n th de La Vall´ ee Poussin mean ). Remark. Of course, without σ m j 1 F j this inequlity does not hold for all f ∈ L 1 . However, the mes. of F j is ”small”, the mes. of F j is ”big” and σ m j 1 F j is close to 1 on a ”big” set. 14 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
The Walsh system expansion with resp. the binary number system i =0 n i 2 i ∈ N , x = � ∞ n = � ∞ x i 2 i +1 ∈ [0 , 1) n i , x i ∈ { 0 , 1 } i =0 i = 0 , 1 , . . . , If x is a dyadic rational number ( x ∈ { p 2 n : p , n ∈ N } ) we choose the expansion which terminates in 0 ’s. Walsh function � ∞ i =0 n i x i n -th Walsh-Paley function: ω n ( x ) := ( − 1) Paley, A remarkable series of orthogonal functions , Proceedings of the London Mathematical Society (1932). Can take +1 and − 1 as a value. 15 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
Dirichlet and Fej´ er kernel functions: Dirichlet and Fej´ er kernel functions n − 1 n − 1 K n := 1 � � D n := ω k , D k , n k =0 k =0 Fourier coefficients, partial sums of Fourier series, Fej´ er means: � 1 ˆ f ( n ) := f ( x ) ω n ( x ) dx ( n ∈ N ) , 0 � 1 n − 1 � ˆ S n f ( y ) := f ( k ) ω k ( y ) = f ( x ∔ y ) D n ( x ) dx 0 k =0 � 1 n − 1 σ n f ( y ) := 1 � S k f ( y ) = f ( x ∔ y ) K n ( x ) dx n 0 k =0 16 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
Fej´ er means: trigonometric system: H. Lebesgue σ n f ( 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? 17 / 23 Gy¨ orgy G´ at Convergence of subsequences of partial sums of trigonometric Fourier
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