Universally divergent Fourier series via Landau s extremal functions

Transcription

1 Comment.Math.Univ.Carolin. 56,2(2015) Universally divergent Fourier series via Landau s extremal functions Gerd Herzog, Peer Chr. Kunstmann Abstract. We prove the existence of functions f A(D), the Fourier series of which being universally divergent on countable subsets of T = D. The proof is based on a uniform estimate of the Taylor polynomials of Landau s extremal functions on T \ {1}. Keywords: Fourier series; universal functions; Landau s extremal functions Classification: 42A16, 30B30, 47B38 1. Introduction Let D denote the unit disc in C and T = D. For f L 1 (T) let = f()e denote the corresponding Fourier series, i.e. e (t) = exp(it), f() = 1 2π 2π 0 f(exp(it)) exp( it) dt ( Z). There is a tremendous amount of classical convergence and divergence results on Fourier series [14]. Moreover, several results on universally divergent Fourier (and trigonometric) series are nown, see for example [3], [5], [7], [10], [11] and the references given there. Roughly speaing, f has a universally divergent Fourier series if the set of restrictions { } ( n ) E f()e : n N = n in a function space Y, of functions over a subset E of T, is dense in Y. In [10] Müller proved that given a countable set E T, then the set of functions f C(T) DOI /

2 160 Herzog G., Kunstmann P.Ch. having the property that { ( n = n f()e ) E : n N } is dense in C E is a dense G δ -subset of C(T). Here C E is the Fréchet space of all functions w : E C endowed with the topology of pointwise convergence. In this paper, we prove that there are even functions f C(T) with universal divergent Fourier series in the sense of Müller such that in addition f( ) = 0 ( N). While Müller s proof is based on the Localization Principle of Fourier series we will use Landau s extremal functions [9, 2]. To formulate our results let H(D) denote the Fréchet space of all analytic functions on D endowed with the compact open topology, let A(D) = C(D) H(D) denote the disc-algebra endowed with the maximum norm, and for f H(D) let S n (f, z) = f () (0) z (n N 0 ).! For a given countable set E T consider the continuous linear operators We will prove L n : A(D) C E, L n (f) = S n (f, ) E. Theorem 1. The set of all f A(D) with the property is a dense G δ -subset of A(D). {L n (f) : n N 0 } is dense in C E Remar. According to a result of Fejér [4], no q-to-one function f A(D) (q N) can share the property in Theorem 1, since for those functions ( ) q S n (f, 1) 1 + f (n N 0 ). 2 Just as Müller s Theorem extends to L p -spaces, we can extend Theorem 1 to certain Banach spaces of analytic functions, such as the Hardy spaces H p (D) (1 p < ), or the little Bloch space (compare [1]), for example. B 0 = {f H(D) : lim z 1 (1 z 2 ) f (z) = 0}, f = f(0) + max z <1 (1 z 2 ) f (z)

3 Universally divergent Fourier series via Landau s extremal functions 161 Theorem 2. Let (X, ) be a Banach space with A(D) X H(D) such that (1) A(D) is dense in X, (2) is weaer than on A(D), (3) convergence in X implies convergence in H(D). Let E T be countable, and let L n : X C E, Ln (f) = S n (f, ) E. Then, the set of all f X with the property is a dense G δ -subset of X. { L n (f) : n N 0 } is dense in C E In [10] Müller also discusses the problem of replacing pointwise convergence by uniform convergence for compact subsets E of T. We note here that the general topological category argument [10, Lemma 2] is also applicable in our setting. Combined with Theorem 1 it proves that there are many and even uncountable compact subsets E T such that the set of all f A(D) with the property {S n (f, ) E : n N 0 } is dense in C(E) is a dense G δ -subset of A(D). Here C(E) denotes the Banach space of all continuous functions w : E C endowed with the maximum norm. It follows from [8, Cor. 3.3] that there are countable compact subsets E T with a single accumulation point such that, for no function f A(D), the set {S n (f, ) E : n N 0 } is dense in C(E). On other hand, recent results in [2] show that, for any compact subset E T of arc length measure 0, the set of all f H p (D), p [1, ), with the property is a dense G δ -subset of H p (D). {S n (f, ) E : n N 0 } is dense in C(E) Remar. After submitting this paper Jürgen Müller informed us that he and George Costais have independently found Theorem 1. Their proof is based on Fejér polynomials F n (z) = 2, n z n, instead of Landau s extremal functions and has not been published up to now.

4 162 Herzog G., Kunstmann P.Ch. 2. Landau s extremal functions In the sequel let that is and a 0 = 1, a = ( ) 1/2 a := ( 1) ( N 0 ), (2 1) ( N), K n (z) := a 0 + a 1 z + a 2 z a n z n (n N 0 ). In [9] Landau proved that 0 / K n (D), that γ n := sup{ S n (f, 1) : f A(D), f 1} (n N 0 ) is a maximum which is attained at f(z) = R n (z) := zn K n (1/z) K n (z) = a n + a n 1 z + a n 2 z a 0 z n a 0 + a 1 z + a 2 z a n z n, and that In particular we have S n (R n, 1) = γ n = a 2 log(n) π (n ). S n (R n, 1) = γ n (n ). Some immediate facts on the rational functions R n are R n (z) = 1 (z T), R n = 1. To utilize the functions R n in the proof of Theorem 1 we prove the following theorem, which seems to us of some interest on its own. It asserts that the functions S n (R n, ), n N 0, have a common majorant on D \ {1}. We shall use this on T \ {1}. Theorem 3. For each z D \ {1} we have S n (R n, z) 4 1 z (n N 0 ). Proof: We fix n N 0, and consider S n (R n, z) = b z. Observe that here b = b,n depend on n, but we shall ignore this in notation. Since the a are real, it is clear that b R, and from R n = 1 we get by

5 Universally divergent Fourier series via Landau s extremal functions 163 Cauchy s formula b 1 ( = 0,...,n). In [13], Wintner proved that in fact b [0, 1] ( = 0,...,n). From (a 0 + a 1 z + a 2 z a n z n )R n (z) = a n + a n 1 z + a n 2 z a 0 z n we obtain (cf. [13]) m a m b = a n m (m = 0,...,n). By setting P := diag ([1,...,1], ) R (n+1) (n+1) (the band matrix with 1 s in the -th subdiagonal and 0 s else), b := (b 0, b 1,..., b n ) R n+1 and a := (a n, a n 1,...,a 0 ) R n+1, the linear system above can be written as ( n ) a P b = a. We claim that ( n ) 1 a P = a 0 P 0 + (a 1 a 0 )P 1 + (a 2 a 1 )P (a n a n 1 )P n. In order to see this we start with the Taylor expansion ( ) α (1 + x) α = x ( x < 1) for α = 1 2 and α = 1 2 We conclude that and tae the Cauchy product 1 = (1 + x) 1/2 (1 + x) 1/2 = ν=0 ( 1/2 ν )( 1/2 ν ( )( ) 1/2 1/2 = δ 0 ( N 0 ). ν ν ν=0 For N we get by a well-nown formula [( ) 1/2 a a 1 = ( 1) + Then I + (a a 1 )P = I + =1 ) x. ( )] ( ) 1/2 1/2 = ( 1). 1 ( ) 1/2 ( 1) P = =1 ( ) 1/2 ( 1) P

6 164 Herzog G., Kunstmann P.Ch. and a P = I + ( ) 1/2 ( 1) P = =1 ( ) 1/2 ( 1) P. We tae the product, respect P P l = P l+ (setting P j = 0 for j > n), and arrive at ( n a P )(I + ) (a a 1 )P =1 ( n ( ) 1/2 )( n ( 1/2 ) = ( 1) P ( 1) )P ( )( ) 1/2 1/2 = ( 1) +l P +l l =,l=0 ( 1) ( ( )( ) 1/2 1/2 ) P = I, ν ν ν=0 and our claim is proved. We now rewrite ( n ) 1 a P = a 0 P 0 + (a 1 a 0 )P 1 + (a 2 a 1 )P (a n a n 1 )P n ( a0 = I 2 P 1 + a 1 4 P a ) n 1 2n P n =: I Q. Now a is an increasing vector with positive entries, therefore each vector a P +1a ( = 0,...,n 1) is an increasing vector. Thus, with c := Qa we have b = a c where a and c are increasing, and from a, b [0, 1], c 0 ( = 0,...,n) we get that all entries of c are in [0, 1]. This proves that b has variation Next, we consider n 1 b +1 b 2. (1 z)s n (R n, z) = (1 z) b z = = b 0 b n z n+1 + n+1 b z b 1 z =1 (b b 1 )z, =1

7 thus, for each z D Universally divergent Fourier series via Landau s extremal functions 165 (1 z)s n (R n, z) b 0 + b n + b b = 4. =1 Remar. We note that the constant 4 in Theorem 3 is most certainly not optimal. Numerical experiments suggest that the vector b is always convex, and that S n (R n, z) 2 1 z 3. Universality criterion (n N 0, z D \ {1}). To prove Theorem 1 we use the universality criterion of Grosse-Erdmann [6, Theorem 1.57]. Theorem 4. Let X be a complete metric space, Y a separable metric space, and T n : X Y, n N 0, continuous maps. Denoting U = U((T n ) n N0 ) as the set of all x X such that the following assertions are equivalent. {T n x : n N 0 } is dense in Y, (1) The family (T n ) n N0 is topologically transitive, i.e. for any pair U X, V Y of nonempty open sets, there is some n N 0 such that T n (U) V. (2) The set U is a dense G δ -subset of X. (3) The set U is dense in X. In the proof of Theorem 1 we shall apply Theorem 4 to the situation X = A(D), Y = C E, T n = L n (n N 0 ) and chec that (1) holds. Here we already observe that the equivalence of (2) and (3) in Theorem 4 shows that Theorem 1 implies Theorem 2. Proof of Theorem 2: Assumption (3) of Theorem 2 implies that L n : X C E is continuous (n N 0 ). By Theorem 1, the set U((L n ) n N0 ) is dense in A(D). Since A(D) is densely and continuously embedded in X, we obtain that U((L n ) n N0 ) is dense in X. Hence also the superset U(( L n ) n N0 ) is dense in X, and is a dense G δ -subset of X by Theorem 4. Open Problem. It would be interesting to now whether U(( L n ) n N0 ) is a dense G δ -set also in case X = H (D). Clearly U((L n ) n N0 ) H (D), but assumption (1) of Theorem 2 is not satisfied.

8 166 Herzog G., Kunstmann P.Ch. We prepare the proof of Theorem 1 and note that if, in the situation of Theorem 4, D is a dense subset of X, and if d and ρ are the metrics on X and Y, respectively, then topological transitivity of (T n ) n N0 is equivalent to the following condition: ε > 0 (x, y) D Y (n, z) N 0 X : d(x, z) < ε ρ(t n z, y) < ε. In our situation, we let D denote the set of all polynomials, which is nown to be a dense subset of X = A(D), [12, p. 366]. Moreover, let E = {z : N} with z z j ( j) and let Y = C E be endowed with the usual Fréchet metric ρ(v, w) = 4. Proof of Theorem 1 =1 1 2 v(z ) w(z ) 1 + v(z ) w(z ). Let ε > 0, p D, n 0 := grad p and w C E. Let m N be such that =m < ε. It is sufficient to find a function f A(D) and some n n 0 such that f < ε L n (f)(z ) = S n (f, z ) = w(z ) p(z ) =: ζ ( = 1,..., m). Once such n and f are nown we set g := f + p and obtain p g < ε and ρ(l n (g), w) = ρ((s n (f, ) + p) E, w) = ρ(s n (f, ) E, w p E ) =m < ε. To construct f and n we set ζ = (ζ 1,...,ζ m ) and we mae the ansatz f(z) = λ 1 R n (z 1 z) + + λ m R n (z m z) where λ 1,..., λ m C have to be chosen. Recalling that R n (z ) = 1 ( = 1,...,m) we already find f λ 1 where 1 denotes the l 1 -norm on C m. Now we consider the m m-matrix Q n with entries q (n) j = S n (R n (z j ), z ) (, j {1,...,m}). Note that if λ = (λ 1,..., λ m ) solves Q n λ = ζ, then S n (f, z ) = ζ ( = 1,...,m).

9 Universally divergent Fourier series via Landau s extremal functions 167 Moreover q (n) = S n(r n (z ), z ) = S n (R n, 1) = γ n, and according to Theorem 3 we have, for j, q (n) j = S n(r n (z j ), z ) ( = 1,...,m), 4 1 z j z = 4 z z j c, where { } 4 c := max : j, = 1,...,m, j z z j does not depend on n. Since γ n (n ) we thus find I Q n γ n 0 (n ). So for large n we have by Neumann s series Q 1 n = 1 ( ( I I Q )) 1 n = 1 ( I Q ) n r, γ n γ n γ n γ n and we conclude that Q 1 n 0 as n. In particular, we can choose n N such that This ends the proof. λ 1 = Q 1 n ζ 1 < ε. r=0 References [1] Anderson J.M., Clunie J., Pommerene Ch., On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), [2] Beise H.-P., Müller J., On the boundary behaviour of Taylor series in Hardy and Bergman spaces, preprint, [3] Bernal-González L., Lineability of universal divergence of Fourier series, Integral Equations Operator Theory 74 (2012), [4] Fejér L., Über die Koeffizientensumme einer beschränten und schlichten Potenzreihe, Acta Math. 49 (1926), [5] Grosse-Erdmann K.-G., Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), [6] Grosse-Erdmann K.-G., Peris Manguillot A., Linear Chaos, Universitext, Springer, London, [7] Kahane J.-P., Baire s category theorem and trigonometric series, J. Anal. Math. 80 (2000), [8] Katsoprinais E., Nestoridis V., Papachristodoulos C., Universality and Cesàro summability Comput. Methods Funct. Theory 12 (2012), [9] Landau E., Gaier D., Darstellung und Begründung einiger neuerer Ergebnisse der Funtionentheorie, Springer, Berlin, 1986.

10 168 Herzog G., Kunstmann P.Ch. [10] Müller J., Continuous functions with universally divergent Fourier series on small subsets of the circle, C.R. Math. Acad. Sci. Paris 348 (2010), [11] Pogosyan N.B., Universal Fourier series, Uspehi Mat. Nau 38 (1983), [12] Rudin W., Real and Complex Analysis, McGraw-Hill Boo Co., New Yor, [13] Wintner A., The theorem of Eneström and the extremal functions of Landau-Schur, Math. Scand. 5 (1957), [14] Zygmund A., Trigonometric Series. Vol. I, II, Cambridge University Press, Cambridge, Institut für Analysis, Karlsruher Institut für Technologie (KIT), D Karlsruhe, Germany Gerd.Herzog2@it.edu, Peer.Kunstmann@it.edu (Received November 13, 2014, revised January 21, 2015)