20 years Universal Taylor Series

Transcription

1 Doubly 20 years Vagia Vlachou, University of Patras Mathematical Analysis in Athens Katavolos and Nestoridis, December 2017

2 Doubly Orbit of a vector X topological vector space T : X X continuous and linear operator The sequence {T n x : n N} is called the orbit of the vector x X.

3 Doubly Why study the orbit? Dynamical Systems (study of a population).

4 Doubly Why study the orbit? Dynamical Systems (study of a population). Convergence-Fixed point Theorem.

5 Doubly Why study the orbit? Dynamical Systems (study of a population). Convergence-Fixed point Theorem. Closed invariant subspace that contains x.

6 Doubly Hypercyclicity X topological vector space T : X X continuous and linear operator T is called hypercyclic if {T n x : n N} is dense in X, for some x X. The term was first used by Beauzamy around 1986.

7 Doubly Examples of Hypercyclic Operators The Doubling map on the circle: T : T T, T (z) = z 2 z 0 = e 2πir, r / Q The orbit {T n z 0 : n N} = {e 4nπir : n N} is dense in T.

8 Doubly Examples of Hypercyclic Operators Irrational Circle Rotation: T : T T, T (z) = e 2πir z, r / Q z 0 = e 2πir The orbit {T n z 0 : n N} = {e 2nπir : n N} is dense in T.

9 Doubly Examples of Hypercyclic Operators Birkhoff s Operator [1929]: T : H(C) H(C), T (f )(z) = f (z + 1). There exists entire function f such that {f (z + n) : n N} is dense in H(C).

10 Doubly Sketch of proof: Approach I, Constructive (P j ) j sequence of polynomials dense in H(C). (D j ) j disjoint closed disks of radius j (E j ) j increasing sequence of closed disks, with D j E j, j.

11 Doubly Sketch of proof: Approach I, Constructive Q 1 = P 1. Q 2 E1 < 1 2 and sup z D 2 Q 2 (z) [P 2 (z c 2 ) Q 1 (z)] < 1 2.

12 Doubly Sketch of proof: Approach I, Constructive Q 1 = P 1. Q 2 E1 < 1 2 and sup Q 2 (z) [P 2 (z c 2 ) Q 1 (z)] < 1 z D 2 2. In general, Q n En 1 < 1 and 2 n 1 sup z D n n k=1 Q k (z) P n (z c n ) < 1 2 n 1.

13 Doubly Sketch of proof: Approach I, Constructive Q 1 = P 1. Q 2 E1 < 1 2 and sup Q 2 (z) [P 2 (z c 2 ) Q 1 (z)] < 1 z D 2 2. In general Q n En 1 < 1 2 n n 1 sup Q k (z) P n (z c n ) < 1,, z D n 2n 1 sup z n k=1 n k=1 Q k (z + c n ) P n (z) < 1 2 n 1.

14 Doubly Sketch of proof: Approach I, Constructive Q 1 = P 1. Q 2 E1 < 1 2 and sup Q 2 (z) [P 2 (z c 2 ) Q 1 (z)] < 1 z D 2 2. In general Q n En 1 < 1 2 n n 1 sup Q k (z) P n (z c n ) < 1,, z D n 2n 1 sup z n k=1 n k=1 Q k (z + c n ) P n (z) < 1 2 n 1. f (z) = + k=1 Q k (z)

15 Doubly Sketch of proof: Approach II, Birkhoff s transitivity theorem X seperable complete metric space without isolated points T : X X continuous T is hypercylic, if and only if, U, V X open, non empty there exists n N with: T n (U) V. The set of hypercyclic vectors is G δ and dense in X.

16 Doubly Sketch of proof: Approach II, Birkhoff s transitivity theorem V = {g H(C) : g f 1 z n1 < r 1 }.

17 Doubly Sketch of proof: Approach II, Birkhoff s transitivity theorem V = {g H(C) : g f 1 z n1 < r 1 } U = {g H(C) : g f 2 z n2 < r 2 }.

18 Doubly Sketch of proof: Approach II, Birkhoff s transitivity theorem V = {g H(C) : g f 1 z n1 < r 1 } U = {g H(C) : g f 2 z n2 < r 2 } There exists n N such that: D(0, n 1 ) D(n, n 2 ) = Runge s theorem: P near f 1 in D(0, n 1 ) P near f 2 (z n) in D(n, n 2 ).

19 Doubly Sketch of proof: Approach III, Hypercyclicity critetion e λ (z) = e λz is Eigenvector of T of eigenvalue e λ.

20 Doubly Sketch of proof: Approach III, Hypercyclicity critetion e λ (z) = e λz is Eigenvector of T of eigenvalue e λ. T (e λ )(z) = e λ (z + 1) = e λ(z+1) = e λ e λ (z).

21 Doubly Sketch of proof: Approach III, Hypercyclicity critetion e λ (z) = e λz is Eigenvector of T of eigenvalue e λ. T (e λ )(z) = e λ (z + 1) = e λ(z+1) = e λ e λ (z). X 0 = span{x X : Tx = λx with λ < 1} Y 0 = span{x X : Tx = λx with λ > 1}

22 Doubly Sketch of proof: Approach III, Hypercyclicity critetion e λ (z) = e λz is Eigenvector of T of eigenvalue e λ. T (e λ )(z) = e λ (z + 1) = e λ(z+1) = e λ e λ (z). X 0 = span{x X : Tx = λx with λ < 1} Y 0 = span{x X : Tx = λx with λ > 1} span{e λ : λ Λ} is dense in H(C) if Λ C has an accumulation point.

23 Doubly Examples of Hypercyclic Operators Differential Operator Maclane [1952]: D : H(C) H(C), D(f ) = f. There exists entire function f such that {f (n) : n N} is dense in H(C).

24 Doubly Examples of Hypercyclic Operators C. Blaire and L. Rubel (1984) proved that there exists a function realising both approximations (constructive approach).

25 Doubly Examples of Hypercyclic Operators Multiple of Bachward Shift (Rolewicz [1969]): 2B is hypercyclic. B : l 2 l 2, B(x 1, x 2, x 3...) = (x 2, x 3,...)

26 Doubly Examples of Hypercyclic Operators 1 Ω = {s C : 2 < Re(s) < 1}

27 Doubly Examples of Hypercyclic Operators 1 Ω = {s C : 2 < Re(s) < 1} K Ω compact

28 Doubly Examples of Hypercyclic Operators 1 Ω = {s C : 2 < Re(s) < 1} K Ω compact Voronin s Theorem [1975]: Every f H (Ω) (holomorphic without zeros) can be uniformly approximated on K by functions in {ζ(s + it) : t > 0}.

29 Doubly Examples of Hypercyclic Operators 1 Ω = {s C : 2 < Re(s) < 1} K Ω compact Voronin s Theorem [1975]: Every f H (Ω) (holomorphic without zeros) can be uniformly approximated on K by functions in {ζ(s + it) : t > 0}. Alex Reich [1980] : Every f H (Ω) (holomorphic without zeros) can be uniformly approximated on K by functions in {ζ(s + in) : n N}.

30 Doubly Hypercyclicity-Universality X topological vector space T : X X continuous and linear operator T is called hypercyclic if {T n x : n N} is dense in X, for some x X. X, Y topological vector spaces T n : X Y, n N continuous and linear operators (T n ) n N is called universal if {T n x : n N} is dense in Y, for some x X.

31 Doubly Examples of Universality Fekete before 1914, there exists formal power series + k=1 a kx k with the property for every h : [ 1.1] R continuous with h(0) = 0 there exists (λ n ) n such that sup x [ 1,1] λ n a k x k h(x) 0. k=1

32 Doubly Examples of Universality Selesnev (1951), there exists formal power series + k=1 a kz k with universal properties in C {0}.

33 Doubly Examples of Universality Chui-Parnes (1971) and W. Luh 1970: There exists f ; D C holomorhic such that {S N (f ), N N} is dense in A(K), for every compact set K M, K (C D) =, M = {K C : K compact set and K c connected } A(K) = {g C(K) : g holomorhic in K o }.

34 Doubly in the disk V. Nestoridis (1996): There exists f : D C holomorhic such that {S N (f ), N N} is dense in A(K), for every compact set K M, K (C D) =.

35 Doubly Preliminaries Fix a simply connected domain Ω C and a point ζ 0 Ω. M Ω c = {K M : K Ω = } If f H(Ω), we use the notation S N (f, ζ 0 )(z) = N n=0 f (n) (ζ 0 ) (z ζ 0 ) n, N = 0, 1,.... n!

36 Doubly A function f H(Ω) belongs to the class U(Ω, ζ 0 ) if {S n (f, ζ 0 ) : n N} is dense in A(K), for every K M Ω c.

37 Doubly A function f H(Ω) belongs to the class U(Ω, ζ 0 ) if {S n (f, ζ 0 ) : n N} is dense in A(K), for every K M Ω c. The class U(Ω, ζ 0 ) is G δ and dense in H(Ω).

38 Doubly A function f H(Ω) belongs to the class U(Ω, ζ 0 ) if {S n (f, ζ 0 ) : n N} is dense in A(K), for every K M Ω c. The class U(Ω, ζ 0 ) is G δ and dense in H(Ω). V. Nestoridis, Universal Taylor series., Ann. Inst. Fourier (Grenoble) 46 (1996), V. Nestoridis, An extension of the notion of universal Taylor series., in Computational Methods and Function Theory 1997 (Nicosia), pp , Ser. Approx. Decompos., 11, World Sci. Publ., River Edge, NJ,1999.

39 Doubly Subclasses of Let {λ n } n N be a sequence of positive integers.

40 Doubly Subclasses of Let {λ n } n N be a sequence of positive integers. A function f H(Ω) belongs to the class U(Ω, ζ 0, {λ n } n N ) if {S λn (f, ζ 0 ) : n N} is dense in A(K), for every K M Ω c.

41 Doubly Subclasses of Let {λ n } n N be a sequence of positive integers. A function f H(Ω) belongs to the class U(Ω, ζ 0, {λ n } n N ) if {S λn (f, ζ 0 ) : n N} is dense in A(K), for every K M Ω c. The class U(Ω, ζ 0, {λ n } n N ) is G δ and dense in H(Ω), if and only if, the sequence {λ n } n N is unbounded.

42 Doubly Subclasses of Let {λ n } n N be a sequence of positive integers. A function f H(Ω) belongs to the class U(Ω, ζ 0, {λ n } n N ) if {S λn (f, ζ 0 ) : n N} is dense in A(K), for every K M Ω c. The class U(Ω, ζ 0, {λ n } n N ) is G δ and dense in H(Ω), if and only if, the sequence {λ n } n N is unbounded. Remark: These classes appeared in the problem of finding a dense vector space inside U(Ω, ζ 0 ) 0.

43 Doubly Disjoint Hypercyclicity-Definition X topological vector space T, S : X X continuous and linear operators T and S are called disjoint hypercyclic if {(T n x, S n x) : n N} is dense in X X, for some x X.

44 Doubly Disjoint Hypercyclicity-Definition X topological vector space T, S : X X continuous and linear operators T and S are called disjoint hypercyclic if {(T n x, S n x) : n N} is dense in X X, for some x X. Remark: If T and S are disjoint hypercyclic, then they are both hypercyclic. The converse is in generall not true.

45 Doubly Disjoint Hypercyclicity-Definition X topological vector space T, S : X X continuous and linear operators T and S are called disjoint hypercyclic if {(T n x, S n x) : n N} is dense in X X, for some x X. Remark: If T and S are disjoint hypercyclic, then they are both hypercyclic. The converse is in generall not true. Luis Bernal-Gonzalez, Disjoint hypercyclic operators, Studia 182 (2007) J. Bès, A. Peris,, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336 (2007)

46 Doubly Disjoint Universality-Definition X topological vector space T n, S n : X X continuous and linear operators (T n ) n and (S n ) n are called disjoint universal if {(T n x, S n x) : n N} is dense in X X, for some x X.

47 Doubly Disjoint Universality-Example G.Costakis, V. Identical approximative sequence for various notions of universality (2005) Disjoint universality for four sequences of operators.

48 Doubly Doubly on the disk Let {λ n } n N be a strictly increasing sequence of positive integers.

49 Doubly Doubly on the disk Let {λ n } n N be a strictly increasing sequence of positive integers. A function f H(D) belongs to the class U CT ({λ n } n N, {n} n N ) {(S λn (f, 0), S n (f, 0)) : n N} is dense in A(K) A(K) for every K M D c.

50 Doubly Doubly on the disk Let {λ n } n N be a strictly increasing sequence of positive integers. A function f H(D) belongs to the class U CT ({λ n } n N, {n} n N ) {(S λn (f, 0), S n (f, 0)) : n N} is dense in A(K) A(K) for every K M D c. U CT ({λ n } n N, {n} n N ) lim sup n λ n n = +.

51 Doubly Doubly on the disk Let {λ n } n N be a strictly increasing sequence of positive integers. A function f H(D) belongs to the class U CT ({λ n } n N, {n} n N ) {(S λn (f, 0), S n (f, 0)) : n N} is dense in A(K) A(K) for every K M D c. U CT ({λ n } n N, {n} n N ) lim sup n λ n n = +. G. Costakis, N. Tsirivas, Doubly universal Taylor series, J. Approx. Theory 180 (2014)

52 Doubly Doubly on simply connected domains Let {λ n } n N be a strictly increasing sequence of positive integers.

53 Doubly Doubly on simply connected domains Let {λ n } n N be a strictly increasing sequence of positive integers. A holomorfic function f belongs to the class U (ζ 0) double (Ω, {λ n} n N, {n} n N ) if {(S λn (f, ζ 0 ), S n (f, ζ 0 )) : n N} is dense in A(K 1 ) A(K 2 ) for every K 1, K 2 M Ω c.

54 Doubly Doubly on simply connected domains Let {λ n } n N be a strictly increasing sequence of positive integers. A holomorfic function f belongs to the class U (ζ 0) double (Ω, {λ n} n N, {n} n N ) if {(S λn (f, ζ 0 ), S n (f, ζ 0 )) : n N} is dense in A(K 1 ) A(K 2 ) for every K 1, K 2 M Ω c. U (ζ 0) double (Ω, {λ λ n n} n N, {n} n N ) lim sup n n = +.

55 Doubly Doubly on simply connected domains Let {λ n } n N be a strictly increasing sequence of positive integers. A holomorfic function f belongs to the class U (ζ 0) double (Ω, {λ n} n N, {n} n N ) if {(S λn (f, ζ 0 ), S n (f, ζ 0 )) : n N} is dense in A(K 1 ) A(K 2 ) for every K 1, K 2 M Ω c. U (ζ 0) double (Ω, {λ λ n n} n N, {n} n N ) lim sup n n = +. N. Chatzigiannakidou and V., Doubly universal Taylor series on simply connected domains (2016).

56 Doubly Multiple on simply connected domain Let {λ (σ) n } n N, σ = 1, 2,..., σ 0 be finite number of sequences of positive integers. A function f H(Ω) belongs to the class U (ζ 0) mult ({λ(1) n } n N, {λ (2) n } n N,..., {λ (σ 0) n } n N ) if {(S λ (1) n (f, ζ 0 ),..., S (σ λ 0 )(f, ζ 0 )) : n N} is dense in A(K 1 )... A(K σ0 ) n for every K 1,..., K σ0 M Ω c. Question: Can we find necessary and sufficient condition so that the above class of functions is non empty?

57 Doubly Ordering of sequences Let {λ (σ) n } n N, σ = 1, 2,..., σ 0, σ 0 N be a finite number of sequences of natural numbers. We say that these sequences are well ordered if lim sup n λ (σ+1) n λ (σ) n lim sup n λ (σ) n λ (σ+1) n, σ = 1, 2,..., σ 0 1. Let {λ (σ) n } n N, σ = 1, 2,..., σ 0 be a finite number of sequences of natural numbers. There exists a rearrangement {λ (π(σ)) n } n N, σ = 1, 2,..., σ 0 which is well ordered.

58 Doubly Main Result The class U (ζ 0) mult ({λ(1) n } n N, {λ (2) n } n N,..., {λ (σ 0) n } n N ) is non-empty, if and only if, there exists a strictly increasing sequence of natural numbers {µ n } n N such that λ (σ+1) lim n λ(1) µ µ n = + and lim n n λ (σ) = +, σ = 1, 2,..., σ 0 1. µ n

59 Doubly Bernstein-Walsh theorem Let K M. If f is analytic in a neighborhood U of K, then: where θ = Notation: lim sup d n (f, K) 1 n θ < 1, n { sup C U exp( g C K (z, )), if c(k) > 0, 0, if c(k) = 0 d n (f, K) = inf{ f p K : p polynomial degp n}.

60 Doubly Bernstein-Walsh type theorem I Let K M. If f n, n = 1, 2,... is a {τ n } locally controlled sequence of functions analytic in a neighborhood U of K, then: lim sup d τn (f n, K) 1 τn θ U < 1, n where θ U is as before and {τ n } n N is any unbounded sequence of natural numbers.

61 Doubly Bernstein-Walsh type theorem II Let K M, 0 / K. If f n, n = 1, 2,... is a {τ n } locally controlled sequence of functions analytic in a neighborhood U of K, then: lim sup d τn,σn (f n, K) 1 τn θ U < 1, n where θ U is as before and τn σ n Notation: If m n +. d n,m (f, K) = inf{ f p K : p polynomial m deg p, degp n}. Remark: If the functions f n are constant zero on the component of K containing zero, then the result is true for every K M.

62 Doubly Crucial Density Result g H(Ω), ε > 0, L Ω compact K 1,..., K σ0 M Ω c p 1,..., p σ0, polynomials.

63 Doubly Crucial Density Result g H(Ω), ε > 0, L Ω compact K 1,..., K σ0 M Ω c p 1,..., p σ0, polynomials.?f H(Ω) and n 0 N f g L < ε S λ (σ) n 0 (f, ζ 0 ) p σ Kσ < ε, σ = 1,..., σ 0.

64 Doubly Sketch of Proof Without loss of generallity, we assume that: λ (σ+1) lim n λ(1) n n = + and lim n λ (σ) n Apply Runge s Theorem to fix p polynomial with: p g L < ε = +, σ = 1, 2,..., σ 0 1. and p p 1 K1 < ε

65 Doubly Sketch of Proof Apply the Bernstein-type theorem II on K 2 ζ 0 L ζ 0 for the functions: { [p 2 (z + ζ 0 ) p(z + ζ 0 )], z K 2 ζ 0 f n (z) = 0, z L ζ 0 This way we obtain a sequence of polynomials Q n (1) such that: The degree of the terms of Q n (1) varies between λ (1) n + 1 and λ (2) n. Q n (1) n (z ζ 0 ) L 0. p(z) + Q n (1) n (z ζ 0 ) p 2 (z) K2 0.

66 Doubly Sketch of Proof Then the polynomial g 1 (z) = p(z) + Q n (1) (z ζ 0 ) for n large enough has the following properties: g 1 g L < ε S λ (σ) n 0 (g 1, ζ 0 ) p σ Kσ < ε, σ = 1, 2. Note that S λ (1) n 0 (g 1, ζ 0 ) = p and S λ (2) n 0 (g 1, ζ 0 ) = g 1.

67 Doubly Sketch of Proof Apply the Bernstein-type theorem II on K 3 ζ 0 L ζ 0 for the functions: { [p 2 (z + ζ 0 ) p(z + ζ 0 ) Q n (1) (z)], z K 2 ζ 0 f n (z) = 0, z L ζ 0 This way we obtain a sequence of polynomials Q n (2) such that: The degree of the terms of Q n (2) varies between λ (2) n + 1 and λ (3) n. Q n (2) n (z ζ 0 ) L 0. p(z) + Q n (1) (z ζ 0 ) + Q n (2) n (z ζ 0 ) p 3 (z) K3 0.

68 Doubly Sketch of Proof Then the polynomial g 2 (z) = p(z) + Q (1) n (z ζ 0 ) + Q (2) n (z ζ 0 ) for n large enough has the following properties: g 2 g L < ε S λ (σ) n 0 (g 2, ζ 0 ) p σ Kσ < ε, σ = 1, 2, 3. After a finite number of steps we obtain the result.

69 Doubly Sketch of Proof-Negative result We argue by a contrudiction and we assume that there exists a function f U (ζ 0) mult ({λ(1) n } n N, {λ (2) n } n N,..., {λ (σ 0) n } n N ). Fix a sequence of sets {E k } such that: (i) E k M Ω c, k = 1, 2,... (ii) k E k is closed and non-thin at. Let ξ Ω c : ξ ζ 0 = R = dist(ζ 0, Ω c ).

70 Doubly Sketch of Proof-Negative result Then: S λ (σ) n k (f, ζ 0 ) Ek {ξ} < 1 k, σ {1, 2,..., σ 0} odd (1) S λ (σ) n k (f, ζ 0 ) 1 Ek {ξ} < 1 k, σ {1, 2,..., σ 0} even. (2) for suitable {n k } k N.

71 Doubly Sketch of Proof-Negative result Case I: lim sup k λ (2) n k λ (1) n k < +. Auxiliary polynomials: ( ) (1) R λ ( ) n k p k (z) = S (2) z ζ λ (f, ζ 0 )(z) S (1) 0 n k λ (f, ζ 0 )(z), k I n k where R = dist(ω c, ζ 0 ) > 0.

72 Doubly Sketch of Proof-Negative result The polynomials p k are near zero on a closed and non-thin at set. J. Müller and A. Yavrian: they are near zero everywhere on C. So, S (2) λ (f, ζ 0 )(ξ) S (1) n k λ (f, ζ 0 )(ξ) is near 0 and we arrive to a n k λ (2) n k contradiction, because this is near 1. Thus lim sup k λ (1) = +, we n k pass to a subsequence and repeat the argument for the next index.

73 Doubly Goodbye Thank you very much for your attention. I hope you enjoyed the talk.