Cuntz algebras, generalized Walsh bases and applications

Transcription

1 Cuntz algebras, generalized Walsh bases and applications Gabriel Picioroaga November 9, 2013 INFAS 1 / 91

2 Basics H, separable Hilbert space, (e n ) n N ONB in H: e n, e m = δ n,m span{e n } = H v = k=1 v, e k e k lim n v n k=1 v, e k e k = 0 2 / 91

3 Basics H, separable Hilbert space, (e n ) n N ONB in H: e n, e m = δ n,m span{e n } = H v = k=1 v, e k e k lim n v n k=1 v, e k e k = 0 In applications: H = L 2 space and v encodes a signal, state, image, measurable function. Good Approximation: least mean square deviation 3 / 91

4 Example Fourier: ( 1 2π e inx ) n Z ONB on L 2 [ π, π] Wavelet : {2 n/2 ψ(2 n t k) n, k Z} ONB in L 2 (R) Walsh: discrete sine-cosine versions, ±1 on dyadic intervals (exp(λ 2πx) λ Λ exponential bases on some L 2 (fractals) 4 / 91

5 Main ideas and layout of the talk: Cuntz relations generate a diversity of bases: Examples, Old and New (generalized Walsh) Zoom in on the new Walsh, study structure properties (How different from the old one is it?) Possible applications of the generalized Walsh. 5 / 91

6 Set Up R- d d expansive. B R d, N = B. IFS: τ b (x) = R 1 (x + b) (x R d, b B) Hutchinson:! attractor (X B, µ B ) invariant for the IFS. µ B is invariant for r : X B X B r(x) = τ 1 b (x), if x τ b(x B ) 6 / 91

7 Set Up R- d d expansive. B R d, N = B. IFS: τ b (x) = R 1 (x + b) (x R d, b B) Hutchinson:! attractor (X B, µ B ) invariant for the IFS. µ B is invariant for r : X B X B Example IFS : τ j (x) = x+j N, j = 0, 1,..., N 1 r(x) = τ 1 b (x), if x τ b(x B ) Attractor: X = [0, 1], with λ the Lebesgue measure r(x) = Nx mod 1 IFS : τ j (x) = x+j N, j = 0, 1,..., N 1 7 / 91

8 Cuntz Relations Definition O N : S i S j = δ i,j I, N 1 i=0 S i S i = I 8 / 91

9 QMFs QMF basis = multiresolution for the wavelet representation associated to a filter m 0. Definition A QMF basis is a set of N QMF s 1 N r(w)=z m 0, m 1,..., m N 1 such that m i (w)m j (w) = δ ij, (i, j {0,..., N 1}, z X ) 9 / 91

10 QMF bases and Cuntz algebra representations Proposition Let (m i ) N 1 i=0 be a QMF basis. The operators on L2 (X, µ) S i (f ) = m i f r, i = 0,..., N 1 are isometries and form a representation of the Cuntz algebra O N. 10 / 91

11 Main Result Theorem H Hilbert space, (S i ) N 1 i=0 Cuntz representation of O N. E orthonormal, X top.space, f : X H norm continuous function and: 11 / 91

12 Main Result Theorem H Hilbert space, (S i ) N 1 i=0 Cuntz representation of O N. E orthonormal, X top.space, f : X H norm continuous function and: 1 E = N 1 i=0 S ie. 2 span{f (t) : t X } = H and f (t) = 1, for all t X. 12 / 91

13 Main Result Theorem H Hilbert space, (S i ) N 1 i=0 Cuntz representation of O N. E orthonormal, X top.space, f : X H norm continuous function and: 1 E = N 1 i=0 S ie. 2 span{f (t) : t X } = H and f (t) = 1, for all t X. 3 on the range of f the Cuntz isometries are like multiplication-dilation operators 4 c 0 X such that f (c 0 ) spane. 13 / 91

14 Main Result Theorem H Hilbert space, (S i ) N 1 i=0 Cuntz representation of O N. E orthonormal, X top.space, f : X H norm continuous function and: 1 E = N 1 i=0 S ie. 2 span{f (t) : t X } = H and f (t) = 1, for all t X. 3 on the range of f the Cuntz isometries are like multiplication-dilation operators 4 c 0 X such that f (c 0 ) spane. 5 If the Ruelle (transfer) operator admits as fixed point a function h constant on f 1 (spane) then h is constant. 14 / 91

15 Main Result Theorem H Hilbert space, (S i ) N 1 i=0 Cuntz representation of O N. E orthonormal, X top.space, f : X H norm continuous function and: 1 E = N 1 i=0 S ie. 2 span{f (t) : t X } = H and f (t) = 1, for all t X. 3 on the range of f the Cuntz isometries are like multiplication-dilation operators 4 c 0 X such that f (c 0 ) spane. 5 If the Ruelle (transfer) operator admits as fixed point a function h constant on f 1 (spane) then h is constant. Then E is an orthonormal basis for H. 15 / 91

16 In applications: f (t) = exp t on L 2 (X B, µ B ) S l (g) = e l g r, (B, L) Hadamard pair S i (g) = m i g r, m i = N N 1 j=0 a ij χ [j/n,(j+1)/n] E = {S l1 S l2 S ln (exp c )}, c extreme cycle point. E = {S i1 S i2 S in (1)} 16 / 91

17 Consequences 1-dimensional: 0 B R, R > 1, 1 R B admits a set L as spectrum. C1. E = {S w (exp c ) : c extreme cycle point} is ONB in L 2 (µ B ) made of piecewise exponential functions. S l1...s ln e c (x) = e l1 (x)e l2 (rx)...e ln (r n 1 x)e c (r n x) C2. When B Z, L Z, and R Z then Λ such that {e λ : λ Λ} is ONB for L 2 (µ B ). 17 / 91

18 Consequences 1-dimensional: 0 B R, R > 1, 1 R B admits a set L as spectrum. C1. E = {S w (exp c ) : c extreme cycle point} is ONB in L 2 (µ B ) made of piecewise exponential functions. S l1...s ln e c (x) = e l1 (x)e l2 (rx)...e ln (r n 1 x)e c (r n x) C2. When B Z, L Z, and R Z then Λ such that {e λ : λ Λ} is ONB for L 2 (µ B ). Example Cantor s (X 1/4, µ 1/4 ) admits exp ONB: R = 4, B = {0, 2}, spectrum L = {0, 1} Example R = 3, B = {0, 2}, L = {0, 3 4 } spectrum of 1 3B: Middle third Cantor set which is known not to admit exponential bases. 18 / 91

19 Consequences C3. Walsh Bases: [0, 1] is the attractor of the IFS: τ 0 x = x 2, τ 1x = x+1 2. rx = 2xmod1. m 0 = 1, m 1 = χ [0,1/2) χ [1/2,1) form a QMF basis. E := {S w 1 : w {0, 1} } is an ONB for L 2 [0, 1], the Walsh basis. Description: For n = lk=0 i k2 k, the n th Walsh function : W n (x) = m i0 (x) m i1 (rx) m il (r l x) = S i0 i 1...i l 1 19 / 91

20 Walsh bases 20 / 91

21 Walsh bases 21 / 91

22 Walsh bases 22 / 91

23 Walsh bases 23 / 91

24 Walsh bases 24 / 91

25 Walsh bases 25 / 91

26 Walsh bases 26 / 91

27 Walsh bases 27 / 91

28 Walsh bases 28 / 91

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30 Walsh bases 30 / 91

31 Generalized Walsh bases C4. Let A = [a ij ] a N N unitary matrix, a 1j = 1 N. m i (x) := N N 1 j=0 a ij χ [j/n,(j+1)/n] (x) r(x) = Nxmod1, n = lk=0 i kn k with i k {0, 1,.., N 1}. The n th generalized Walsh function : W n,a (x) = m i0 (x) m i1 (rx) m il (r l x) The set (W n,a ) n N is ONB in L 2 [0, 1]. 31 / 91

32 Generalized Walsh bases Example We will graph a few generalized Walsh functions that correspond to 4 4 matrix A = / 91

33 Generalized Walsh bases 33 / 91

34 Generalized Walsh bases 34 / 91

35 Generalized Walsh bases 35 / 91

36 Generalized Walsh bases 36 / 91

37 Generalized Walsh bases 37 / 91

38 Generalized Walsh bases 38 / 91

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44 Generalized Walsh bases 44 / 91

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47 Generalized Walsh bases 47 / 91

48 Generalized Walsh bases 48 / 91

49 Example We will graph a few generalized Walsh functions that correspond to 3 3 matrix A = / 91

50 Generalized Walsh bases 50 / 91

51 Generalized Walsh bases 51 / 91

52 Generalized Walsh bases 52 / 91

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54 Generalized Walsh bases 54 / 91

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56 Generalized Walsh bases 56 / 91

57 Generalized Walsh bases 57 / 91

58 Generalized Walsh bases 58 / 91

59 Generalized Walsh bases 59 / 91

60 Generalized Walsh bases 60 / 91

61 Some differences The classic Walsh functions form a group : W n (x) W m (x) = W n m (x) Figure: Graph of W 4 7,A (W n,a) n does not form a group 61 / 91

62 Convergence properties Theorem For f L 1 [0, 1] the sequence converges a.e. to f (x). Corollary S N q(x) = N q 1 n=0 f, W n,a W n,a (x) If f L 1 [0, 1] is continuous in a neighborhood of x = a then S N q f uniformly inside an interval centered at a. 62 / 91

63 Approximation issues Example f (x) = { 0, x [0, 1/16) [1/8, 3/16) [1/4, 1/2) 1, x [1/16, 1/8) [3/16, 1/4) [1/2, 1] With generalized Walsh ONB to the unitary matrix A = / 91

64 Figure: Graph of f and S 27 (f ) 64 / 91

65 Figure: Graph of f and S 36 (f ) 65 / 91

66 Figure: Graph of f and S 60 (f ) 66 / 91

67 Figure: Graph of f and S 81 (f ) 67 / 91

68 Figure: Graph of f and S 100 (f ) 68 / 91

69 Figure: Graph of f and S 200 (f ) 69 / 91

70 Figure: Graph of f and S 241 (f ) 70 / 91

71 Figure: Graph of f and S 300 (f ) 71 / 91

72 Symmetric encryption Corollary If f : [0, 1] C is constant on the interval I j := [j/n q, (j + 1)/N q ) for some j {0, 1,.., N q 1}, then for all x I j : f (x) = N q 1 n=0 f, W n,a W n,a (x) f (x) = v j, x I j, j = 0,..., N q 1. The sequence f, W n,a encrypts f with respect to a secret matrix A. 72 / 91

73 Example f = abcadbcad is encoded as a n = [ , , , , , , , , e 1] A = / 91

74 Example Given the previous sequence a n and slightly perturbed matrix à = 0.2 Figure: Graph of a n W n,ã 74 / 91

75 Continuity w.r.t. matrix entries For a fixed sequence (a n ) Nq 1 n=0 the map R N2 A N q 1 n=0 a n W n,a is continuous To strengthen the encryption : f f, W n,a + extra, e.g. ( 1) n M sin(1/a 2 ) 75 / 91

76 Previous example, now with entry a = Figure: Graph of a n W n, Ã 76 / 91

77 Encryption/Compression with Cuntz QMF basis m i (x) := N N 1 j=0 a ij χ [j/n,(j+1)/n] (x) S i (f ) = m i f r S i (f ) = 1 N N 1 k=0 m i( x+k N ) f ( x+k N ) signal f [S i (f )] i=0,n 1 (i.e. f encrypted). Compress [S i (f )] i=0,n / 91

78 Example Figure: Signal f, piece wise constant on tri-adic intervals 78 / 91

79 Example Figure: First frequency band, signal S 0 f 79 / 91

80 Example Figure: 2 nd frequency band, signal S 1 f 80 / 91

81 Example Figure: 3 rd frequency band, signal S 2 f 81 / 91

82 A cryptographic protocol H 1 space of messages, H 2 the space of encrypted messages Assume plenty of operators A : H 1 H 2 and B : H 1 H 2 such that B 1 A B 1 A = I H1 Ping-pong messaging (also Eve is eavesdropping): 1) Alice to Bob: w 1 = A(v) H 2 2) Bob to Alice: w 2 = B 1 A(v) H 1 3) Alice to Bob: w 3 = AB 1 A(v) H 2 4) Bob applies B 1 to w / 91

83 Bad choices A(f )(x) = f (x + a), B(f ) = f (x + b) f can be detected from its translations A(f )(x) = f (x a ), B(f )(x) = f (x b ) dilation/compression, some of f could be guessed, issues with the domain More generally f G, G Abelian group: Ax = ax, Bx = bx. Previous ping-pong: w 1 = af, w 2 = b 1 w 1 Eve can figure out b = w2 1 w 1 w 3 = a 1 w 2 = b 1 f Eve multiplies by b and reveals f 83 / 91

84 Transforms commutation A, B unitary N N matrices having constant 1/ N first row. W A : L 2 [0, 1] l 2 (N), W A (f ) = f, W n,a n 0 The inverse tranform (only needed for finite sequences ) : W 1 A ((a n) n ) = n a n W n,a Question: Given f under what conditions for A and B does the ping-pong protocol work? W 1 B W A W 1 B W A(f ) = f 84 / 91

85 Theorem If row l,b, row k,a = row l,a, row k,b for all k, l in {1, 2, 3,..., N} then f piecewise constant on consecutive N-adic intervals : W 1 B W A W 1 B W A(f ) = f 85 / 91

86 Theorem If row l,b, row k,a = row l,a, row k,b for all k, l in {1, 2, 3,..., N} then f piecewise constant on consecutive N-adic intervals : W 1 B N = 3 one equation is relevant: W A W 1 B W A(f ) = f 86 / 91

87 Protocol set up Alice has A = [a i,j ] j=1,n i=1,n Bob receives from Alice: with real number entries. N a kj x lj = j=1 N a lj x kj, 1 < l < k N masking coefficients j=1 87 / 91

88 Protocol set up Alice has A = [a i,j ] j=1,n i=1,n Bob receives from Alice: with real number entries. N a kj x lj = j=1 N a lj x kj, 1 < l < k N masking coefficients j=1 B = [x i,j ] j=1,n i=1,n must be unitary: x 1,j = 1/ N, j = 1,..., N N j=1 x i,j 2 = 1, i = 2,..., N N j=1 x i,j = 0, i = 2,..., N N k=1 x i,k x j,k = 0, 1 < i < j N System of N 2 N quadratics with N 2 N unknowns. 88 / 91

89 Question Are there infinitely many A for which the previous system has infinitely many solutions? Study their Grobner bases. 89 / 91

90 Question Are there infinitely many A for which the previous system has infinitely many solutions? Study their Grobner bases. Example There are infintely many B transform commuting with A = / 91

91 Thank you! Bibliography: 1) D.Dutkay, G.Picioroaga, M.S. Song Orthonormal bases generated by Cuntz algebras to appear in JMAA 2) D.Dutkay, G.Picioroaga Generalized Walsh bases and applications, to appear in ACAP 3) S.Harding, G.Picioroaga Continuity properties of a generalized Walsh system and applications to cryptography -undergraduate research project 91 / 91