Construction of Orthogonal and Biorthogonal Product Systems Bal - - PowerPoint PPT Presentation

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1/18

Construction of Orthogonal and Biorthogonal Product Systems

Bal´ azs Kir´ aly 6th Workshop on Fourier Analysis and Related Fields, P´ ecs, Hungary 2017.

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2/18 Rademacher and Walsh Systems

First Back

2/18 Rademacher and Walsh Systems

• The Rademacher functions rn (n ∈ N) can be derived from the basic function r by

dilation: rn(x) := r(2nx) (x ∈ [0, 1), n ∈ N) r(x) :=

• 1, x ∈ [k, k + 1

2), k ∈ Z,

−1, x ∈ [k + 1

2, k + 1), k ∈ Z.

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2/18 Rademacher and Walsh Systems

• The Rademacher functions rn (n ∈ N) can be derived from the basic function r by

dilation: rn(x) := r(2nx) (x ∈ [0, 1), n ∈ N) r(x) :=

• 1, x ∈ [k, k + 1

2), k ∈ Z,

−1, x ∈ [k + 1

2, k + 1), k ∈ Z.

• The Walsh system is the product system of the Rademacher system i.e.

wm =

∞

• k=0

rmk

k ,

where m =

∞

• k=0

mk · 2k, mk ∈ {0, 1}.

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2/18 Rademacher and Walsh Systems

• The Rademacher functions rn (n ∈ N) can be derived from the basic function r by

dilation: rn(x) := r(2nx) (x ∈ [0, 1), n ∈ N) r(x) :=

• 1, x ∈ [k, k + 1

2), k ∈ Z,

−1, x ∈ [k + 1

2, k + 1), k ∈ Z.

• The Walsh system is the product system of the Rademacher system i.e.

wm =

∞

• k=0

rmk

k ,

where m =

∞

• k=0

mk · 2k, mk ∈ {0, 1}. The wm (m ∈ N) Walsh system is a complete orthonormal system with respect to the scalar product f, g =

1

• f(t) · g(t)dt.

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3/18 Haar System

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3/18 Haar System

• The Haar-system was presented in 1910 by Alfr´

ed Haar.

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3/18 Haar System

• The Haar-system was presented in 1910 by Alfr´

ed Haar.

• The original definition

h0(x) = χ[0,1) =

• 1 x ∈ [0, 1)

0 otherwise, hm(x) = hn,k(x) = 2

n 2 h(2nx − k) =

   2

n 2

x ∈ [ k

2n, 2k+1 2n+1 )

−2

n 2

x ∈ [2k+1

2n+1 , k+1 2n )

• therwise,

where m = 2n + k and n ∈ N, 0 ≤ k < 2n.

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3/18 Haar System

• The Haar-system was presented in 1910 by Alfr´

ed Haar.

• The original definition

h0(x) = χ[0,1) =

• 1 x ∈ [0, 1)

0 otherwise, hm(x) = hn,k(x) = 2

n 2 h(2nx − k) =

   2

n 2

x ∈ [ k

2n, 2k+1 2n+1 )

−2

n 2

x ∈ [2k+1

2n+1 , k+1 2n )

• therwise,

where m = 2n + k and n ∈ N, 0 ≤ k < 2n.

• The Haar system was the first and the simplest wavelet.

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3/18 Haar System

• The Haar-system was presented in 1910 by Alfr´

ed Haar.

• The original definition

h0(x) = χ[0,1) =

• 1 x ∈ [0, 1)

0 otherwise, hm(x) = hn,k(x) = 2

n 2 h(2nx − k) =

   2

n 2

x ∈ [ k

2n, 2k+1 2n+1 )

−2

n 2

x ∈ [2k+1

2n+1 , k+1 2n )

• therwise,

where m = 2n + k and n ∈ N, 0 ≤ k < 2n.

• The Haar system was the first and the simplest wavelet.
• Wavelet construction

Haar-functions can be derived from the basic function h(x) := h1(x) =      1, (0 ≤ x < 1/2), −1, (1/2 ≤ x < 1), 0, (1 ≤ x < ∞). by translation and dilation: hn,k(x) := 2n/2h(2nx − k) (x ∈ R, 0 ≤ k < 2n, n ∈ N).

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4/18 Haar Scaling Functions

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4/18 Haar Scaling Functions

• The χn,k characteristic functions of dyadic intervals are known as the Haar scaling

functions.

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4/18 Haar Scaling Functions

• The χn,k characteristic functions of dyadic intervals are known as the Haar scaling

functions.

• These functions can be derived similarly to Haar-functions starting from χ[0,1) so

χn,k(x) = χ(2nx − k) (x ∈ R, 0 ≤ k < 2n, n ∈ N).

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4/18 Haar Scaling Functions

• The χn,k characteristic functions of dyadic intervals are known as the Haar scaling

functions.

• These functions can be derived similarly to Haar-functions starting from χ[0,1) so

χn,k(x) = χ(2nx − k) (x ∈ R, 0 ≤ k < 2n, n ∈ N).

• For the Haar functions and for the Haar scaling functions the following are true

χn,k = χn+1,2k + χn+1,2k+1, (0 ≤ k < 2n, n ∈ N), hn,k = 2n/2(χn+1,2k − χn+1,2k+1) (0 ≤ k < 2n, n ∈ N).

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4/18 Haar Scaling Functions

• The χn,k characteristic functions of dyadic intervals are known as the Haar scaling

functions.

• These functions can be derived similarly to Haar-functions starting from χ[0,1) so

χn,k(x) = χ(2nx − k) (x ∈ R, 0 ≤ k < 2n, n ∈ N).

• For the Haar functions and for the Haar scaling functions the following are true

χn,k = χn+1,2k + χn+1,2k+1, (0 ≤ k < 2n, n ∈ N), hn,k = 2n/2(χn+1,2k − χn+1,2k+1) (0 ≤ k < 2n, n ∈ N).

• The Haar-Fourier analysis and synthesis are based on these equations.

(O(2N) operations)

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4/18 Haar Scaling Functions

• The χn,k characteristic functions of dyadic intervals are known as the Haar scaling

functions.

• These functions can be derived similarly to Haar-functions starting from χ[0,1) so

χn,k(x) = χ(2nx − k) (x ∈ R, 0 ≤ k < 2n, n ∈ N).

• For the Haar functions and for the Haar scaling functions the following are true

χn,k = χn+1,2k + χn+1,2k+1, (0 ≤ k < 2n, n ∈ N), hn,k = 2n/2(χn+1,2k − χn+1,2k+1) (0 ≤ k < 2n, n ∈ N).

• The Haar-Fourier analysis and synthesis are based on these equations.

(O(2N) operations)

• The 2n-th Dirichlet kernel of the Walsh-system is of the form

D2n(x, y) =

2n−1

• k=0

wk(x) · wk(y) =

n−1

• j=0

(1 + rj(x)rj(y)) (x, y ∈ [0, 1), n ∈ N)

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4/18 Haar Scaling Functions

• The χn,k characteristic functions of dyadic intervals are known as the Haar scaling

functions.

• These functions can be derived similarly to Haar-functions starting from χ[0,1) so

χn,k(x) = χ(2nx − k) (x ∈ R, 0 ≤ k < 2n, n ∈ N).

• For the Haar functions and for the Haar scaling functions the following are true

χn,k = χn+1,2k + χn+1,2k+1, (0 ≤ k < 2n, n ∈ N), hn,k = 2n/2(χn+1,2k − χn+1,2k+1) (0 ≤ k < 2n, n ∈ N).

• The Haar-Fourier analysis and synthesis are based on these equations.

(O(2N) operations)

• The 2n-th Dirichlet kernel of the Walsh-system is of the form

D2n(x, y) =

2n−1

• k=0

wk(x) · wk(y) =

n−1

• j=0

(1 + rj(x)rj(y)) (x, y ∈ [0, 1), n ∈ N)

• The Haar-functions and the Haar scaling functions can be expressed as

hn,k(x) = 2−n/2rn(x)D2n(x, k2−n) χn,k(x) = 2−nD2n(x, k2−n) (x ∈ [0, 1), 0 ≤ k < 2n, n ∈ N).

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5/18 Generalization of Product system

• Let us fix the number p ∈ N∗∗ := {2, 3, · · · } and the non-empty set X. Let us start

from the following finite collection of the systems φn = (ϕ(i)

n , 0 ≤ i < p),

(0 ≤ n < N ≤ ∞, ϕ(i)

n : X → C)

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5/18 Generalization of Product system

• Let us fix the number p ∈ N∗∗ := {2, 3, · · · } and the non-empty set X. Let us start

from the following finite collection of the systems φn = (ϕ(i)

n , 0 ≤ i < p),

(0 ≤ n < N ≤ ∞, ϕ(i)

n : X → C)

• Then

Φm =

N−1

• k=0

ϕ(mk)

k

, m =

N−1

• k=0

mk · pk mk ∈ {0, 1, . . . , p − 1} is the generalized product system of the systems φ.

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5/18 Generalization of Product system

• Let us fix the number p ∈ N∗∗ := {2, 3, · · · } and the non-empty set X. Let us start

from the following finite collection of the systems φn = (ϕ(i)

n , 0 ≤ i < p),

(0 ≤ n < N ≤ ∞, ϕ(i)

n : X → C)

• Then

Φm =

N−1

• k=0

ϕ(mk)

k

, m =

N−1

• k=0

mk · pk mk ∈ {0, 1, . . . , p − 1} is the generalized product system of the systems φ.

• It is really the generalization of idea of the product system, because in special case

when p = 2 and ϕ(0)

m = 1, ϕ(1) m = rm we reobtain the Walsh system.

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6/18 Biorthogonality with respect to a p-fold map

• Let us fix the number p ∈ N∗∗ := {2, 3, · · · } and the set X = ∅. The map A : X → X′

is called p-fold map on set X if every x ∈ X′ has exactly p preimages, i.e. the set A−1(x) = {x0, x1, . . . , xp−1} x ∈ X′. has p elements.

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6/18 Biorthogonality with respect to a p-fold map

• Let us fix the number p ∈ N∗∗ := {2, 3, · · · } and the set X = ∅. The map A : X → X′

is called p-fold map on set X if every x ∈ X′ has exactly p preimages, i.e. the set A−1(x) = {x0, x1, . . . , xp−1} x ∈ X′. has p elements.

• Let

f (j), g(j) : X → C, j = 0, 1, . . . , p − 1. and ρ : X → (0, +∞) is a positive weight-function. The system F = (f (0), f (1), . . . f (p−1)) and the system G = (g(0), g(1), . . . g(p−1)) is called (A, ρ)-biorthogonal if for every x ∈ X′ ()

• t∈A−1(x)

f (i)(t) · g(j)(t)ρ(t) =

p−1

• k=0

f (i)(xk) · g(j)(xk)ρ(xk) = δij if F = G the system F is called (A, ρ)-orthonormal.

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6/18 Biorthogonality with respect to a p-fold map

• Let us fix the number p ∈ N∗∗ := {2, 3, · · · } and the set X = ∅. The map A : X → X′

is called p-fold map on set X if every x ∈ X′ has exactly p preimages, i.e. the set A−1(x) = {x0, x1, . . . , xp−1} x ∈ X′. has p elements.

• Let

f (j), g(j) : X → C, j = 0, 1, . . . , p − 1. and ρ : X → (0, +∞) is a positive weight-function. The system F = (f (0), f (1), . . . f (p−1)) and the system G = (g(0), g(1), . . . g(p−1)) is called (A, ρ)-biorthogonal if for every x ∈ X′ ()

• t∈A−1(x)

f (i)(t) · g(j)(t)ρ(t) =

p−1

• k=0

f (i)(xk) · g(j)(xk)ρ(xk) = δij if F = G the system F is called (A, ρ)-orthonormal.

• We will prove, that condition is equivalent with

(♣)

p−1

• i=0

f (i)(xk) · g(i)(xℓ)ρ(xk) = δkℓ (0 ≤ k, ℓ < p)

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Proof: Let us introduce the following matrices: F :=     f (0)(x0) f (0)(x1) . . . f (0)(xp−1) f (1)(x0) f (1)(x1) . . . f (1)(xp−1) . . . . . . . . . f (p−1)(x0) f (p−1)(x1) . . . f (p−1)(xp−1)    

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Proof: Let us introduce the following matrices: F :=     f (0)(x0) f (0)(x1) . . . f (0)(xp−1) f (1)(x0) f (1)(x1) . . . f (1)(xp−1) . . . . . . . . . f (p−1)(x0) f (p−1)(x1) . . . f (p−1)(xp−1)     G :=     g(0)(x0) g(0)(x1) . . . g(0)(xp−1) g(1)(x0) g(1)(x1) . . . g(1)(xp−1) . . . . . . . . . g(p−1)(x0) g(p−1)(x1) . . . g(p−1)(xp−1)    

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7/18

Proof: Let us introduce the following matrices: F :=     f (0)(x0) f (0)(x1) . . . f (0)(xp−1) f (1)(x0) f (1)(x1) . . . f (1)(xp−1) . . . . . . . . . f (p−1)(x0) f (p−1)(x1) . . . f (p−1)(xp−1)     G :=     g(0)(x0) g(0)(x1) . . . g(0)(xp−1) g(1)(x0) g(1)(x1) . . . g(1)(xp−1) . . . . . . . . . g(p−1)(x0) g(p−1)(x1) . . . g(p−1)(xp−1)     and R :=

p−1

diag

k=0

ρ(xk).

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7/18

Proof: Let us introduce the following matrices: F :=     f (0)(x0) f (0)(x1) . . . f (0)(xp−1) f (1)(x0) f (1)(x1) . . . f (1)(xp−1) . . . . . . . . . f (p−1)(x0) f (p−1)(x1) . . . f (p−1)(xp−1)     G :=     g(0)(x0) g(0)(x1) . . . g(0)(xp−1) g(1)(x0) g(1)(x1) . . . g(1)(xp−1) . . . . . . . . . g(p−1)(x0) g(p−1)(x1) . . . g(p−1)(xp−1)     and R :=

p−1

diag

k=0

ρ(xk). Then the condition ()

• t∈A−1(x)

f (i)(t) · g(j)(t)ρ(t) =

p−1

• k=0

f (i)(xk) · g(j)(xk)ρ(xk) = δij can be written as (FRG

T)ij = δij

⇔ FRG

T = I.

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7/18

Proof: Let us introduce the following matrices: F :=     f (0)(x0) f (0)(x1) . . . f (0)(xp−1) f (1)(x0) f (1)(x1) . . . f (1)(xp−1) . . . . . . . . . f (p−1)(x0) f (p−1)(x1) . . . f (p−1)(xp−1)     G :=     g(0)(x0) g(0)(x1) . . . g(0)(xp−1) g(1)(x0) g(1)(x1) . . . g(1)(xp−1) . . . . . . . . . g(p−1)(x0) g(p−1)(x1) . . . g(p−1)(xp−1)     and R :=

p−1

diag

k=0

ρ(xk). Then the condition ()

• t∈A−1(x)

f (i)(t) · g(j)(t)ρ(t) =

p−1

• k=0

f (i)(xk) · g(j)(xk)ρ(xk) = δij can be written as (FRG

T)ij = δij

⇔ FRG

T = I.

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8/18

I = IT =

• FRG

TT

= GRTF T = GRF T

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8/18

I = IT =

• FRG

TT

= GRTF T = GRF T

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8/18

I = IT =

• FRG

TT

= GRTF T = GRF T

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8/18

I = IT =

• FRG

TT

= GRTF T = GRF T

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8/18

I = IT =

• FRG

TT

= GRTF T = GRF T It means that

• GRF T

ℓk = δℓk

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8/18

I = IT =

• FRG

TT

= GRTF T = GRF T It means that

• GRF T

ℓk = δℓk

written with the elements of matrices we get (♣)

p−1

• i=0

f (i)(xk) · g(i)(xℓ)ρ(xk) = δkℓ (0 ≤ k, ℓ < p)

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9/18 The Discrete Set Xn

Assume that for the elements An : X → X′, n ∈ N∗ of the sequence of p-fold maps A−1

n (x) ⊂ X′ (x ∈ X′, n ∈ N∗).

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9/18 The Discrete Set Xn

Assume that for the elements An : X → X′, n ∈ N∗ of the sequence of p-fold maps A−1

n (x) ⊂ X′ (x ∈ X′, n ∈ N∗).

By composition of these maps we get T0(x) := x (x ∈ X), Tn = An ◦ An−1 ◦ . . . ◦ A1 = An ◦ Tn−1 (n ∈ N∗). Starting from a fixed element x0 ∈ X′ we can define the preimage of this element in map Tn, this discrete set is denoted by Xn := T −1

n (x0) (n ∈ N).

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9/18 The Discrete Set Xn

Assume that for the elements An : X → X′, n ∈ N∗ of the sequence of p-fold maps A−1

n (x) ⊂ X′ (x ∈ X′, n ∈ N∗).

By composition of these maps we get T0(x) := x (x ∈ X), Tn = An ◦ An−1 ◦ . . . ◦ A1 = An ◦ Tn−1 (n ∈ N∗). Starting from a fixed element x0 ∈ X′ we can define the preimage of this element in map Tn, this discrete set is denoted by Xn := T −1

n (x0) (n ∈ N).

The set Xn has pn elements.

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9/18 The Discrete Set Xn

Assume that for the elements An : X → X′, n ∈ N∗ of the sequence of p-fold maps A−1

n (x) ⊂ X′ (x ∈ X′, n ∈ N∗).

By composition of these maps we get T0(x) := x (x ∈ X), Tn = An ◦ An−1 ◦ . . . ◦ A1 = An ◦ Tn−1 (n ∈ N∗). Starting from a fixed element x0 ∈ X′ we can define the preimage of this element in map Tn, this discrete set is denoted by Xn := T −1

n (x0) (n ∈ N).

The set Xn has pn elements. If x0 is fixed-point of every maps An, i.e. An(x0) = x0, (n ∈ N), then Xn ⊃ Xn−1 ⊃ . . . ⊃ X1.

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9/18 The Discrete Set Xn

Assume that for the elements An : X → X′, n ∈ N∗ of the sequence of p-fold maps A−1

n (x) ⊂ X′ (x ∈ X′, n ∈ N∗).

By composition of these maps we get T0(x) := x (x ∈ X), Tn = An ◦ An−1 ◦ . . . ◦ A1 = An ◦ Tn−1 (n ∈ N∗). Starting from a fixed element x0 ∈ X′ we can define the preimage of this element in map Tn, this discrete set is denoted by Xn := T −1

n (x0) (n ∈ N).

The set Xn has pn elements. If x0 is fixed-point of every maps An, i.e. An(x0) = x0, (n ∈ N), then Xn ⊃ Xn−1 ⊃ . . . ⊃ X1. The point x0 has p preimages in map An: A−1

n (x0) = {x0, x1, . . . , xp−1}. The set Xn can be

written in form Xn = Xn−1 ∪ X1

n−1 ∪ X2 n−1 ∪ . . . ∪ Xp−1 n−1

where Xj

n−1 = Tn−1(xj) (j = 1, 2, . . . , p − 1).

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10/18

• Set Xn has pn elements.

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10/18

• Set Xn has pn elements.
• The elements of the Xn = {xn

k : 0 ≤ k < pn} will be indexed in the following way

A−1({xn

k}) = {xn+1 pk , xn+1 pk+1, . . . , xn+1 pk+p−1}.

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10/18

• Set Xn has pn elements.
• The elements of the Xn = {xn

k : 0 ≤ k < pn} will be indexed in the following way

A−1({xn

k}) = {xn+1 pk , xn+1 pk+1, . . . , xn+1 pk+p−1}.

• Aj(xn

ℓ ) = xn−j [ℓp−j] (0 ≤ ℓ < pn, 0 ≤ j ≤ n).

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10/18

• Set Xn has pn elements.
• The elements of the Xn = {xn

k : 0 ≤ k < pn} will be indexed in the following way

A−1({xn

k}) = {xn+1 pk , xn+1 pk+1, . . . , xn+1 pk+p−1}.

• Aj(xn

ℓ ) = xn−j [ℓp−j] (0 ≤ ℓ < pn, 0 ≤ j ≤ n).

• Let us fix N ∈ N∗. Let us define the following subsets of the set XN:

In,k := An−N({xn

k})

(0 ≤ k < pn, 0 ≤ n ≤ N). The set In,k has pN−n elements and is called p-adic interval of the set XN.

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10/18

• Set Xn has pn elements.
• The elements of the Xn = {xn

k : 0 ≤ k < pn} will be indexed in the following way

A−1({xn

k}) = {xn+1 pk , xn+1 pk+1, . . . , xn+1 pk+p−1}.

• Aj(xn

ℓ ) = xn−j [ℓp−j] (0 ≤ ℓ < pn, 0 ≤ j ≤ n).

• Let us fix N ∈ N∗. Let us define the following subsets of the set XN:

In,k := An−N({xn

k})

(0 ≤ k < pn, 0 ≤ n ≤ N). The set In,k has pN−n elements and is called p-adic interval of the set XN.

• The p-adic intervals of the set XN have similar properties as the intervals

[k2−n, (k + 1)2−n): I0,0 = A−N({x0

0}) = XN, IN,k = A0({xN k }) = {xN k },

In+1,2k ∪ In+1,2k+1 ∪ . . . ∪ In+1,2k+p−1 = In,k (0 ≤ k < pn, 0 ≤ n < N).

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10/18

• Set Xn has pn elements.
• The elements of the Xn = {xn

k : 0 ≤ k < pn} will be indexed in the following way

A−1({xn

k}) = {xn+1 pk , xn+1 pk+1, . . . , xn+1 pk+p−1}.

• Aj(xn

ℓ ) = xn−j [ℓp−j] (0 ≤ ℓ < pn, 0 ≤ j ≤ n).

• Let us fix N ∈ N∗. Let us define the following subsets of the set XN:

In,k := An−N({xn

k})

(0 ≤ k < pn, 0 ≤ n ≤ N). The set In,k has pN−n elements and is called p-adic interval of the set XN.

• The p-adic intervals of the set XN have similar properties as the intervals

[k2−n, (k + 1)2−n): I0,0 = A−N({x0

0}) = XN, IN,k = A0({xN k }) = {xN k },

In+1,2k ∪ In+1,2k+1 ∪ . . . ∪ In+1,2k+p−1 = In,k (0 ≤ k < pn, 0 ≤ n < N).

• Two of these subsets are always disjoint or one of them includes the other.

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10/18

The set Xn in case p = 3:

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11/18 The Discrete Biorthogonal Product System

• Denote Fn := (f (0)

n , f (1) n , . . . , f (p−1) n

) and Gn := (g(0)

n , g(1) n , . . . , g(p−1) n

) two sequences of systems and ρ : X → (0, +∞) (n ∈ N) a positive weight function, where f (i)

n , g(i) n : X → C (i = 0, 1, . . . , p − 1) (n ∈ N).

First Back

11/18 The Discrete Biorthogonal Product System

• Denote Fn := (f (0)

n , f (1) n , . . . , f (p−1) n

) and Gn := (g(0)

n , g(1) n , . . . , g(p−1) n

) two sequences of systems and ρ : X → (0, +∞) (n ∈ N) a positive weight function, where f (i)

n , g(i) n : X → C (i = 0, 1, . . . , p − 1) (n ∈ N).

• Assume that system Fn and system Gn are (An+1, ρn) biorthogonals for every index

0 ≤ n < N.

First Back

11/18 The Discrete Biorthogonal Product System

• Denote Fn := (f (0)

n , f (1) n , . . . , f (p−1) n

) and Gn := (g(0)

n , g(1) n , . . . , g(p−1) n

) two sequences of systems and ρ : X → (0, +∞) (n ∈ N) a positive weight function, where f (i)

n , g(i) n : X → C (i = 0, 1, . . . , p − 1) (n ∈ N).

• Assume that system Fn and system Gn are (An+1, ρn) biorthogonals for every index

0 ≤ n < N.

• By composition construct the following systems

φ(i)

n := f (i) n ◦ Tn,

γ(i)

n := g(i) n ◦ Tn (0 ≤ i < p) (n ∈ N).

and from them Φn := (φ(i)

n , 0 ≤ i < p), Γn := (γ(i) n , 0 ≤ i < p),

(n ∈ N).

First Back

11/18 The Discrete Biorthogonal Product System

• Denote Fn := (f (0)

n , f (1) n , . . . , f (p−1) n

) and Gn := (g(0)

n , g(1) n , . . . , g(p−1) n

) two sequences of systems and ρ : X → (0, +∞) (n ∈ N) a positive weight function, where f (i)

n , g(i) n : X → C (i = 0, 1, . . . , p − 1) (n ∈ N).

• Assume that system Fn and system Gn are (An+1, ρn) biorthogonals for every index

0 ≤ n < N.

• By composition construct the following systems

φ(i)

n := f (i) n ◦ Tn,

γ(i)

n := g(i) n ◦ Tn (0 ≤ i < p) (n ∈ N).

and from them Φn := (φ(i)

n , 0 ≤ i < p), Γn := (γ(i) n , 0 ≤ i < p),

(n ∈ N).

• The systems were constructed the previous way are called Rademacher-like functions.

First Back

11/18 The Discrete Biorthogonal Product System

• Denote Fn := (f (0)

n , f (1) n , . . . , f (p−1) n

) and Gn := (g(0)

n , g(1) n , . . . , g(p−1) n

) two sequences of systems and ρ : X → (0, +∞) (n ∈ N) a positive weight function, where f (i)

n , g(i) n : X → C (i = 0, 1, . . . , p − 1) (n ∈ N).

• Assume that system Fn and system Gn are (An+1, ρn) biorthogonals for every index

0 ≤ n < N.

• By composition construct the following systems

φ(i)

n := f (i) n ◦ Tn,

γ(i)

n := g(i) n ◦ Tn (0 ≤ i < p) (n ∈ N).

and from them Φn := (φ(i)

n , 0 ≤ i < p), Γn := (γ(i) n , 0 ≤ i < p),

(n ∈ N).

• The systems were constructed the previous way are called Rademacher-like functions.
• The product systems of these systems are called Walsh-like systems and can be written

in form ψm :=

N−1

• i=0

φmi

i

ηm :=

N−1

• i=0

γmi

i

(0 ≤ m < pN) m =

N−1

• i=0

mipi,

First Back

12/18

Theorem 1 The pn-th mixed kernel of the system (ψm, 0 ≤ m < pN) and system (ηm, 0 ≤ m < pN) can be written in product form Dpn(x, t) :=

pn−1

• k=0

ψk(x)ηk(t) =

n−1

• i=0

(φ(0)

i (x)γ(0) i (t) + φ(1) i (x)γ(1) i (t) + . . . + φ(p−1) i

(x)γ(p−1)

i

(t)), where x, t ∈ X and 0 ≤ n < N.

First Back

12/18

Theorem 1 The pn-th mixed kernel of the system (ψm, 0 ≤ m < pN) and system (ηm, 0 ≤ m < pN) can be written in product form Dpn(x, t) :=

pn−1

• k=0

ψk(x)ηk(t) =

n−1

• i=0

(φ(0)

i (x)γ(0) i (t) + φ(1) i (x)γ(1) i (t) + . . . + φ(p−1) i

(x)γ(p−1)

i

(t)), where x, t ∈ X and 0 ≤ n < N. This theorem can be proven by induction for n.

First Back

13/18

Theorem 2 The product system ψm :=

N−1

• i=0

φmi

i

(0 ≤ m < pN) m =

N−1

• i=0

mipi, and product system ηm :=

N−1

• i=0

γmi

i

(0 ≤ m < pN) m =

N−1

• i=0

mipi are biorthogonal with respect to the discrete scalarproduct f, g :=

• x∈XN

f(x)g(x)σN(x), and (⋆) σN(x)DpN(x, t) = δx,t (x, t ∈ X), where σN(x) :=

N−1

• i=0

ρi(Ti(x)) (x ∈ XN).

First Back

14/18

Proof : First the statement (⋆) σN(x)DpN(x, t) = δx,t (x, t ∈ X), will be shown.

First Back

14/18

Proof : First the statement (⋆) σN(x)DpN(x, t) = δx,t (x, t ∈ X), will be shown. For this we investigate two cases (when x = t and when x = t).

First Back

14/18

Proof : First the statement (⋆) σN(x)DpN(x, t) = δx,t (x, t ∈ X), will be shown. For this we investigate two cases (when x = t and when x = t). For the proof of biortogonality we use the trick with matrix-products what we saw previ-

• usly.

First Back

14/18

Proof : First the statement (⋆) σN(x)DpN(x, t) = δx,t (x, t ∈ X), will be shown. For this we investigate two cases (when x = t and when x = t). For the proof of biortogonality we use the trick with matrix-products what we saw previ-

• usly.

Consequence 1 For any function f the pN-th partial sum of Fourier series is (SpNf)(x) =

pN−1

• k=0

f, ηkψk(x) =

• t∈XN

f(t)

pN−1

• k=0

ψk(x)ηk(t)σN(t) =

• t∈XN

f(t)DpN(x, t)σN(t). From this (SpNf)(x) = f(x) x ∈ XN.

First Back

15/18 Generalised Haar-scaling Functions

First Back

15/18 Generalised Haar-scaling Functions

• The pn+1-th Dirichlet kernel can be written in the product form

Dpn(x, t) :=

pn−1

• k=0

ψk(x)ηk(t) =

n−1

• i=0

(φ(0)

i (x)γ(0) i (t) + φ(1) i (x)γ(1) i (t) + . . . + φ(p−1) i

(x)γ(p−1)

i

(t)).

First Back

15/18 Generalised Haar-scaling Functions

• The pn+1-th Dirichlet kernel can be written in the product form

Dpn(x, t) :=

pn−1

• k=0

ψk(x)ηk(t) =

n−1

• i=0

(φ(0)

i (x)γ(0) i (t) + φ(1) i (x)γ(1) i (t) + . . . + φ(p−1) i

(x)γ(p−1)

i

(t)).

• Using the function

L(x, y) = φ(0)(x)γ(0)(y) + φ(1)(x)γ(1)(y) + . . . + φ(p−1)(x)γ(p−1)(y) (x, y ∈ X) the Dirichlet-kernel DpN can be written as DpN(x, y) :=

N−1

• j=0

L(AN−1−j(x), AN−1−j(y)) (x, y ∈ X).

First Back

15/18 Generalised Haar-scaling Functions

• The pn+1-th Dirichlet kernel can be written in the product form

Dpn(x, t) :=

pn−1

• k=0

ψk(x)ηk(t) =

n−1

• i=0

(φ(0)

i (x)γ(0) i (t) + φ(1) i (x)γ(1) i (t) + . . . + φ(p−1) i

(x)γ(p−1)

i

(t)).

• Using the function

L(x, y) = φ(0)(x)γ(0)(y) + φ(1)(x)γ(1)(y) + . . . + φ(p−1)(x)γ(p−1)(y) (x, y ∈ X) the Dirichlet-kernel DpN can be written as DpN(x, y) :=

N−1

• j=0

L(AN−1−j(x), AN−1−j(y)) (x, y ∈ X).

• Let us introduce the following analogues of the scaling functions χn,k:

In,k(x) := p−n

n−1

• j=0

L(AN−1−j(x), AN−1−j(xn

k)),

(x ∈ X, 0 ≤ k < pn, n = 1, 2, · · · , N).

First Back

16/18

• The following theorem is true and it means that similarly to Haar scaling functions

In,k is the characteristic function of the set In,k. Theorem:The scaling functions In,k, which were made from orthonormal functions, satisfies

• n the set XN

In,k = χIn,k (0 ≤ k < pn, n = 1, 2, . . . , N), thus in the points of the set XN the following scaling equation is true In+1,pk(x) + In+1,pk+1(x) + . . . + In+1,pk+p−1(x) = In,k(x), (x ∈ XN, 0 ≤ k < pn, n = 1, 2, · · · , N).

First Back

16/18

• The following theorem is true and it means that similarly to Haar scaling functions

In,k is the characteristic function of the set In,k. Theorem:The scaling functions In,k, which were made from orthonormal functions, satisfies

• n the set XN

In,k = χIn,k (0 ≤ k < pn, n = 1, 2, . . . , N), thus in the points of the set XN the following scaling equation is true In+1,pk(x) + In+1,pk+1(x) + . . . + In+1,pk+p−1(x) = In,k(x), (x ∈ XN, 0 ≤ k < pn, n = 1, 2, · · · , N).

• The proof of this theorem is based on the fact what says two point of set Xn are same
• r have a common preimage.

First Back

16/18

• The following theorem is true and it means that similarly to Haar scaling functions

In,k is the characteristic function of the set In,k. Theorem:The scaling functions In,k, which were made from orthonormal functions, satisfies

• n the set XN

In,k = χIn,k (0 ≤ k < pn, n = 1, 2, . . . , N), thus in the points of the set XN the following scaling equation is true In+1,pk(x) + In+1,pk+1(x) + . . . + In+1,pk+p−1(x) = In,k(x), (x ∈ XN, 0 ≤ k < pn, n = 1, 2, · · · , N).

• The proof of this theorem is based on the fact what says two point of set Xn are same
• r have a common preimage.
• We showed that all factor in the product form of function In,k are p if x ∈ In,k.

First Back

16/18

• The following theorem is true and it means that similarly to Haar scaling functions

In,k is the characteristic function of the set In,k. Theorem:The scaling functions In,k, which were made from orthonormal functions, satisfies

• n the set XN

In,k = χIn,k (0 ≤ k < pn, n = 1, 2, . . . , N), thus in the points of the set XN the following scaling equation is true In+1,pk(x) + In+1,pk+1(x) + . . . + In+1,pk+p−1(x) = In,k(x), (x ∈ XN, 0 ≤ k < pn, n = 1, 2, · · · , N).

• The proof of this theorem is based on the fact what says two point of set Xn are same
• r have a common preimage.
• We showed that all factor in the product form of function In,k are p if x ∈ In,k.
• The mentioned product has a zero factor in case x /

∈ In,k.

kezd˝

• lap

vissza

16/18 Haar-like System

• We can introduce the Haar-like functions by the following equation

Hn,k :=

p−1

• ℓ=0

(−1)ℓIn+1,pk+ℓ (0 ≤ k < pn, n = 0, 1, · · · , N − 1). Theorem: If p = 2q (q ∈ N) then the discretized system of Hn,k (0 ≤ k < pn, n = 0, 1, · · · , N − 1) is a discrete orthogonal Haar-like system with respect to the scalar product f, g := p−N

x∈XN

f(x)g(x), exactly Hn,k, Hm,ℓ = p−n · δn,m · δk,ℓ, (0 ≤ k < pn, 0 ≤ n < N, 0 ≤ ℓ < pm, 0 ≤ m < N).

• The previous theorem can be proven by simply computation. For the proof we used

the mentioned special properties of the generalized p-adic intervals.

First Back

17/18 References

[1.] Alexits, G., Konvergenzprobleme der Orthogonalreihen, Akad´ emiai Kiad´

• (Budapest, 1960)

[2.] Alexits, G., Sur la sommabilit´ e des series orthogonales, Acta Math. Acad. Sci. Hungar., 4 (1953), 181–188. [3.] Haar, A., On the theory of orthogonal function systems,

• Math. Annalen 69. (1910), 331–371.

[4.] Kaczmarz, S., ¨ Uber ein Orthogonalsystem.

• Comt. Rend. Congres Math. (Warsaw,1929)

[5.] Kir´ aly, B., Construction of Haar-like Systems, PU.M.A. Pure Mathematics And Applications, 17 (2010), 343–347. [6.] Kir´ aly, B., Construction of Walsh-like Systems, Annales Univ. Sci. Budapest., Sec. Comp., 33 (2010), 261–272. [7.] Schipp, F., On a generalization of the Haar system. Acta Math. Acad. Sci. Hung.,, 33(1-2), (1979), 183–188. [8.] Schipp, Wade, Simon, Walsh series, an introduction to dyadic harmonic analysis. Adam Hilger, Bristol, New York (1989) [9.] Walnut, D. F., An Introduction to Wavelet Analysis. Birkh¨ aser, Boston, Basel, Berlin.

18/18

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