Right side derivative and conjugate function

- link to interactive notebooks session.

import numpy as np

import numerical_function_spaces.orlicz_spaces as osm

Denote by \(p_{+}\) (p_plus) the right-hand side derivative of \(\Phi \) with the domain restricted to the interval \(\left[ 0,\infty \right) .\) For any Orlicz function \(\Phi\) we define function \(\Psi \) by the formula

\[\begin{equation*} \Psi \left( u\right) =\sup \left\{ \left\vert u\right\vert v-\Phi \left( v\right) :v\geq 0\right\} \text{ for all } u\in\mathbb{R} \end{equation*}\]

Function \(\Psi\) is called conjugated to \(\Phi \) in the sense of Young. It is also known as Legendre transform.

def Orlicz_function(u):
    # return np.where(u <=1, 0, u - 1)
    # or
    Phi = np.zeros(len(u))
    for i in range(len(u)):
        if u[i] <= 1:
            Phi[i] = 0
        else:
            Phi[i] = u[i] - 1
    return Phi

p_plus = osm.right_side_derivative(Orlicz_function, u_max=3, du=0.2)

u = np.arange(0, 3, 0.2)
for i in range(len(u)):
    print(f"p_plus({u[i]})= {p_plus[i]}")

p_plus(0.0)= 0.0
p_plus(0.2)= 0.0
p_plus(0.4)= 0.0
p_plus(0.6000000000000001)= 0.0
p_plus(0.8)= 0.0
p_plus(1.0)= 1.0000000000000009
p_plus(1.2000000000000002)= 0.9999999999999998
p_plus(1.4000000000000001)= 0.9999999999999998
p_plus(1.6)= 0.9999999999999998
p_plus(1.8)= 0.9999999999999998
p_plus(2.0)= 1.0000000000000009
p_plus(2.2)= 1.0000000000000009
p_plus(2.4000000000000004)= 0.9999999999999987
p_plus(2.6)= 1.0000000000000009
p_plus(2.8000000000000003)= 0.9999999999999987

Psi = osm.conjugate_function(Orlicz_function, u_max=5, du=0.5)

b_Psi = 1.5 ?
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
1-rd  extending of the Phi domain to check b_Psi
u_max =  50.5
Counting Phi for extended domain
YES: b_Psi = 1.5

u = np.arange(0, 5, 0.5)
for i in range(len(u)):
    print(f"Psi({u[i]})= {Psi[i]}")

Psi(0.0)= 0.0
Psi(0.5)= 0.5
Psi(1.0)= 1.0
Psi(1.5)= inf
Psi(2.0)= inf
Psi(2.5)= inf
Psi(3.0)= inf
Psi(3.5)= inf
Psi(4.0)= inf
Psi(4.5)= inf

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=100, du=0.01, max_u_on_plots=5)

b_Psi = 1.01 ?
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
1-rd  extending of the Phi domain to check b_Psi
u_max =  1010.0
Counting Phi for extended domain
YES: b_Psi = 1.01

_images/b221c132833773e3bd51e85bcb7b62c0a11d6a9341c34faaa51ea893d14c55b9.png

Orlicz_function = osm.Orlicz_function_L_1_cap_L_inf

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=10, du=0.01, max_u_on_plots=10)

_images/144b3e50f9974a96186a395071d7a720a232f58da045ecaed6fbe45fed0ead7b.png

def Orlicz_function(u):
    res = np.zeros(len(u))
    for i in range(len(u)):
        n = -1
        while True:
            if u[i] > n and u[i] <= n + 1:  # below two congjugated functions?
                res[i] = (n + 1) * u[i] - (n + 1) * n / 2
                # res[i] = n * u[i] - (n) * (n + 1) / 2
                break
            n = n + 1
    return res

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=10, du=0.01, max_u_on_plots=10)

_images/54bb87d8161babbea980437a5cd769cde7be192d8b79462c43bf8a2bbebdd398.png

def Orlicz_function(u):
    return np.where(u < 1, u / (1 - u), np.inf)

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=10, du=0.01, max_u_on_plots=10)

/tmp/ipykernel_678/1153624846.py:2: RuntimeWarning: divide by zero encountered in divide
  return np.where(u < 1, u / (1 - u), np.inf)

_images/d2d6dc94545575c51b72b8bef49b2705652a96e9ebcdf279189bfd69e3568645.png

def Orlicz_function(u):
    # return np.where(u <= 2, u / 2, np.where(u <= 4, (u / 2) ** (u / 2), np.inf))
    # or
    Phi = np.zeros(len(u))
    for i in range(len(u)):
        if u[i] <= 2:
            Phi[i] = u[i] / 2
        elif u[i] <= 4:
            Phi[i] = (u[i] / 2) ** (u[i] / 2)
        else:
            Phi[i] = np.inf
    return Phi

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=10, du=0.01, max_u_on_plots=10)

_images/341019f1cc03d695f8164f08acc4cc50b1754a060eaf740081d8fa5862e5d4fb.png

def Orlicz_function(u):
    return 3 * (u - np.log(u + 1))

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=10, du=0.001, max_u_on_plots=10)

b_Psi = 2.728 ?
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
1-rd  extending of the Phi domain to check b_Psi
u_max =  101.0
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 2.728 new_b_Psi 2.971
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
2-rd  extending of the Phi domain to check b_Psi
u_max =  329.4900436770063
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 2.971 new_b_Psi 2.991
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
3-rd  extending of the Phi domain to check b_Psi
u_max =  742.8137844855637
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 2.991 new_b_Psi 2.996
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
4-th extending of the Phi domain to check b_Psi
u_max =  1395.211836892324
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 2.996 new_b_Psi 2.998
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
5-th extending of the Phi domain to check b_Psi
u_max =  2350.7829260210424
Counting Phi for extended domain

new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 2.998 new_b_Psi 2.999
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
6-th extending of the Phi domain to check b_Psi
u_max =  3685.555749754256
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 2.999 new_b_Psi 3.0
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
7-th extending of the Phi domain to check b_Psi
u_max =  5489.618683519976
Counting Phi for extended domain
YES: b_Psi = 3.0

_images/9fad4397d1c38e38fc457c9cbbb7e13d0da4ba33cb5a2fa5e00eed2b735b09a1.png

def Orlicz_function(u):
    # return np.where(u <= 1, 1 - np.sqrt(1 - u ** 2), np.inf)
    # or
    Phi = np.zeros(len(u))
    for i in range(len(u)):
        if u[i] <= 1:
            Phi[i] = 1 - np.sqrt(1 - u[i] ** 2)
        else:
            Phi[i] = np.inf
    return Phi

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=3, du=0.01, max_u_on_plots=10)

_images/5b7fda167a208d4fc3683b6e2c7eb2bcf3c06edb4176fdfa543ed3bcdaf9e968.png

def Orlicz_function(u):
    return 3 * u * np.log(u + 2) - np.log(u + 1)

osm.plot_Phi_p_plus_Psi(Orlicz_function, u_max=30, du=0.01, max_u_on_plots=10)

b_Psi = 13.18 ?
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
1-rd  extending of the Phi domain to check b_Psi
u_max =  303.0
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 13.18 new_b_Psi 20.14
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
2-rd  extending of the Phi domain to check b_Psi
u_max =  988.470131031019
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 20.14 new_b_Psi 23.69
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
3-rd  extending of the Phi domain to check b_Psi
u_max =  2228.441353456691
Counting Phi for extended domain

new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 23.69 new_b_Psi 26.13
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
4-th extending of the Phi domain to check b_Psi
u_max =  4185.635510676972
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 26.13 new_b_Psi 28.02
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
5-th extending of the Phi domain to check b_Psi
u_max =  7052.348778063128
Counting Phi for extended domain

new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 28.02 new_b_Psi 29.59
trying to extend domain of Phi to check b_Psi - it may take a lot of time - press CTRL+C to exit (may not work) or interrupt cell
6-th extending of the Phi domain to check b_Psi
u_max =  11056.66724926277
Counting Phi for extended domain
new_b_Psi > b_Psi - problem with domain and/or slowly increasing of Phi
old_b_Psi 29.59 new_b_Psi 30.0

_images/ca252b7d483aa6abbc4a501c8c52e69fd19f5115461708ff00dc9c96ce584f9e.png