Distance between two functions in term of a third function

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I am wondering if it is possible to define a distance between two real-valued functions that express the concept of "the space between them that can be filled by a third function", where the third function is fixed.

Consider functions defined on a domain $mathcalDsubseteq mathbbR$, fix a function $s$ (in my mind something gaussian-shapedw with the peak inside $mathcalD$). For simplicity $int_mathcalD s(x) dx=1$. See figure.

I would define the distance between $f$, $g:mathcalDtomathbbR$ as:

$$
d(f, g) = left|mathrmArgmin_N_s int_mathcalDdx left|f(x) + N_s s(x) - g(x) right|right| = left|mathrmArgmin_N_s d_L1(f+N_s s, g)right|
$$

Is it possible to find a set of hypothesis that makes $d$ a proper distance? I am open also to small changes to the definition of $d$. I am interested when the functions are positive, or when they are as probability density functions, but not necessary normalized to 1.

For sure $d(f, g)geq 0$, I think it is easy to prove $d(g, f)=d(f, g)$:

$$d(g, f) = left|mathrmArgmin_N_s d_L1(g+N_s s, f)right|
=left|mathrmArgmin_N_s d_L1(f-N_s s, g)right|=left|-mathrmArgmin_N_s d_L1(f+N_s s, g)right| = d(f, g)$$

I am not sure about $d(f, g)=0Rightarrow f=g$ and the triangular property.

A less local definition may be:

$$
d(f, g) = left|mathrmArgmin_N_s max_muin C int_mathcalDdx left|f(x) + N_s s(x, mu) - g(x) right|right|
$$

with $mathcalCsubseteq mathcalD$ where $mu$ is something related to the location of $s$, for example the expected value of $s$.

real-analysis metric-spaces hilbert-spaces geometric-functional-analysis

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edited Jul 24 at 9:09

asked Jul 24 at 8:46

Ruggero Turra

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up vote
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down vote

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I am wondering if it is possible to define a distance between two real-valued functions that express the concept of "the space between them that can be filled by a third function", where the third function is fixed.

Consider functions defined on a domain $mathcalDsubseteq mathbbR$, fix a function $s$ (in my mind something gaussian-shapedw with the peak inside $mathcalD$). For simplicity $int_mathcalD s(x) dx=1$. See figure.

I would define the distance between $f$, $g:mathcalDtomathbbR$ as:

$$
d(f, g) = left|mathrmArgmin_N_s int_mathcalDdx left|f(x) + N_s s(x) - g(x) right|right| = left|mathrmArgmin_N_s d_L1(f+N_s s, g)right|
$$

Is it possible to find a set of hypothesis that makes $d$ a proper distance? I am open also to small changes to the definition of $d$. I am interested when the functions are positive, or when they are as probability density functions, but not necessary normalized to 1.

For sure $d(f, g)geq 0$, I think it is easy to prove $d(g, f)=d(f, g)$:

$$d(g, f) = left|mathrmArgmin_N_s d_L1(g+N_s s, f)right|
=left|mathrmArgmin_N_s d_L1(f-N_s s, g)right|=left|-mathrmArgmin_N_s d_L1(f+N_s s, g)right| = d(f, g)$$

I am not sure about $d(f, g)=0Rightarrow f=g$ and the triangular property.

A less local definition may be:

$$
d(f, g) = left|mathrmArgmin_N_s max_muin C int_mathcalDdx left|f(x) + N_s s(x, mu) - g(x) right|right|
$$

with $mathcalCsubseteq mathcalD$ where $mu$ is something related to the location of $s$, for example the expected value of $s$.

real-analysis metric-spaces hilbert-spaces geometric-functional-analysis

share|cite|improve this question

edited Jul 24 at 9:09

asked Jul 24 at 8:46

Ruggero Turra

1386

add a comment | 

up vote
2
down vote

favorite

1

up vote
2
down vote

favorite

1

1

I am wondering if it is possible to define a distance between two real-valued functions that express the concept of "the space between them that can be filled by a third function", where the third function is fixed.

Consider functions defined on a domain $mathcalDsubseteq mathbbR$, fix a function $s$ (in my mind something gaussian-shapedw with the peak inside $mathcalD$). For simplicity $int_mathcalD s(x) dx=1$. See figure.

I would define the distance between $f$, $g:mathcalDtomathbbR$ as:

$$
d(f, g) = left|mathrmArgmin_N_s int_mathcalDdx left|f(x) + N_s s(x) - g(x) right|right| = left|mathrmArgmin_N_s d_L1(f+N_s s, g)right|
$$

Is it possible to find a set of hypothesis that makes $d$ a proper distance? I am open also to small changes to the definition of $d$. I am interested when the functions are positive, or when they are as probability density functions, but not necessary normalized to 1.

For sure $d(f, g)geq 0$, I think it is easy to prove $d(g, f)=d(f, g)$:

$$d(g, f) = left|mathrmArgmin_N_s d_L1(g+N_s s, f)right|
=left|mathrmArgmin_N_s d_L1(f-N_s s, g)right|=left|-mathrmArgmin_N_s d_L1(f+N_s s, g)right| = d(f, g)$$

I am not sure about $d(f, g)=0Rightarrow f=g$ and the triangular property.

A less local definition may be:

$$
d(f, g) = left|mathrmArgmin_N_s max_muin C int_mathcalDdx left|f(x) + N_s s(x, mu) - g(x) right|right|
$$

with $mathcalCsubseteq mathcalD$ where $mu$ is something related to the location of $s$, for example the expected value of $s$.

real-analysis metric-spaces hilbert-spaces geometric-functional-analysis

share|cite|improve this question

edited Jul 24 at 9:09

asked Jul 24 at 8:46

Ruggero Turra

1386

I am wondering if it is possible to define a distance between two real-valued functions that express the concept of "the space between them that can be filled by a third function", where the third function is fixed.

Consider functions defined on a domain $mathcalDsubseteq mathbbR$, fix a function $s$ (in my mind something gaussian-shapedw with the peak inside $mathcalD$). For simplicity $int_mathcalD s(x) dx=1$. See figure.

I would define the distance between $f$, $g:mathcalDtomathbbR$ as:

$$
d(f, g) = left|mathrmArgmin_N_s int_mathcalDdx left|f(x) + N_s s(x) - g(x) right|right| = left|mathrmArgmin_N_s d_L1(f+N_s s, g)right|
$$

Is it possible to find a set of hypothesis that makes $d$ a proper distance? I am open also to small changes to the definition of $d$. I am interested when the functions are positive, or when they are as probability density functions, but not necessary normalized to 1.

For sure $d(f, g)geq 0$, I think it is easy to prove $d(g, f)=d(f, g)$:

$$d(g, f) = left|mathrmArgmin_N_s d_L1(g+N_s s, f)right|
=left|mathrmArgmin_N_s d_L1(f-N_s s, g)right|=left|-mathrmArgmin_N_s d_L1(f+N_s s, g)right| = d(f, g)$$

I am not sure about $d(f, g)=0Rightarrow f=g$ and the triangular property.

A less local definition may be:

$$
d(f, g) = left|mathrmArgmin_N_s max_muin C int_mathcalDdx left|f(x) + N_s s(x, mu) - g(x) right|right|
$$

with $mathcalCsubseteq mathcalD$ where $mu$ is something related to the location of $s$, for example the expected value of $s$.

real-analysis metric-spaces hilbert-spaces geometric-functional-analysis

share|cite|improve this question

edited Jul 24 at 9:09

asked Jul 24 at 8:46

Ruggero Turra

1386

share|cite|improve this question

share|cite|improve this question

edited Jul 24 at 9:09

edited Jul 24 at 9:09

edited Jul 24 at 9:09

asked Jul 24 at 8:46

Ruggero Turra

1386

asked Jul 24 at 8:46

Ruggero Turra

1386

asked Jul 24 at 8:46

Ruggero Turra

1386

1386

add a comment | 

add a comment | 

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