There is alot of work to do for " fake function theory ".

It seems to be growing superexponentially

So far the main idea was to approximate a nonentire function by an entire one with all derivatives positive at 0. And then considering alternative solutions and zero's of those functions.

Recently we also started considering fake function theory to approximate a nonentire function with an entire one , WITHOUT the restriction of the signs of the derivatives. ( mick's fake ln(x^2+1) ).

Least squares methods can be used to measure the " quality " of the fake function.

Connections to many other fields of math seem to occur.

BUT there is ALSO a second possible deviation from the main idea :

Approximate an entire function with not all derivatives nonnegative at 0 with an entire function with all derivatives positive at 0.

I call this " hyperfake function theory ".

It is a subset of " fake function theory ".

A fake function can be hyper or not , and all hyperfake are fakes.

I assume series multisections to be important.

Natural questions :

A) Is exp(x)/2 the best hyperfake function for sinh(x) ?

Is it unique in some sense ?

In this example there are an infinite amount of negative derivatives (for the given function) but perhaps even more intresting is the case when there are only a finite number of negative derivatives.

Maybe call that " Ultrafake function theory ".

This leads to imho the following natural question :

Ultrafake exercise 1

B) find the fake function for exp(x) - (x^3 / 5).

Note : ultrafake function theory seems to make more sense when the given function f(z) satisfies :

x > 0 => f(x) > 0 , f ' (x) > 0 and f " (x) > 0.

( the selfreference could not be more subtle, right !? )

...

Not completely sure about the consistancy of ultrafake ...

Mainly because exp(x) - ultrafake(exp(x) - (x^3 / 5)) should be a Taylor series with all + derivatives but grow about O(x^3) ??

Do fake function theory algorithms give an error for exp(x) - (x^3 / 5) ?

Anyways hyperfake seems a good concept.

regards

tommy1729

" purpose is the biggest gift after money , love and health "

tommy1729

It seems to be growing superexponentially

So far the main idea was to approximate a nonentire function by an entire one with all derivatives positive at 0. And then considering alternative solutions and zero's of those functions.

Recently we also started considering fake function theory to approximate a nonentire function with an entire one , WITHOUT the restriction of the signs of the derivatives. ( mick's fake ln(x^2+1) ).

Least squares methods can be used to measure the " quality " of the fake function.

Connections to many other fields of math seem to occur.

BUT there is ALSO a second possible deviation from the main idea :

Approximate an entire function with not all derivatives nonnegative at 0 with an entire function with all derivatives positive at 0.

I call this " hyperfake function theory ".

It is a subset of " fake function theory ".

A fake function can be hyper or not , and all hyperfake are fakes.

I assume series multisections to be important.

Natural questions :

A) Is exp(x)/2 the best hyperfake function for sinh(x) ?

Is it unique in some sense ?

In this example there are an infinite amount of negative derivatives (for the given function) but perhaps even more intresting is the case when there are only a finite number of negative derivatives.

Maybe call that " Ultrafake function theory ".

This leads to imho the following natural question :

Ultrafake exercise 1

B) find the fake function for exp(x) - (x^3 / 5).

Note : ultrafake function theory seems to make more sense when the given function f(z) satisfies :

x > 0 => f(x) > 0 , f ' (x) > 0 and f " (x) > 0.

( the selfreference could not be more subtle, right !? )

...

Not completely sure about the consistancy of ultrafake ...

Mainly because exp(x) - ultrafake(exp(x) - (x^3 / 5)) should be a Taylor series with all + derivatives but grow about O(x^3) ??

Do fake function theory algorithms give an error for exp(x) - (x^3 / 5) ?

Anyways hyperfake seems a good concept.

regards

tommy1729

" purpose is the biggest gift after money , love and health "

tommy1729