How to interpolate/extrapolate a complex function?
$begingroup$
Let us assume that we have $f:mathbb{R} to mathbb{R}$. Also let us assume that $x_1in mathbb{R}$ and $x_2in mathbb{R}$ are given too. With this we can calculate $y_1 = f(x_1)$ and $y_2 = f(x_2)$.
Now if we want to interpolate/extrapolate an in-between value we could take a value like e.g. $alpha in mathbb{R}$ (but for interpolation $alpha$ would be in range $[0,1]$).
With this we could calculate the interpolated values $x_{irp},y_{irp} in mathbb{R}$ like:
$$x_{irp}=alpha cdot x_1 + (1- alpha) cdot x_2$$
$$y_{irp}=alpha cdot y_1 + (1- alpha) cdot y_2$$
Now my question:
Does the same approach also works for complex numbers too? In this case instead of $mathbb{R}$ we would have $mathbb{C}$.
(I know indeed that for multidimensional input and output this approach works, but for complex numbers too?)
complex-numbers interpolation
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
$endgroup$
add a comment |
$begingroup$
Let us assume that we have $f:mathbb{R} to mathbb{R}$. Also let us assume that $x_1in mathbb{R}$ and $x_2in mathbb{R}$ are given too. With this we can calculate $y_1 = f(x_1)$ and $y_2 = f(x_2)$.
Now if we want to interpolate/extrapolate an in-between value we could take a value like e.g. $alpha in mathbb{R}$ (but for interpolation $alpha$ would be in range $[0,1]$).
With this we could calculate the interpolated values $x_{irp},y_{irp} in mathbb{R}$ like:
$$x_{irp}=alpha cdot x_1 + (1- alpha) cdot x_2$$
$$y_{irp}=alpha cdot y_1 + (1- alpha) cdot y_2$$
Now my question:
Does the same approach also works for complex numbers too? In this case instead of $mathbb{R}$ we would have $mathbb{C}$.
(I know indeed that for multidimensional input and output this approach works, but for complex numbers too?)
complex-numbers interpolation
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
$endgroup$
add a comment |
$begingroup$
Let us assume that we have $f:mathbb{R} to mathbb{R}$. Also let us assume that $x_1in mathbb{R}$ and $x_2in mathbb{R}$ are given too. With this we can calculate $y_1 = f(x_1)$ and $y_2 = f(x_2)$.
Now if we want to interpolate/extrapolate an in-between value we could take a value like e.g. $alpha in mathbb{R}$ (but for interpolation $alpha$ would be in range $[0,1]$).
With this we could calculate the interpolated values $x_{irp},y_{irp} in mathbb{R}$ like:
$$x_{irp}=alpha cdot x_1 + (1- alpha) cdot x_2$$
$$y_{irp}=alpha cdot y_1 + (1- alpha) cdot y_2$$
Now my question:
Does the same approach also works for complex numbers too? In this case instead of $mathbb{R}$ we would have $mathbb{C}$.
(I know indeed that for multidimensional input and output this approach works, but for complex numbers too?)
complex-numbers interpolation
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
$endgroup$
Let us assume that we have $f:mathbb{R} to mathbb{R}$. Also let us assume that $x_1in mathbb{R}$ and $x_2in mathbb{R}$ are given too. With this we can calculate $y_1 = f(x_1)$ and $y_2 = f(x_2)$.
Now if we want to interpolate/extrapolate an in-between value we could take a value like e.g. $alpha in mathbb{R}$ (but for interpolation $alpha$ would be in range $[0,1]$).
With this we could calculate the interpolated values $x_{irp},y_{irp} in mathbb{R}$ like:
$$x_{irp}=alpha cdot x_1 + (1- alpha) cdot x_2$$
$$y_{irp}=alpha cdot y_1 + (1- alpha) cdot y_2$$
Now my question:
Does the same approach also works for complex numbers too? In this case instead of $mathbb{R}$ we would have $mathbb{C}$.
(I know indeed that for multidimensional input and output this approach works, but for complex numbers too?)
complex-numbers interpolation
complex-numbers interpolation
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
asked Nov 26 '18 at 13:47
PiMathCLanguagePiMathCLanguage
256
256
add a comment |
add a comment |
1 Answer
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$begingroup$
Yes, exactly the same method will work (with exactly the same limitations that this is a really bad estimate in general).
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
$endgroup$
$begingroup$
Is there a better approach too? I mean like for the $mathbb{R}$ numbers with a square line approximation? How would this one work for $mathbb{C}$?
$endgroup$
– PiMathCLanguage
Nov 26 '18 at 13:51
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes, exactly the same method will work (with exactly the same limitations that this is a really bad estimate in general).
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
$endgroup$
$begingroup$
Is there a better approach too? I mean like for the $mathbb{R}$ numbers with a square line approximation? How would this one work for $mathbb{C}$?
$endgroup$
– PiMathCLanguage
Nov 26 '18 at 13:51
add a comment |
$begingroup$
Yes, exactly the same method will work (with exactly the same limitations that this is a really bad estimate in general).
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
$endgroup$
$begingroup$
Is there a better approach too? I mean like for the $mathbb{R}$ numbers with a square line approximation? How would this one work for $mathbb{C}$?
$endgroup$
– PiMathCLanguage
Nov 26 '18 at 13:51
add a comment |
$begingroup$
Yes, exactly the same method will work (with exactly the same limitations that this is a really bad estimate in general).
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
$endgroup$
Yes, exactly the same method will work (with exactly the same limitations that this is a really bad estimate in general).
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
answered Nov 26 '18 at 13:49
user3482749user3482749
4,047818
4,047818
$begingroup$
Is there a better approach too? I mean like for the $mathbb{R}$ numbers with a square line approximation? How would this one work for $mathbb{C}$?
$endgroup$
– PiMathCLanguage
Nov 26 '18 at 13:51
add a comment |
$begingroup$
Is there a better approach too? I mean like for the $mathbb{R}$ numbers with a square line approximation? How would this one work for $mathbb{C}$?
$endgroup$
– PiMathCLanguage
Nov 26 '18 at 13:51
$begingroup$
Is there a better approach too? I mean like for the $mathbb{R}$ numbers with a square line approximation? How would this one work for $mathbb{C}$?
$endgroup$
– PiMathCLanguage
Nov 26 '18 at 13:51
$begingroup$
Is there a better approach too? I mean like for the $mathbb{R}$ numbers with a square line approximation? How would this one work for $mathbb{C}$?
$endgroup$
– PiMathCLanguage
Nov 26 '18 at 13:51
add a comment |
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