GOAL:

I would like to approximate some positive, scalar function, f(x,y) > 0, on a 2D field of finite size i.e. x=[a,b],y=[c,d]

OBSTACLE:

I am familiar with the set of basis functions used in the Fourier series but using the Fourier series requires too many terms. Infinitely many:)

Instead, I desire to find a collection of functions that can be used to approximate f(x,y) with a "finite" number of terms.

QUESTION:

How can I approximate f(x,y) with a collection of functions G(x) and H(y) (see definition of G and H below), such that the number of elements in G or H is finite?

G(x) = [g1(x) ... gn(x)]

H(y) = [h1(y) ... hn(y)]

*the elements in G and H do not have to be of the same form ( g1(x) = sin(nx),g2(x) = sin(mx) )

Thanks in advance:)

EDIT:

I should clarify that G(x) and H(y) need to be combined in an inner product fashion i.e.

f(x,y) = < G(x),H(y) > = g1(x)h1(y) + g2(x)h2(y) + ... gn(x)hn(y)

GOAL:

I would like to approximate some positive, scalar function, $f(x,y) > 0$, on a 2D field of finite size i.e. $x=[a,b],y=[c,d]$

OBSTACLE:

I am familiar with the set of basis functions used in the Fourier series but using the Fourier series requires too many terms. Infinitely many:)

Instead, I desire to find a collection of functions that can be used to approximate $f(x,y)$ with a "finite" number of terms.

QUESTION:

How can I approximate $f(x,y)$ with a collection of functions $G(x)$ and $H(y)$ (see definition of $G$ and $H$ below), such that the number of elements in $G$ or $H$ is finite?

$$G(x) = [g_1(x)\ldots g_n(x)]\\ H(y) = [h_1(y)\ldots h_n(y)]$$

The elements in $G$ and $H$ do not have to be of the same form ($g_1(x)=\sin(nx),g_2(x)=\sin(mx)$)

Thanks in advance:)

EDIT:

I should clarify that $G(x)$ and $H(y)$ need to be combined in an inner product fashion i.e.

$$f(x,y) = < G(x),H(y) > = g_1(x)h_1(y) + g_2(x)h_2(y) + ... g_n(x)h_n(y)$$

GOAL:

I would like to approximate some positive, scalar function, f(x,y) > 0, on a 2D field of finite size i.e. x=[a,b],y=[c,d]

OBSTACLE:

I am familiar with the set of basis functions used in the Fourier series but using the Fourier series requires too many terms. Infinitely many:)

Instead, I desire to find a collection of functions that can be used to approximate f(x,y) with a "finite" number of terms.

QUESTION:

How can I approximate f(x,y) with a collection of functions G(x) and H(y) (see definition of G and H below), such that the number of elements in G or H is finite?

G(x) = [g1(x) ... gn(x)] H(y) = [h1(y) ... hn(y)]

*the elements in G and H do not have to be of the same form ( g1(x) = sin(nx),g2(x) = sin(mx) )

Thanks in advance:)

GOAL:

I would like to approximate some positive, scalar function, f(x,y) > 0, on a 2D field of finite size i.e. x=[a,b],y=[c,d]

OBSTACLE:

I am familiar with the set of basis functions used in the Fourier series but using the Fourier series requires too many terms. Infinitely many:)

Instead, I desire to find a collection of functions that can be used to approximate f(x,y) with a "finite" number of terms.

QUESTION:

How can I approximate f(x,y) with a collection of functions G(x) and H(y) (see definition of G and H below), such that the number of elements in G or H is finite?

G(x) = [g1(x) ... gn(x)] 

H(y) = [h1(y) ... hn(y)]

*the elements in G and H do not have to be of the same form ( g1(x) = sin(nx),g2(x) = sin(mx) )

Thanks in advance:)

EDIT:

I should clarify that G(x) and H(y) need to be combined in an inner product fashion i.e.

f(x,y) = < G(x),H(y) > = g1(x)h1(y) + g2(x)h2(y) + ... gn(x)hn(y)