Multi-dimensional Chebyshev polynomials?

In summary, the speaker is looking for an extension of Chebyshev polynomials to multiple dimensions in order to fit unknown functions in a general way without imposing pre-conceived ideas about the form. They are hoping for someone to point them in the direction of the generating functions of Chebyshev polynomials in multiple dimensions, and they mention an old paper that addresses this problem.
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Wallace
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I was hoping someone could point me in the direction of a suitable extension of Chebyshev polynomials to mutple dimensions?

I find Chebyshev polynomials useful in situations when I need to fit some function in a general way, imposing as little pre-concieved ideas about the form as possible but still allowing an analytic form of the fitted function. This can be acheived by numerically searching for the values of co-efficients in front of Chebyshev polynomials of increasing order.

However, I want to extend this to some unknown function of multiple dimensions. Does anyone know what the generating functions of Chebyshev polynomials in multiple dimensions are?
 
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1. What are multi-dimensional Chebyshev polynomials?

Multi-dimensional Chebyshev polynomials are a set of mathematical functions used in multiple dimensions to approximate other functions. They are built from the one-dimensional Chebyshev polynomials, which are commonly used for approximating functions in one dimension.

2. What is the significance of using multi-dimensional Chebyshev polynomials?

Multi-dimensional Chebyshev polynomials are especially useful for approximating functions in higher dimensions, as they have desirable properties such as orthogonality, which allows for more accurate approximations compared to other methods.

3. How are multi-dimensional Chebyshev polynomials calculated?

Multi-dimensional Chebyshev polynomials are calculated using a recurrence relation, which involves recursively applying the one-dimensional Chebyshev polynomials in each dimension. This results in a set of polynomials in multiple dimensions, with each polynomial being a product of one-dimensional Chebyshev polynomials.

4. What are the applications of multi-dimensional Chebyshev polynomials?

Multi-dimensional Chebyshev polynomials have various applications, including signal processing, image and video compression, and solving partial differential equations. They are also used in physics, engineering, and other scientific fields to approximate complex functions in multiple dimensions.

5. Are there any limitations to using multi-dimensional Chebyshev polynomials?

While multi-dimensional Chebyshev polynomials have many advantages, they also have limitations. They are most effective for approximating smooth functions, and their accuracy can decrease for functions with discontinuities or sharp peaks. Additionally, the calculation of multi-dimensional Chebyshev polynomials can be computationally intensive for higher dimensions.

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