Assume the truncated discrete Fourier series of a function $f(x)$:
$$
f(x)=\sum_{n=-N}^N C_n e^{2\pi n x i /L}
$$
where $L$ is the period of $f(x)$. There is a rule for $f(x)$ which claims,
$$
\forall x \in [0, L] :\ \ f(x)>0 \iff T_f \succeq 0
$$
where $T_f$ is the Toeplitz matrix.
Here my question is how can we extend this result for multidimensional Fourier series: For example: consider the function $f(x_1,x_2,...,x_n) : [0,L_1] \times [0,L_2] \times ... [0,L_n]\to \mathbb{R}$ which can be formulated as:
$$
f(x_1,x_2,...,x_n)= \sum_{m_1=-N}^N\sum_{m_2=-N}^N... \sum_{m_n=-N}^N C_{m_1m_2...m_n} e^{2\pi i(\frac{m_1 x_1}{L_1}+\frac{m_2 x_2}{L_2}+...+\frac{m_n x_n}{L_n})}
$$
How can we extend the mentioned result for one dimensional Discrete fourier series to this multidimensional one? I guess we should talk about tensors in this case. Am I correct?
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Thanks Hayat, But there is no expansion for multidimensional fourier series in the book.
The question seems to be incorrect. A Fourier series is a set of complex numbers that are neither positive nor negative.
In addition [1 1; -1 1] is not symmetric but it is positive definite .
In addition cvx can also handles complex valued positive definite matrices.
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