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I'm working with convolution operations and encountered a theoretical question that I hope someone here can help clarify.

Consider a Gaussian function $g(x)$ and a delta distribution $\delta(x - (a+bi))$ where $a,b > 0$ and $i$ is the imaginary unit. The delta distribution is centered at a complex location $(a+bi)$.

My question is: What is the result of the convolution between $g(x)$ and $\delta(x - (a+bi))$? Specifically, is the result a Gaussian function shifted along the complex plane, positioned on the line $\text{Im} = b$, centered at $\text{Re} = a$?

In mathematical terms, given: $$g(x) = \exp\left(-\frac{x^2}{2\sigma^2}\right) $$ and $$ \delta(x - (a+bi)), $$ what is $g(x) * \delta(x - (a+bi))$? Is it correct to interpret the convolution as yielding a Gaussian centered at $a$ on the line $\text{Im} = b$ in the complex plane? How is this operation defined and interpreted when the delta distribution is centered at a complex number?

Thank you for your insights!

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This is nothing but the distribution of the random variable $a+ib+X$ concentrated on a horizontal line of the complex plane, or if you prefer, the random variable $(a+X,b)$ of $R^2.$

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