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Let $f(x+iy): \mathbb{c}\to\mathbb{c}$ be a function on the complex plane.

Let $T_{a+ib}: f(x+iy)\to f(x-a+i(y-b))$ be the translation operator.

For a real-valued one dimensional function we can express a translation as a convolution with a shifted $\delta$. For example, $$T_ng(x)=g(x)*\delta(x-n)$$ or, assuming the Fourier Transform exists and the convolution theorem is valid, $$T_ng(x)=FT^{-1}(\hat{\delta}(x-n)\hat{g}(x))$$

Question pt1: For the complex valued function $f(x+iy)$, how should we properly define a complex valued delta distribution and express the translation operator $T_{a+ib}$ in terms of a convolution with a $\delta$ distribution? I can't really find any information on $\delta$ in the complex plane. Is the idea to treat it as a double convolution with two $\delta$s, one for the real part, and another for the imaginary part: $$T_{a+ib}f(x+iy)=f(x-a+i(y-b))=$$$$\delta_{a+ib}*f(x+iy)=\delta_{ib}*(\delta_{a}*f(x+iy))=f(x-a+i(y-b))$$

Question pt2: How do we express $T_{a+ib}$ in terms of a multiplication with the Fourier Transform of $f(x+iy)$ and $\delta_{a+ib}$ (assuming the Fourier transform exists for both)?

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What you said in the case of $\mathbb R$ can be easily generalized to the case of $\mathbb R^n$. Complex numbers are a particular case when $n=2$.

Define $\delta(f)=f(0)$. This define some distribution. We get

$$(g*\delta(x-a))(t)=\int\limits_{\mathbb R^n}g(x)\delta(t-x-a)=g(t-a).$$

As $FT(f*g)=FT(f)*FT(g)$ we get

$g(t-a)=FT^{-1}(FT(g(t-a)))=FT^{-1}(FT(g*\delta(x-a)))=FT^{-1}(FT(g)FT(\delta(x-a)))=FT^{-1}(FT(g)e^{-2\pi i(\xi, a)}).$

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