# Convolution with delta in complex plane

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)?

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

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