## Will irrational parameters make a problem not well-defined on complexity

Given a set $$𝑁=\{𝑎_1,⋯,𝑎_𝑛\}$$ where all $$𝑎_𝑖$$s are rational positive numbers and $$\sum_{i\in N}a_i=1$$, find a subset 𝑆⊆𝑁 such that $$(\sqrt{2\sum_{i\in S}a_i}-1)^2$$ is minimized. Does the appearance of √ make the problem ill-defined with regrading to complexity? If well-defined, it is NP-hard, right?

## For which tempered distributions is the fractional derivative well-defined?

Let $$\gamma \geq 0$$ and consider the fractional derivative operator defined in Fourier domain by $$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\varphi\} (\omega),$$ where $$\varphi \in \mathcal{S}(\mathbb{R})$$ is a smooth and rapidly decaying function.

Of course, the definition can be extended to much more functions than $$\varphi \in \mathcal{S}(\mathbb{R})$$, including some, but not all, tempered distributions. It is for instance possible to extend $$\mathrm{D}^{\gamma}$$ to any compactly supported distribution (as for any convolution operator from $$\mathcal{S}(\mathbb{R})$$ to $$\mathcal{S}'(\mathbb{R})$$).

My question is the following: Is there a good notion of the “domain of definition” of the operator $$\mathrm{D}^{\gamma}$$, understood as the largest topological vector space $$\mathcal{S}(\mathbb{R}) \subseteq \mathcal{X} \subseteq \mathcal{S}'(\mathbb{R})$$ such that $$\mathrm{D}^{\gamma} : \mathcal{X} \rightarrow \mathcal{S}'(\mathbb{R})$$ is well-defined and continuous? Or at least, if the question is somehow meaningless, any natural construction that will include many tempered distributions in a satisfactory* manner?

*To give a bit of context, I am especially interested by the fractional case where $$\gamma \notin \mathbb{N}$$. The question is obvious for $$\gamma = n \in \mathbb{N}$$, since one can select $$\mathcal{X} = \mathcal{S}'(\mathbb{R})$$. However. when $$\gamma$$ is purely fractional, there is no hope to define the product $$(\mathrm{i} \omega)^{\gamma} \mathcal{F}\{u\} (\omega)$$ when $$u \in \mathcal{S}'(\mathbb{R})$$ is too irregular around the origin, which means morally that $$u$$ growth too fast at infinity. “In a satisfactory manner” would be a way of specifying properly a good “growth property” of $$u \in \mathcal{X}$$.