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?

# Tag: welldefined

## 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}$ .