= Solution
The <closure property of positive-semidefinite kernels> gives three constructions, each seen directly from the <feature maps> $\varphi_1,\varphi_2$.
* \b[Nonnegative sums:] $k=\alpha k_1+\beta k_2$, for $\alpha,\beta\geq0$. Its <feature map> is $(\sqrt\alpha\varphi_1(x),\sqrt\beta\varphi_2(x))$ in the <Hilbert space> direct sum; expanding its <inner product> gives $k$.
* \b[Pointwise products:] $k(x,x')=k_1(x,x')k_2(x,x')$. In the Hilbert <tensor product>, use $\varphi(x)=\varphi_1(x)\otimes\varphi_2(x)$: the <inner product> of two such elementary tensors is the product of their two <inner products>.
* \b[Input-dependent weighting:] $k(x,x')=a(x)k_1(x,x')a(x')$ for any real-valued function $a$. Its <feature map> is $a(x)\varphi_1(x)$, so signs of $a$ are allowed. Nonnegative scalar multiplication is the special case of constant $a$; the weighting factor $a(x)a(x')$ is itself a rank-one <positive-semidefinite kernel>.
Another useful consequence is pullback: if $h:\mathcal X\to\mathcal Z$ and $k_0$ is a <positive-semidefinite kernel> on $\mathcal Z$, then $k_0(h(x),h(x'))$ has <feature map> $\varphi_0\circ h$.
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