Consider a continuous symmetric real function $k$ on $I\times I$, where $I$ is a compact real interval. Let $K$ be the integral operator whose kernel is $k$. Assume that $K$ is strictly positive definite. Mercer's theorem states that given there is a orthonormal basis $\{\phi_n\}$ of $L^2(I)$ which is made of eigenfunctions of $K$ and such that $$\sum_{n=0}^\infty \lambda_n\phi_n(x)\overline{\phi_n(y)} = k(x,y)$$ with uniform convergence.
Moreover all the $\phi_n$ are continuous on $I$ and hence bounded.
Is there any condition on $k$ which assures that all the $\phi_n$ are uniformly bounded, i.e. $$\sup_{n\in \mathbb{N}}\sup_{x\in I} |\phi_n(x)|<\infty\,?$$