Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv 1$. Let $f \in L^2 (\pi )$ be expressible in this basis as $f = \sum_{k \geqslant 0} f_k \phi_k$.
In some calculations, it has become relevant for me to deal with (and bound) quantities of the form
\begin{align} Q^k = \int \pi (dx) |f(x)|^2 |\phi_k (x)|^2 \quad \text{for } k \geqslant 1, \end{align}
and ideally, I would like a bound of the form $Q^k \leqslant c^k \sum_{k \geqslant 0} | f_k |^2$ for some explicit sequence $\{ c^k \}_{k \geqslant 0}$.
In principle, $Q^k$ is a quadratic form in the $\{ f_k \}_{k \geqslant 0}$, and so this should be helpful. However, the nature of this quadratic form is in general a bit mysterious; it should end up involving quantities like
\begin{align} Q^k_{ij} &= \int \pi (dx) \phi_i (x) \overline{\phi}_j (x) |\phi_k (x)|^2 \\ &= \int \pi (dx) \left( \phi_i (x) \overline{\phi}_k (x) \right) \cdot \overline{\left( \phi_j (x) \overline{\phi}_k (x) \right)} \quad \text{for } i, j \geqslant 0 \end{align}
and these should depend quite strongly on the properties of the basis $\Phi$.
Ultimately, I think that a resolution to this problem is only going to be possible once a basis is fixed, and fixed to be something fairly tractable. In the case where $\Phi$ is a Fourier basis, for example, things are quite nice, and one can take $c^k \equiv 1$. I think it could also be possible in other cases where $\phi_a \overline{\phi_b}$ can be written as linear combinations of other $\phi_c$; beyond that, it could be quite tricky.
My question is: are there other orthonormal bases for which bounds of this form should be tractable? I would be particularly happy if there were families of classical orthogonal polynomials for which this is possible, but I'm not sure where to look for such results. Any relevant references would be gladly received as well.
