Let $ pi $ to be a measure of probability on a space $ mathcal {X} $and leave $ Phi = { phi_k } _ {k geqslant 0} $ be an orthonormal base (possibly with complex value) for $ L ^ 2 ( pi) $, with $ phi_0 equiv 1 $. Let $ f in L ^ 2 ( pi) $ be expressible on this basis as $ f = sum_ {k geqslant 0} f_k phi_k $.

In some calculations, it became relevant for me to process the quantities (and linked) 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 form limit $ Q ^ k leqslant c ^ k sum_ {k geqslant 0} | f_k | ^ $ 2 for an explicit sequence $ {c ^ k } _ {k geqslant 0} $.

In principle, $ Q ^ k $ is a quadratic form in the $ {f_k } _ {k geqslant 0} $, so it should be useful. However, the nature of this quadratic form is generally somewhat 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 base $ Phi $.

In the end, I think that a solution to this problem will only be possible once a base has been set, and that it will be something relatively treatable. In the case where $ Phi $ is a Fourier basis, for example, things are pretty good, and we 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 others $ phi_c $; beyond that, it could be quite tricky.

**My question is**: are there *other* orthonormal bases for which the limits of this form should be traceable? I would be particularly happy if there were families of classical orthogonal polynomials for which this is possible, but I don't know where to look for such results. All relevant references will also be received with pleasure.