Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta\_1, \ldots, \theta\_k) \mapsto e^{i \sum\_j c\_j \theta\_j}$ for integral $c\_j$'s. This basis has also the nice property that each element in it is point-wise bounded by $1$ in absolute value. In general, an $L^2$ function takes the form $g = \sum\_n q\_n f\_n$ where $f\_n$ are the basis functions described above. $g$ can have arbitrary bad $L^\infty$ bound, since the only requirement is that $\sum q\_n^2 = 1$, but $\sum\_n q\_n$ can be $\infty$. 


Now my question is, for the unitary or orthogonal group, what is the smallest uniform $L^\infty$ bound one can achieve on an orthogonal basis on the space of class functions? For the unitary groups, any orthogonal basis of class functions forms an orthogonal basis in the space of symmetric functions of the eigenvalues under the inner product such that the Schur polynomials are orthonormal. So the question can be phrased in terms of symmetric functions. For the orthogonal groups, one orthogonal (not normalized) basis of class functions comes from $\prod\_{i=1}^n (Tr X^i)^{\alpha\_i}$ where for some odd $i$, $\alpha\_i$ is odd (I am not entirely sure about this, but see [this paper](http://stat.stanford.edu/~cgates/PERSI/papers/random_matrices.pdf) theorem 4).


Edit: what about all $L^2$ functions in general? Can one get a uniform $L^\infty$ bound on the basis of size $e^{\mathcal{O}(n)}$ for $SO(n)$ or $U(n)$?



L^2 basis of class functions on a compact Lie group that are point-wise small
There is a sharp comparison of sup norm to $L^2$ norm on compact topological groups $K$. (I saw a proof of the version of this for orthogonal groups or spheres in Stein-Weiss "Fourier Analysis on Euclidean Spaces", but the argument succeeds generally. E.g., section 7 of <http://www.math.umn.edu/~garrett/m/mfms/notes_c/spheres_I.pdf>)


The assertion (as alluded-to by @Alain Valette) is that the sup of the ratio of sup norm to $L^2$ norm is $\sqrt{dim V}$ if the total measure is normalized to 1, where $V$ is a (e.g.) left $K$-stable collection of functions on $K$. This estimate is *sharp*, in that it is attained.


Indeed, the argument does not depend on the space of functions $V$ being an irreducible repn, and it is easy to do tangible experiments on products of circles to see the phenomenon manifest.


Edit: (Thx @Denis... for noting broken link.) And, yes, the Stein-Weiss trick does not refer to basis elements, but to a subspace, so does not answer the first, original question exactly. Only the question-at-the-end. However, on another hand, the *pointwise* interaction of orthonormal basis elements is illuminated by the inevitability of that estimate. 


