  In this paper we formulate Donoho and Logan's large sieve principle for the
continuous wavelet transform on the Hardy space, adapting the concept of
maximum Nyquist density to the hyperbolic geometry of the underlying space. The
results provide deterministic guarantees for $L_{1}$-minimization methods and
hold for a class of mother wavelets which constitute an orthonormal basis of
the Hardy space and can be associated with higher hyperbolic Landau levels.
Explicit calculations of the basis functions reveal a connection with the
Zernike polynomials. We prove a novel local reproducing formula for the spaces
in consideration and use it to derive concentration estimates of the large
sieve type for the corresponding wavelet transforms. We conclude with a
discussion of optimality of localization in the analytic case by building on
recent results of Kulikov, Ramos and Tilli based on the groundbreaking methods
of Nicola and Tilli. This leads to a sharp uncertainty principle and a local
Lieb inequality for the wavelet transform.

Functional Analysis