In this paper we formulate Donoho and Logan's large sieve principle for the 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.