We study decomposition of functions in the Hardy space H 2 (D) into linear combinations of the basic functions (modified Blaschke products) in the systemwhere the points a n 's in the unit disc D are adaptively chosen in relation to the function to be decomposed. The chosen points a n 's do not necessarily satisfy the usually assumed hyperbolic non-separability conditionin the traditional studies of the system. Under the proposed procedure functions are decomposed into their intrinsic components of successively increasing non-negative analytic instantaneous frequencies, whilst fast convergence is resumed. The algorithm is considered as a variation and realization of greedy algorithm.Keywords Rational orthonormal system · Blaschke product · Complex hardy space · Analytic signal · Instantaneous frequency · Mono-components · Adaptive decomposition of functions · Greedy algorithm Mathematics Subject Classifications (2010) 42A50 · 32A30 · 32A35 · 46J15

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We propose a function decomposition model, called intrinsic mono-component decomposition (IMD). It is a continuation of the recent study on adaptive decomposition of functions into mono-components (MCs). It is a further improvement of two recent results of which one is adaptive decomposition of functions into modified inner functions, and the other is decomposition by using adaptive Takenaka-Malmquist systems. The proposed new decomposition model is of less restriction and thus gains more adaptivity. The theory is valid to both the unit circle and the real line contexts. Copyright

One-dimensional adaptive Fourier decomposition, abbreviated as 1-D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher order Szegö kernels of the context. In the present paper we study multi-dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product-TM Systems; and the other uses Product-Szegö Dictionaries. With the Product-TM Systems approach we prove that at each selection of a pair of parameters the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product-Szegö dictionary approach we show that Pure Greedy Algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre-Orthogonal Greedy Algorithm (P-OGA). We prove its convergence and convergence rate estimation, allowing a weak type version of P-OGA as well. The convergence rate estimation of the proposed P-OGA evidences its advantage over Orthogonal Greedy Algorithm (OGA). In the last part we analyze P-OGA in depth and introduce the concept P-OGA-Induced Complete Dictionary, abbreviated as Complete Dictionary . We show that with the Complete Dictionary P-OGA is applicable to the Hardy H 2 space on 2-torus.

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