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Positive-instantaneous frequency and approximation

Frontiers of mathematics in China 2022年第3期17卷 337-371页

作者： Tao QIAN macao Center for mathematical Science Macao University of Science and TechnologyMacaoChina

Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is *** desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in *** approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal *** method decomposes any signal into a few basic signals that possess positive instantaneous *** theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values,and in particular,promotes kernel approximation for multi-variate *** article mainly serves as a *** also gives two important technical proofs of which one for a general convergence result(Theorem 3.4),and the other for necessity of multiple kernel(Lemma 3.7).Expositorily,for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with *** function F has the form F=f+iHf,where H stands for the Hilbert transformation of the *** develop fast converging expansions of F in orthogonal terms of the form F=∑k=1^(∞)c_(k)B_(k),where B_(k)'s are also Hardy space functions but with the additional properties B_(k)(t)=ρ_(k)(t)e^(iθ_(k)(t)),ρk≥0,θ′_(k)(t)≥0,*** original real-valued function f is accordingly expanded f=∑k=1^(∞)ρ_(k)(t)cosθ_(k)(t)which,besides the properties ofρ_(k)andθ_(k)given above,also satisfies H(ρ_(k)cosθ_(k))(t)ρ_(k)(t)sinρ_(k)(t).Real-valued functions f(t)=ρ(t)cosθ(t)that satisfy the conditionρ≥0,θ′(t)≥0,H(ρcosθ)(t)=ρ(t)sinθ(t)are called *** f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of *** above described positive-instantaneou

关键词： möbius transform blaschke product mono-component hilbert transform hardy space inner and outer functions adaptive fourier decomposition rational orthogonal system nevanlinna factorization beurling-lax theorem reproducing kernel hilbert space several complex variables Clifford alge-bra pre-orthogonal AFD

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