We prove the following extension of the Wiener–Wintner Theorem and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows (X,μ, Tt) and f ∈ L(X,μ), there is a set Xf ⊂ X of probability one, so that for all x ∈ Xf we have lim s↓0 ∫ s<|t|<1/s e f(Tt x) dt t exists for all θ. The proof is by way of establishing an appropriate oscillation inequality which ...

Hilbert-Huang Transform (HHT) is a data analysis tool, first developed in 1998, which can be used to extract the periodic components embedded within oscillatory data. This thesis is dedicated to the understanding, application, and development of this tool. First, the background theory of HHT will be described and compared with other spectral analysis tools. Then, a number of applications will b...

Let e be a homogeneous subset of R in the sense of Carleson. Let μ be a finite positive measure on R and Hμ(x) its Hilbert transform. We prove that if limt→∞ t|e ∩ {x | |Hμ(x)| > t}| = 0, then μs(e) = 0, where μs is the singular part of μ.

In the literature adaptive Fourier decomposition is abbreviated as AFD that addresses adaptive rational approximation, or alternatively adaptive Takenaka-Malmquist system approximation. The AFD type approximations may be characterized as adaptive approximations by linear combinations of parameterized Szegö and higher order Szegö kernels. This note proposes two kinds of such analytic approximati...

This paper describe the features extraction algorithm for electrocardiogram (ECG) signal using Huang Hilbert Transform and Wavelet Transform. ECG signal for an individual human being is different due to unique heart structure. The purpose of feature extraction of ECG signal would allow successful abnormality detection and efficient prognosis due to heart disorder. Some major important features ...

This paper presents random residue sequences derived from the number theoretic Hilbert (NHT) transform and their correlation properties. The autocorrelation of a NHT derived sequence is zero for all non-zero shifts which illustrates that these are self-orthogonal sequences. The cross correlation function between two sequences may be computed with respect to the moduli of the either sequence. Th...

If /(OG^P) £>!> then f(x)^Lpy and a considerable literature is devoted to studying the relationship of such pairs of "conjugate" functions to the theory of functions analytic in a half-plane. More to the point of the present note is a series of papers studying the Hubert transform along strictly real variable lines ([2, 3 ] ; further bibliography in [2]). Much less is known about f(x) when f(t)...

For any dimension n ≥ 2, we consider the maximal directional Hilbert transform HU on R associated with a direction set U ⊆ Sn−1: HUf(x) := 1 π sup v∈U ∣∣∣p.v.∫ f(x− tv) dt t ∣∣∣. The main result in this article asserts that for any exponent p ∈ (1,∞), there exists a positive constant Cp,n such that for any finite direction set U ⊆ Sn−1, ||HU ||p→p ≥ Cp,n √ log #U, where #U denotes the cardinali...

Hf (x)= lim →0+ H f (x). It is well known that this limit exists a.e. for all f ∈ L, 1 ≤ p < ∞. In this paper, we will consider the oscillation and variation of this family of operators as goes to zero, which gives extra information on their convergence as well as an estimate on the number of λ-jumps they can have. For earlier results on oscillation and variation operators in analysis and ergod...

Real-variable methods are used to prove the Marcinkiewicz Interpolation Theorem, boundedness of the dyadic and Hardy-Littlewood maximal operators, and the Calderón-Zygmund Covering Lemma. The Hilbert transform is defined, and its boundedness is investigated. All results lead to a final theorem on the pointwise convergence of the truncated Hilbert transform