Econometrics

An Introduction to Wavelets by CHARLES K. CHUI (Eds.)

By CHARLES K. CHUI (Eds.)

An advent to Wavelets is the 1st quantity in a brand new sequence, WAVELET research AND ITS functions. this can be an introductory treatise on wavelet research, with an emphasis on spline wavelets and time-frequency research. one of the easy themes coated during this publication are time-frequency localization, vital wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. moreover, the writer offers a unified therapy of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the one prerequisite being a easy wisdom of functionality idea and genuine research. it really is compatible as a textbook for a starting path on wavelet research and is directed towards either mathematicians and engineers who desire to find out about the topic. experts may perhaps use this quantity as a helpful supplementary studying to the sizeable literature that has already emerged during this box.

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In this case, its Fourier series again describes the spectral behaviour of the digital signal, and the domain of definition of a Fourier series is again the real line IR, which is the frequency domain. However, since Fourier series are 27-periodic, the frequency domain IR in this situation is usually identified with the unit circle. To a mathematician, this identification is more satisfactory, since the "dual group" of ZZ is the "circle group". The importance of both the Fourier transform and the Fourier series stems not only from the significance of their physical interpretations, such as timefrequency analysis of signals, but also from the fact that Fourier analytic techniques are extremely powerful.

Convergence is absolute, and f/ € l i ( 0 , 2 ) with Proof. The almost everywhere absolute convergence will be established, once we have 2. 5), follows immediately from the simple observation that In view of this lemma, we may consider the Fourier series of f, namely: where 2 Hence, if the Fourier series of converges to f, then the two quantities and can be equated. Unfortunately, since f is only l1 L1(0, 2), its Fourier series may even diverge everywhere. 7) are the same. We first settle for a very general statement.

Let a > 0. Then In particular, the Fourier transform of the Gaussian function Proof. 13), they must agree on the complex plane C. 2. Continuous-time convolution and the delta function Let / and g be functions in L1(IR). Then the (continuous-time) convolution of / and g is also an L1 (IR) function h defined by It is clear that h e L1(IR), and in fact, since 28 2. 1) yields: That is, the convolution operation is "commutative". Since / * g is in (IR) we can again convolve / * g with another function u € i1(IR); that is, we may consider (/ * g) * u.

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