By CHARLES K. CHUI (Eds.)
An creation to Wavelets is the 1st quantity in a brand new sequence, WAVELET research AND ITS functions. this is often an introductory treatise on wavelet research, with an emphasis on spline wavelets and time-frequency research. one of the easy issues lined during this publication are time-frequency localization, critical wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. additionally, the writer provides a unified therapy of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the one prerequisite being a simple wisdom of functionality concept and actual research. it's compatible as a textbook for a starting direction on wavelet research and is directed towards either mathematicians and engineers who desire to know about the topic. experts may well use this quantity as a helpful supplementary studying to the enormous literature that has already emerged during this box.
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Additional info for An Introduction to Wavelets
5 in the previous section. 2) with p = 1. 16). 5. 9). 18) converges to f ( x ) as a —> 0+. 18) tends to (F-1f)(x), we have 32 2. Fourier Analysis 2,3. Fourier transform of square-integrable functions In this section, we introduce the definition of the Fourier transform of functions in L2(IR). To do so, the notion of "autocorrelation" is needed. 9. ), so that F(x) is a finite value for every x IR. In fact, we can say a little more as in the following. 10. Let F be the autocorrelation function of f € L 2 (IR).
6. 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.
The width 2/a of the time-window is called the "width of the time-frequency window", and the width 1/a of the frequency window is called the "height of the time-frequency window". 3. Observe that the width of the time-frequency window is unchanged for observing the spectrum at all frequencies. 3. 3. Gabor window.
An Introduction to Wavelets by CHARLES K. CHUI (Eds.)