By CHARLES K. CHUI (Eds.)
An advent to Wavelets is the 1st quantity in a brand new sequence, WAVELET research AND ITS purposes. this can be an introductory treatise on wavelet research, with an emphasis on spline wavelets and time-frequency research. one of the easy subject matters lined during this ebook are time-frequency localization, imperative wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. furthermore, the writer provides a unified remedy of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the one prerequisite being a uncomplicated wisdom of functionality thought and actual 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 know about the topic. experts might use this quantity as a worthy supplementary examining to the tremendous literature that has already emerged during this box.
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Additional info for An Introduction to Wavelets
And W2(f; •) is a nondecreasing function, it follows that, for (N + I) 2M, as N —» . 20. By using the preceding result on the density of trigonometric polynomials in L 2 ( 0 , 2 ) , we can now establish the following main result of this section. 21. 6) is an isometric isomorphism of l2 onto L 2 ( 0 , 2 ) . In other words, F* maps I2 oneone onto L2(0, 27), such that the Parseval Identity holds, where Ck = Ck(f) is the kth Fourier coefficient of f. Proof. 19 already says that F* maps £2 into L 2 ( 0 , 2 ) .
The Nth degree trigonometric polynomial is of special importance. It is called the "Dirichlet kernel" of degree N. 11) is certainly meaningful if / € Ll(0, 2). On the other hand, if / is any function in lp(0, 2), 1 < p <, then we can define the "inverse discrete Fourier transform" F*-1 of / by: That is, T*- l takes / L p ( 0 , 2 ) to a bi-infinite sequence (c This sequence, of course, defines a Fourier series and is called the sequence of "Fourier coefficients" of the Fourier series. A fundamental question is if this series "converges" to the original function /.
The standard notation is supp /. We will see that for all m = 1,2,.... In addition to having minimum supports, the B-wavelets enjoy many other important properties. We only mention three of them m here. o. wavelet. Secondly, efficient algorithms for computing m and all its derivatives are available. Finally, the B-wavelets m are symmetric for even m and antisymmetric for odd m, meaning: In applications to signal analysis, symmetry and antisymmetry of the wavelet functions are very important. For instance, they are essential to avoid distortion in reconstruction of compressed data.
An Introduction to Wavelets by CHARLES K. CHUI (Eds.)