By CHARLES K. CHUI (Eds.)

ISBN-10: 0121745651

ISBN-13: 9780121745653

ISBN-10: 0121745848

ISBN-13: 9780121745844

ISBN-10: 0585470901

ISBN-13: 9780585470900

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**

**Example text**

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.)

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