Wavelets in Closed Forms
Almost all the known orthonormal wavelets, except for the Haar and Shannon wavelets, cannot be expressed in closed form. By a closed-form expression we mean a representation of the function in terms of elementary functions, such as trigonometric, exponential, or rational functions, or in terms of special functions of mathematical physics, such as gamma, Bessel, or hypergeometric functions. We outline a procedure for obtaining orthonormal wavelets, as well as nonorthogonal, but interpolating, wavelets. We then apply this procedure to specific cases and obtain a number of orthonormal wavelets and nonorthogonal, but interpolating, wavelets in closed-form expressions.
Unable to display preview. Download preview PDF.
- A. Erdelyi, W. Magnus, F. Oberhettinger, and E Tricomi. Tables of Integral Transforms, Vols. I, II, McGraw-Hill, New York, 1954.Google Scholar
- I. Gradshteyn and I. Ryzhik Tables of Integrals, Series,and Products, Academic Press, New York, 1965.Google Scholar
- W. Jones. A unified approach to orthogonally multiplexed communications using wavelet basis and digital filter banks, PhD. Thesis, University of Ohio, August, 1994.Google Scholar
- B. Ja. Levin. Distribution of Zeros of Entire Functions, Translation Math. Monographs, Vol. 5, American Mathematical Society, Providence, RI, 1964.Google Scholar
- Y. Meyer. Ondelettes et Operateurs, Vol. I, Hermann, Paris, 1990.Google Scholar
- J. Proakis. Digital Communications, 3rd ed., McGraw-Hill, New York, 1995.Google Scholar
- W. H. Young. On infinite integrals involving a generalization of the sine and cosine functions, Quart. J. Math. 4 1912, 161–177.Google Scholar
- G. Walter. Translation and dilation invariance in orthogonal wavelets, Appl. Comput. Harmonic Anal. 1994, 344–349.Google Scholar
- G. Walter. Wavelets and Other Orthogonal Systems with Applications, CRC Press, Boca Raton, FL, 1994.Google Scholar
- A. I. Zayed. Function and Generalized Function Transformations, CRC Press, Boca Raton, FL, 1996.Google Scholar
- A. I. Zayed. Advances in Shannon’s Sampling Theory, CRC Press, Boca Raton, FL, 1993.Google Scholar