Abstract
In this chapter we present the basic properties of complex vector spaces and the Fourier transform. Sections 1.1 and 1.2 prepare the subject through the standard definitions of linear independence, bases, coordinates, inner product, and norm. In Section 1.3 we introduce linear transformations in vector spaces, emphasizing the conceptual difference between passive and active ones: the former refer to changes in reference coordinates, while the latter imply a “physical” process actually transforming the points of the space. Permutations of reference axes and the Fourier transformation are prime examples of coordinate changes (Section 1.4), while the second-difference operator in particular and self-adjoint operators in general (Section 1.5) will be important in applications. We give, in Section 1.6, the elements of invariance group considerations for a finite N-point lattice. Finally, in Section 1.7 we examine the axes of a transformation and develop the properties of self-adjoint and unitary operators.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer Science+Business Media New York
About this chapter
Cite this chapter
Wolf, K.B. (1979). Concepts from Complex Vector Analysis and the Fourier Transform. In: Integral Transforms in Science and Engineering. Mathematical Concepts and Methods in Science and Engineering, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0872-1_1
Download citation
DOI: https://doi.org/10.1007/978-1-4757-0872-1_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-0874-5
Online ISBN: 978-1-4757-0872-1
eBook Packages: Springer Book Archive