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I've come across laplace transforms as a method to solve differential/integral equations by diagonalizing the derivative operator. I've also seen generating functions as a similar way to transform a recurrence relation into an algebraic problem by diagonalizing the shift operator. The laplace transform of a convolution is a product of the laplace transforms for each function and similarly the generating function of a convolution is a product of generating function. A different application of transforms would be how the dirichlet convolution can become a product when using Dirichlet generating functions. Yesterday I was reading about multiplication algorithms and discovered one fast way to multiply numbers is based on doing a discrete fourier transform because a product of two numbers can be viewed as a convolution of two polynomials where the x in the polynomials would be the base you use to represent the numbers.

These examples of using transforms to simplify a problem I've seen from various courses in math, but I've never really studied the transforms for there own sake. What would be a good textbook/other source on transform theory that deals with things like the similarities of various integral transforms and generating functions?

Edit: My current math background is essentially a standard undergraduate math courses (introductory algebra, analysis, topology, discrete math, etc).

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  • $\begingroup$ signal processing for the practical side, functional analysis for the theoretical part, and analysis of PDE is in the middle. $\endgroup$ – reuns Nov 24 '16 at 12:23
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The choice of the appropriate book depends on whether you are looking for basic or advanced references. For a basic approach, I would suggest you the following two books:

  • "An introduction to Transform Theory" (1971) by David V. Widder, if you want a general introduction on the topic. Albeit rather old, this book provides a very good description of the basic theory and main applications of the integral transforms, dealing primarily with the Laplace transform and its utility for the solution of ordinary differential equations, but also with several other issues, including the general convolution transform;

  • "An Introduction to Laplace Transforms and Fourier Series" (2014) by Phil Dyke, if you want a more recent nice book giving a general introduction to this topic. It also contains a number of well-explained examples and solutions, which make it easy to read.

For an advanced approach, specific references for single types of transform might be preferrable. I would suggest you the following ones:

  • "The Laplace Transform: Theory and Applications" (1999) by Joel Schiff (already cited in the previous answer), if you search a deep and complete assessment of the applications of the Laplace transform. The link above is a pdf version;

  • "The Fourier Transform and its Applications" (2007) by Brad Osgood, if you want an updated, advanced book on the theory and applications of the Fourier transform. The link above is a pdf version;

  • "Hilbert transforms" (2009) by Frederick W. King, a very comprehensive book on this topic, that also deals with many practical applications in physical sciences.

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  • $\begingroup$ Is there any material you'd recommend on why generating functions feel so similar to transforms? Or an alternative question is there a general meaning of convolution that contains discrete/continuous convolution, dirichlet convolution, etc as special cases and how different transforms tend to simplify convolution? $\endgroup$ – Mehdi2277 Nov 25 '16 at 2:54
  • $\begingroup$ You can find many details on these general issues particularly in the first two references above. The strict relation between generating functions and transforms (for example, the moment generating function and the Laplace transform) might be generally seen in the light of what both they are, i.e. functions that encode the properties of an original function into a different form that is more useful for certain purposes. $\endgroup$ – Anatoly Nov 26 '16 at 8:24
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If you would like to study Laplace transform in depth, there is no better book than Joel Schiff's book on Laplace Transform: http://www.springer.com/gp/book/9780387986982

If you would like to study a large variety of transforms and see how they relate to each other I highly recommend

Advanced Engineering Mathematics with MATLAB, Third Edition By Dean G. Duffy https://www.amazon.ca/Advanced-Engineering-Mathematics-MATLAB-Third/dp/1439816247 In this book they talk about Hilbert Transform, Z-transform, Fourier, Laplace, everything you can think of.

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