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2018 | Book

Fourier Series on Hilbert Spaces Read chapter Read first chapter

Fourier Analysis of Economic Phenomena

Author: Prof. Dr. Toru Maruyama

Publisher: Springer Singapore

Book Series : Monographs in Mathematical Economics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This is the first monograph that discusses in detail the interactions between Fourier analysis and dynamic economic theories, in particular, business cycles.Many economic theories have analyzed cyclical behaviors of economic variables. In this book, the focus is on a couple of trials: (1) the Kaldor theory and (2) the Slutsky effect. The Kaldor theory tries to explain business fluctuations in terms of nonlinear, 2nd-order ordinary differential equations (ODEs). In order to explain periodic behaviors of a solution, the Hopf-bifurcation theorem frequently plays a key role. Slutsky's idea is to look at the periodic movement as an overlapping effect of random shocks. The Slutsky process is a weakly stationary process, the periodic (or almost periodic) behavior of which can be analyzed by the Bochner theorem. The goal of this book is to give a comprehensive and rigorous justification of these ideas. Therefore, the aim is first to give a complete theory that supports the Hopf theorem and to prove the existence of periodic solutions of ODEs; and second to explain the mathematical structure of the Bochner theorem and its relation to periodic (or almost periodic) behaviors of weakly stationary processes.Although these two targets are the principal ones, a large number of results from Fourier analysis must be prepared in order to reach these goals. The basic concepts and results from classical as well as generalized Fourier analysis are provided in a systematic way.Prospective readers are assumed to have sufficient knowledge of real, complex analysis. However, necessary economic concepts are explained in the text, making this book accessible even to readers without a background in economics.

Table of Contents

Frontmatter
Chapter 1. Fourier Series on Hilbert Spaces
Abstract
Let e 1, e 2, …, e l be the standard basis of an l-dimensional Euclidean space consisting of l unit vectors. Then any vector x can be expressed as
$$\displaystyle x=\sum _{i=1}^l c_ie_i $$
and such an expression is determined uniquely. The coefficients c 1, c 2, ⋯, c l are computed as c i = 〈x, e i〉 (inner product).
Toru Maruyama
Chapter 2. Convergence of Classical Fourier Series
Abstract
We have discussed basic contents of the theory of Fourier series on a general Hilbert space. We now proceed to the classical problem concerning the Fourier series expansion of an integrable function with respect to the trigonometric functions. If we choose \(\mathfrak {L}^2([-\pi , \pi ], \mathbb {C})\) as a Hilbert space and
$$\displaystyle \frac {1}{\sqrt {2\pi } }, \frac {1}{\sqrt {\pi }} \cos x , \; \frac {1}{\sqrt {\pi } } \sin x , \; \cdots ,\; \frac {1}{\sqrt {\pi } } \cos nx , \; \frac {1}{\sqrt {\pi } } \sin nx , \; \cdots ; \; n=1,2, \cdots $$
as a complete orthonormal system, the Fourier series of \(f\in \mathfrak {L}^2([-\pi , \pi ],\mathbb {C})\) is given in the form
$$\displaystyle \frac {a_0}{2} + \sum _{n=1}^\infty (a_n \cos nx + b_n \sin nx) , $$
where
$$\displaystyle a_n = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \cos nx \; dx ,\quad b_n = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \sin nx \; dx . $$
This Fourier series converges to f in \(\mathfrak {L}^2\)-norm.
Toru Maruyama
Chapter 3. Fourier Transforms (I)
Abstract
The objects of classical theory of Fourier series discussed in the preceding chapter are periodic functions. Is it possible to construct an analogous theory for nonperiodic functions? It is the theory of Fourier transforms which answers this question positively.
Toru Maruyama
Chapter 4. Fourier Transforms (II)
Abstract
In the previous chapter, we observed a peculiar relation between the smoothness and the rapidity of vanishing at infinity of a function f, as well as its Fourier transform \(\hat {f}\). Based upon this observation, we introduce an important function space \(\mathfrak {S}\), which is invariant under the Fourier transforms. We then proceed to \(\mathfrak {L}^2\)-theory of Fourier transforms due to M. Plancherel. As a simple application of Plancherel’s theory, we discuss how to solve integral equations of convolution type. Finally, a tempered distribution is defined as an element of \(\mathfrak {S}'\), and its Fourier transform is examined in detail.
Toru Maruyama
Chapter 5. Summability Kernels and Spectral Synthesis
Abstract
We have discussed the (C, 1)-summability of Fourier series in Chapter 2. We recapitulate this topic primarily in the framework of Fourier series in complex form.
Toru Maruyama
Chapter 6. Fourier Transforms of Measures
Abstract
So far, we have studied Fourier transforms or Fourier coefficients of functions defined on \(\mathbb {R}\) or \(\mathbb {T}\). Inverse procedures to recover original functions from given Fourier transforms or Fourier coefficients were also discussed (spectral synthesis). However, there are many functions to which the methods of classical Fourier analysis can not be applied. In this chapter, we develop the theory of Fourier transforms of measures as a similar but new method to overcome such difficulties.
Toru Maruyama
Chapter 7. Spectral Representation of Unitary Operators
Abstract
The main topic of this chapter is the spectral representation of unitary operators as well as one-parameter groups of unitary operators. That is, the problem is how to represent such objects by certain analogues of Fourier transforms. We are going to describe theories based upon the Herglotz–Bochner theorem already discussed in Chap. 6. However, the Herglotz–Bochner theorem can be conversely deduced from the spectral representation theorem of unitary operators.
Toru Maruyama
Chapter 8. Fourier Analysis of Periodic Weakly Stationary Processes
Abstract
During the decade around 1930, the world economy was thrown into a serious depression that nobody had previously experienced.
Toru Maruyama
Chapter 9. Almost Periodic Functions and Weakly Stationary Stochastic Processes
Abstract
It is a basic idea for the classical theory of Fourier series to express periodic functions as compositions of harmonic waves. This idea can be successfully extended to nonperiodic functions by means of Fourier transforms. However, we will be confronted with a lot of obstacles when we consider \(\mathfrak {L}^p\)-function spaces in the case p > 2.
Toru Maruyama
Chapter 10. Fredholm Operators
Abstract
A bounded linear operator acting between Banach spaces is called a Fredholm operator if the dimension of its kernel and the codimension of its image are both finite. An equation defined by a Fredholm operator sometimes enjoys a nice property which reduces the difficulties associated with infinite dimension to the finite dimensional problem. The object of this chapter is to study the basic elements of Fredholm operators, which will be made use of in the next chapter in the context of bifurcation theory.
Toru Maruyama
Chapter 11. Hopf Bifurcation Theorem
Abstract
The Hopf bifurcation theorem provides an effective criterion for finding periodic solutions for ordinary differential equations. Although various proofs of this classical theorem are known, there seems to be no easy way to arrive at the goal.
Toru Maruyama
Backmatter
Metadata
Title
Fourier Analysis of Economic Phenomena
Author
Prof. Dr. Toru Maruyama
Copyright Year
2018
Publisher
Springer Singapore
Electronic ISBN
978-981-13-2730-8
Print ISBN
978-981-13-2729-2
DOI
https://doi.org/10.1007/978-981-13-2730-8

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