Fourier analysis on the roots of unity

Algebraic theory of finite fourier transforms

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $\Phi_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $\Phi_n(x)$ at the other roots of unity. This allows one to explicitly evaluate $\Phi_n(e^{2\pi i/m})$ with $m\in \{3,4,5,6,8,10,12\}$. We use this evaluation with $m=5$ to give a simple reproof of a result of Vaughan (1975) on the maximum coefficient (in absolute value) of $\Phi_n(x)$. We also obtain a formula for $\Phi_n'(e^{2\pi i/m}) / \Phi_n(e^{2\pi i/m})$ with $n \ne m$, which is effectively applied to $m \in \{3,4,6\}$. Furthermore, we compute the resultant of two cyclotomic polynomials in a novel very short way.

Fourier inversions of operations identify those with certain symmetries. A general method with invertible matrices over semirings sets up many different inversive schemes, such as disjunctive normal form, algebraic normal form and classical Fourier analysis. First, unary functions are inverted, and then, with tensor products, the same is done for operations of more than one argument. Classical Fourier analysis is applied to inverting operations on a set of k elements by replacing it by the set of \({k^{\rm th}}\) roots of unity. The Fourier transform of an operation preserving rotations of the roots has many coefficients that are zero, in fact, at most \({1/k}\) are non-zero. A related result holds for operations preserving reflections of the roots.

We come now to reality. The truth is that the digital computer has totally defeated the analog computer. The input is a sequence of numbers and not a continuous function. The output is another sequence of numbers, whether it comes from a digital filter or a finite element stress analysis or an image processor. The question is whether the special ideas of Fourier analysis still have a part to play, and the answer is absolutely yes . G. Strang [1986, p. 290] First we consider the easiest kind of Fourier analysis – that on the additive group ℤ/ n ℤ, the integers modulo n . This is an abelian group of order n and it is cyclic (generated by the congruence class 1 mod n ). Thus it is the simplest possible group for Fourier analysis. Yet it seems to have the most applications. As we saw in the last chapter, it may be viewed as the multiplicative group of n th roots of unity. This can be drawn as n equally spaced points on a circle of radius 1. Thus ℤ/ n ℤ is a finite analogue of the circle (or even of the real line). The discrete Fourier transform on ℤ/ n ℤ, or DFT, arises whenever anyone needs to compute the classical Fourier series and integrals of sines and cosines. In fact, the first application of the discrete Fourier transform was perhaps A.-C. Clairaut's use of it in 1754 to compute an orbit, which can be considered as a finite Fourier series of cosines.

Fourier analysis on the roots of unity

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1. Introduction: The nature of the problems.- The combinatorial structures in question.- Group rings, characters, Fourier analysis.- Number theoretic tools.- Algebraic-combinatorial tools. 2. The field descent: The fixing theorem.- Prescribed absolute value.- Bounding the absoute value.- The modulus equation and the class group. 3. Exponent bounds: Self-conjugacy exponent bounds.- Field descent exponent bounds. 4. Two-weight irreducible cyclic bounds: A necessary and sufficient condition.- All two-weight irreducible cyclic codes?- Partial proof of Conjecture 4.2.4.- Two-intersection sets and sub-difference sets

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of L(1,χ)=∑n=1∞χ(n)n. On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula.

* Author writes in a clear and engaging style * Contains never before published elementary proofs * Author provides new results and detailed exposition * Self-contained, and suitable for use in a classroom setting or for self-study * A highly creative contribution to the theory of modular forms and dirichlet series The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method. After a review of elementary Fourier analysis, the author presents completely new results with new methods, though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications, new formulas for the second factor of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However, all definitions and facts are clearly stated, and precise references are given. The notion of nearly holomorphic modular forms is introduced and applied to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of elliptic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight. The book will be of interest to graduate students and researchers who are interested in special values of L-functions, class number formulae, arithmetic properties of modular forms (especially their values), and the arithmetic properties of Dirichlet series. It treats in detail, from an elementary viewpoint, the simplest cases of a fundamental area of ongoing research, the only prerequisite being a basic course in algebraic number theory.