THREE-VARIABLE ALTERNATING TRIGONOMETRIC FUNCTIONS AND CORRESPONDING FOURIER TRANSFORMS

Abstract. The common trigonometric functions admit generalizations to any higher dimension. In this paper, we restrict ourselves to three dimensional generalization only, focusing on alternating case in detail. Many specific properties of this new class of special functions are studied, such as the orthogonalities, both the continuous one and the discrete one on the 3D lattice of any density, discrete and continuous Fourier transforms, and others. Rapidly increasing precision of the interpolation with increasing density of the 3D lattice is shown in an example.


Introduction
There are many applications of functions which are symmetric or antisymmetric with respect to the symmetry group S n in mathematics.They appear for example in quantum theory or in theory of integrable systems.In [1] (anti)symmetric multivariate exponential functions were defined and corresponding Fourier transforms were given.These functions were examined in detail in dimension two and three in [2].The natural question which arises is the restriction of the symmetry to the subgroup A n of S n consisting of transformations w with det w = 1.Such functions were examined in [3] for A n involving the general number n, and in [4] in greater detail concerning the smallest possible n, namely n = 3.
The exponential functions are not the only functions which admit symmetric, antisymmetric and alternating generalizations to higher dimensions.Their odd and even counterparts, sines and cosines, admit similar generalizations and fulfil many analogical properties as pure exponentials.Symmetric and antisymmetric case in general was covered in [5], alternating case was covered in [6].Two-dimensional symmetric and antisymmetric generalizations were studied in detail in [7,8].
The aim of this paper is to study in detail the alternating generalization of sine and cosine functions in dimension three.
(Anti)symmetric multivariate sine and cosine functions, considered in [5,7,8], as well as alternating multivariate sine and cosine functions, studied in [6] and in this paper, are closely related to symmetric and antisymmetric orbit functions studied in [9][10][11].Because the definition of the considered functions and orbit functions are similar (roughly speaking, the exponentials are replaced by sines and cosines), they satisfy the same or similar properties, namely symmetry relations, reductions to the dominant or semidominant forms, periodicity etc.The discrete Fourier transforms of sine and cosine functions can be derived with the help of relations valid for exponential functions.[8] The restriction of the functions to three variables allows us to be more specific about the details of their properties in the following sections.This is most notable when describing their discretization and orthogonality relations.
The paper consists of several parts.After the first part, devoted to the definition of three dimensional alternating sine and cosine functions and their basic properties, come the parts describing their continuous orthogonality relations, their decomposition rules and the two parts treating four kinds of discrete cosine and sine transforms.

Continuous orthogonality
The alternating trigonometric functions sin (k,l,m) resp.cos (k,l,m) are pairwise orthogonal on F ( Ãaff 3 ).Let us denote Then we have Let us have a function f : R 3 → R with the following properties: rotational symmetry, that is, y, z) for all r, s, t ∈ Z, and odd in each variable, that is, Let f be sufficiently smooth.Then it can be expanded in terms of the alternating sine functions, respectively, using formulas Similarly, any function f : R 3 → R with the symmetries and sufficiently smooth can be expanded in terms of the alternating cosine functions.The expansion is done using the formulas

Product decomposition
The product of two alternating cosines, cos (λ,µ,ν) cos (λ ,µ ,ν ) (we omit general argument (x, y, z) here) can be decomposed into the sum of alternating cosines using the following formulas.The decomposition can serve e. g. for the derivation of recursion relations for the considered functions.

AMDCT-1
For given N ∈ N let us define a lattice Table 1.Integral error estimates for the approximations in Fig. 4.This lattice is chosen such way that it fulfills the whole space when symmetries (3) and ( 4) are applied to it.Evidently, we have a partial freedom which parts of boundary to include in the lattice.The lattice (9) contains 1 3 (N 3 + 3N 2 + 5N + 3) points.For example, Another example, for N = 5, is shown in Fig. 2. The alternating cosines cos (k,l,m) are pairwise orthogonal on this lattice, namely we have 0≤r,s,t≤N, r≥s≥t or s>r>t where otherwise, Let f be a real function defined on F N .Then it can be expanded into a sum of alternating cosines, where the coefficients a (k,l,m) are given by the formula As an interpolation example, we take f (x, y, z) = cos 2πx cos 2πy cos 2πz cos 10(x + y + z), (10) and compute four interpolations f N for N = 1, 2, 3, 4 using alternating cosines.The graph of the function (10) is shown in Fig. 3 and four approximations in Fig. 4. By visual inspection, approximations f N get closer to f as N increases.This is verified by computing integral error estimates shown in Table 1.

AMDCT-2
Alternating discrete cosine transform of second kind uses a lattice This lattice contains Let f be a real function defined on FN .Then it can be expanded into a sum of alternating cosines, where the coefficients b (k,l,m) are given by the formula

AMDCT-3
Alternating discrete cosine transform of the third kind uses the lattice Instead of alternating cosines cos (k,l,m) , shifted cosines cos (k+1/2,l+1/2,m+1/2) are used.They have similar properties as normal alternating cosines with integer arguments.For scalar product we obtain 0≤r,s,t≤N −1, r≥s≥t or s>r>t Let g be a real function defined on F N .Then it can be expanded into a sum of shifted alternating cosines, where the coefficients c (k,l,m) are given by the formula

AMDCT-4
Alternating discrete cosine transform of the fourth kind uses the lattice FN and shifted cosines as in the third case.We have 0≤r,s,t≤N −1, r≥s≥t or s>r>t Let f be a real function defined on FN .Then it can be expanded into a sum where the coefficients d (k,l,m) are given by the formula

AMDST-1
For given N ∈ N let us define a lattice This lattice contains 1 3 (N 3 − 3N 2 + 5N − 3) points.For example, The alternating sines sin (k,l,m) are pairwise orthogonal on this lattice, namely we have 1≤r,s,t≤N −1, r≥s≥t or s>r>t Let f (S) be a real function defined on F (S) N .Then it can be expanded into a sum of alternating sines, where the coefficients a (S) (k,l,m) are given by the formula As an interpolation example, we take f (S) (x, y, z) = sin 2πx sin 2πy sin 2πz sin 10(x + y + z), (12) and compute four interpolations f (S) N for N = 1, 2, 3, 4 using alternating sines.Picture of ( 12) is shown in Fig. 5 and four approximations in Fig. 6.
By visual inspection, approximations f (S) N get closer to (12) as N increases.This is verified by computing integral error estimates shown in Table 2.

AMDST-2
Alternating discrete sine transform of second kind uses the same lattice as AMDCT-2, that is where FN is given by (11). vol.
no. / Three-Variable Alternating Trigonometric Functions Table 2. Integral error estimates for the approximations in Fig. 6.

AMDST-3
Alternating discrete sine transform of the third kind uses the lattice Instead of alternating sines sin (k,l,m) , shifted sines sin (k+1/2,l+1/2,m+1/2) are used.They have similar properties as normal alternating sines with integer arguments.For scalar product we obtain 1≤r,s,t≤N, r≥s≥t or s>r>t where c p = 1 2 for p = N and c p = 1 otherwise and 0 Let g (S) be a real function defined on F N (S) .Then it can be expanded into a sum of shifted alternating sines,

AMDST-4
Alternating discrete sine transform of the fourth kind uses the lattice F (S) N given by ( 13) and shifted sines as in the third case.We have

Conclusion
We have presented a detailed study of alternating three dimensional trigonometric functions and their properties including product decomposition, continuous and discrete orthogonality and interpolation problem.Practical computational aspects of the formalism presented in this paper need further investigation, namely a fast Fourier transform analog and comparing to the usual discrete Fourier transform in three dimensions.

Figure 3 .
Figure 3.The function (10) used in the interpolation example.