Generalized Trigonometric Functions Φₙᵏ and Ψₙᵏ: Bounded Exponentials, the Polar Decomposition exp(ixT) = Uₙ · Vₙ, and Directional Boundedness of y⁽ⁿ⁾ = y
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2026
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| _version_ | 1866901072549773312 |
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| author | EL MAHYAOUI, Nabil |
| author_facet | EL MAHYAOUI, Nabil |
| contents | <p>The generalized exponential components E<sub>n</sub><sup>k</sup> carry a rich algebraic structure—a ℤ/nℤ convolution algebra, a cyclic derivative, a DFT extraction formula—all traceable to the matrix identity e<sup>xT</sup> = Σ E<sub>n</sub><sup>k</sup>(x) T<sup>k</sup>. However, E<sub>n</sub><sup>k</sup> grows exponentially for n ≥ 3.</p> <p><strong>Part I</strong> classifies all bounded solutions of y<sup>(n)</sup> = y on rays, lines, and sectors in the complex plane, proving: a DFT support condition for boundedness; a <em>trigonometric rigidity theorem</em> (every bounded restriction to a line has frequency ±1, regardless of n); exactly n distinguished lines at angle π/n; and a discrete Stokes phenomenon.</p> <p><strong>Part II</strong> decomposes the generator iT into its skew-Hermitian part A<sub>n</sub> = i(T+T<sup>−1</sup>)/2 and its Hermitian part H<sub>n</sub> = i(T−T<sup>−1</sup>)/2. Since [A<sub>n</sub>, H<sub>n</sub>] = 0, the naive bounded attempt e<sup>ixT</sup> admits a <em>polar decomposition</em> e<sup>ixT</sup> = U<sub>n</sub>(x) · V<sub>n</sub>(x), where U<sub>n</sub> = exp(xA<sub>n</sub>) is unitary (oscillation) and V<sub>n</sub> = exp(xH<sub>n</sub>) is Hermitian positive definite (growth). Defining Φ<sub>n</sub><sup>k</sup> = [U<sub>n</sub>]<sub>0,k</sub> and Ψ<sub>n</sub><sup>k</sup> = [V<sub>n</sub>]<sub>0,k</sub>, the factorisation becomes E<sub>n</sub><sup>k</sup>(ix) = Σ Φ<sub>n</sub><sup>ℓ</sup> Ψ<sub>n</sub><sup>k−ℓ</sup> at the component level.</p> <p>The three families form a coherent triple: Σ E<sub>n</sub><sup>k</sup> = e<sup>x</sup>, Σ Φ<sub>n</sub><sup>k</sup> = e<sup>ix</sup>, Σ Ψ<sub>n</sub><sup>k</sup> = 1. The Φ<sub>n</sub><sup>k</sup> inherit the full E<sub>n</sub><sup>k</sup> algebraic machinery: matrix identity U<sub>n</sub> = Σ Φ<sub>n</sub><sup>k</sup> T<sup>k</sup>, DFT extraction, convolution addition theorem, unitarity Σ|Φ<sub>n</sub><sup>k</sup>|² = 1, cross-unitarity, reproducing kernel Σ Φ<sub>n</sub><sup>k</sup>(x)Φ̄<sub>n</sub><sup>k</sup>(y) = Φ<sub>n</sub><sup>0</sup>(x−y), det U<sub>n</sub> = 1, and convergence to Bessel functions Φ<sub>n</sub><sup>k</sup> → i<sup>k</sup> J<sub>k</sub>(−x) at exponential rate. For n = 2, Φ<sub>2</sub><sup>0</sup> = cos x and Φ<sub>2</sub><sup>1</sup> = i sin x: the Φ<sub>n</sub><sup>k</sup> are the correct generalization of the trigonometric functions to arbitrary n.</p> <p>Sharp supremum bounds, moment expansions, the spectral theory of A<sub>n</sub> (Chebyshev distribution), and a multiplication obstruction theorem are developed. This paper provides foundational bounded machinery for the E<sub>n</sub><sup>k</sup> program.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19060087 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | Generalized Trigonometric Functions Φₙᵏ and Ψₙᵏ: Bounded Exponentials, the Polar Decomposition exp(ixT) = Uₙ · Vₙ, and Directional Boundedness of y⁽ⁿ⁾ = y EL MAHYAOUI, Nabil generalized exponentials bounded solutions polar decomposition trigonometric functions Bessel functions circulant matrices directional boundedness <p>The generalized exponential components E<sub>n</sub><sup>k</sup> carry a rich algebraic structure—a ℤ/nℤ convolution algebra, a cyclic derivative, a DFT extraction formula—all traceable to the matrix identity e<sup>xT</sup> = Σ E<sub>n</sub><sup>k</sup>(x) T<sup>k</sup>. However, E<sub>n</sub><sup>k</sup> grows exponentially for n ≥ 3.</p> <p><strong>Part I</strong> classifies all bounded solutions of y<sup>(n)</sup> = y on rays, lines, and sectors in the complex plane, proving: a DFT support condition for boundedness; a <em>trigonometric rigidity theorem</em> (every bounded restriction to a line has frequency ±1, regardless of n); exactly n distinguished lines at angle π/n; and a discrete Stokes phenomenon.</p> <p><strong>Part II</strong> decomposes the generator iT into its skew-Hermitian part A<sub>n</sub> = i(T+T<sup>−1</sup>)/2 and its Hermitian part H<sub>n</sub> = i(T−T<sup>−1</sup>)/2. Since [A<sub>n</sub>, H<sub>n</sub>] = 0, the naive bounded attempt e<sup>ixT</sup> admits a <em>polar decomposition</em> e<sup>ixT</sup> = U<sub>n</sub>(x) · V<sub>n</sub>(x), where U<sub>n</sub> = exp(xA<sub>n</sub>) is unitary (oscillation) and V<sub>n</sub> = exp(xH<sub>n</sub>) is Hermitian positive definite (growth). Defining Φ<sub>n</sub><sup>k</sup> = [U<sub>n</sub>]<sub>0,k</sub> and Ψ<sub>n</sub><sup>k</sup> = [V<sub>n</sub>]<sub>0,k</sub>, the factorisation becomes E<sub>n</sub><sup>k</sup>(ix) = Σ Φ<sub>n</sub><sup>ℓ</sup> Ψ<sub>n</sub><sup>k−ℓ</sup> at the component level.</p> <p>The three families form a coherent triple: Σ E<sub>n</sub><sup>k</sup> = e<sup>x</sup>, Σ Φ<sub>n</sub><sup>k</sup> = e<sup>ix</sup>, Σ Ψ<sub>n</sub><sup>k</sup> = 1. The Φ<sub>n</sub><sup>k</sup> inherit the full E<sub>n</sub><sup>k</sup> algebraic machinery: matrix identity U<sub>n</sub> = Σ Φ<sub>n</sub><sup>k</sup> T<sup>k</sup>, DFT extraction, convolution addition theorem, unitarity Σ|Φ<sub>n</sub><sup>k</sup>|² = 1, cross-unitarity, reproducing kernel Σ Φ<sub>n</sub><sup>k</sup>(x)Φ̄<sub>n</sub><sup>k</sup>(y) = Φ<sub>n</sub><sup>0</sup>(x−y), det U<sub>n</sub> = 1, and convergence to Bessel functions Φ<sub>n</sub><sup>k</sup> → i<sup>k</sup> J<sub>k</sub>(−x) at exponential rate. For n = 2, Φ<sub>2</sub><sup>0</sup> = cos x and Φ<sub>2</sub><sup>1</sup> = i sin x: the Φ<sub>n</sub><sup>k</sup> are the correct generalization of the trigonometric functions to arbitrary n.</p> <p>Sharp supremum bounds, moment expansions, the spectral theory of A<sub>n</sub> (Chebyshev distribution), and a multiplication obstruction theorem are developed. This paper provides foundational bounded machinery for the E<sub>n</sub><sup>k</sup> program.</p> |
| title | Generalized Trigonometric Functions Φₙᵏ and Ψₙᵏ: Bounded Exponentials, the Polar Decomposition exp(ixT) = Uₙ · Vₙ, and Directional Boundedness of y⁽ⁿ⁾ = y |
| topic | generalized exponentials bounded solutions polar decomposition trigonometric functions Bessel functions circulant matrices directional boundedness |
| url | https://doi.org/10.5281/zenodo.19060087 |