Births on April 27

Marc-Antoine Parseval, French mathematician and theorist (d. 1836)

Marc-Antoine Parseval des Chênes (1755–1836) was a notable French mathematician whose name is indelibly linked to what is universally known today as Parseval's Theorem. This foundational principle of analysis and signal processing remarkably presaged the concept of the unitarity of the Fourier transform, a cornerstone of modern mathematics and engineering. His life spanned a tumultuous period in French history, marked by revolution, political upheaval, and intellectual ferment, influencing both his personal trajectory and his scientific contributions.

A Life Amidst Revolution and Enlightenment

Born on April 27, 1755, in Rosières-aux-Salines, France, Marc-Antoine Parseval des Chênes hailed from an aristocratic family. This background placed him squarely in opposition to the seismic shifts brought about by the French Revolution, which began in 1789. As a staunch monarchist, his political convictions led to his imprisonment in 1792 during the height of revolutionary fervor. Even after the Revolution, his criticisms of the evolving political landscape continued, particularly against the government of Napoleon Bonaparte. Such was the perceived threat of his published poetry, which openly critiqued the regime, that Parseval was compelled to flee the country to escape further persecution. Amidst this tumultuous backdrop, his personal life also saw significant change; he married Ursule Guerillot in 1795, a union that, perhaps reflective of the instability of the era, ended in divorce soon after.

Parseval's Enduring Mathematical Legacy

At the heart of Parseval's fame lies the theorem that bears his name. Parseval's Theorem is a fundamental result in Fourier analysis that states the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of the magnitude of its Fourier coefficients or transform. In simpler terms, it expresses the conservation of energy or power of a signal across its different representations (time domain vs. frequency domain). This concept was groundbreaking, effectively presaging the idea of the "unitarity" of the Fourier transform, meaning that the transform preserves the inner product (and thus energy) of functions, making it an isometry between function spaces. It is a cornerstone for applications ranging from physics and engineering to signal processing and quantum mechanics, ensuring that information and energy are conserved during transformations.

Parseval first articulated this significant theorem in his second memoir, dated April 5, 1799, though he did not provide a formal proof, considering it to be "self-evident"—a testament to his intuitive grasp of its fundamental truth. He further elaborated on its implications and applications in his July 5, 1801, memoir, where he employed it effectively to solve various differential equations, demonstrating its practical utility. Interestingly, the theorem found its way into print even before Parseval's collected works. It was first published in 1800 as part of Sylvestre François Lacroix's seminal work, Traité des différences et des séries, appearing on page 377, a detail that underscores the independent recognition and dissemination of mathematical ideas during that period.

Academic Pursuits and Published Works

Despite the profound impact of his eponymous theorem, Parseval’s journey through official academic circles was marked by a curious blend of recognition and rejection. He was nominated an impressive five times to the prestigious French Academy of Sciences, a span covering nearly three decades from 1796 to 1828. Yet, despite these multiple nominations, he was never formally elected as a member. This suggests a complex interplay of scientific merit, political affiliations, and possibly personal character that influenced academic appointments in post-revolutionary France.

His mathematical output, while influential, was surprisingly limited in terms of formal publication. His only known mathematical works were collected and published in 1806 under the rather lengthy and formal title: Mémoires présentés à l'Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences mathématiques et physiques. (Savants étrangers.) This collection brought together five earlier monographs that covered a range of advanced mathematical topics pertinent to the scientific challenges of the day:

• "Mémoire sur la résolution des équations aux différences partielles linéaires du second ordre," (presented May 5, 1798): Addressing the resolution of second-order linear partial differential equations, a critical area in mathematical physics.
• "Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaires du second ordre, à coefficients constants," (presented April 5, 1799): Focusing on series and the complete integration of specific partial differential equations, where his famous theorem was first stated.
• "Intégration générale et complète des équations de la propagation du son, l'air étant considéré avec ses trois dimensions," (presented July 5, 1801): A deep dive into the general and complete integration of equations governing sound propagation, considering air in three dimensions, where he further applied his theorem.
• "Intégration générale et complète de deux équations importantes dans la mécanique des fluides," (presented August 16, 1803): Tackling the integration of two significant equations central to fluid mechanics, a field vital for engineering and physics.
• "Méthode générale pour sommer, par le moyen des intégrales définies, la suite donnée par le théorème de M. Lagrange, au moyen de laquelle il trouve une valeur qui satisfait à une équation algébrique ou transcendante," (presented May 7, 1804): Describing a general method for summing series using definite integrals, building upon Lagrange's theorem to find solutions for algebraic or transcendental equations.

FAQs about Marc-Antoine Parseval des Chênes

What is Parseval's Theorem?

Parseval's Theorem is a fundamental result in Fourier analysis that states the total energy (or power) of a signal is the same whether computed in the time domain or the frequency domain. It quantifies the relationship between a function and its Fourier transform in terms of their respective energies, effectively showing that the Fourier transform is an energy-preserving operation.

Why is Marc-Antoine Parseval famous?

He is primarily famous for Parseval's Theorem, a pivotal concept in mathematics and signal processing. Although he had a tumultuous life and limited formal academic recognition during his time, this theorem has become a cornerstone for understanding and manipulating signals and functions in various scientific and engineering disciplines.

Did Parseval himself prove his theorem?

No, Parseval stated the theorem in his 1799 memoir but did not provide a formal proof, describing it as "self-evident." Subsequent mathematicians later provided rigorous proofs and expanded on its applications, cementing its status as a fundamental principle.

What was Parseval's political stance?

Marc-Antoine Parseval des Chênes was a staunch monarchist who openly opposed the French Revolution and later the government of Napoleon Bonaparte. His political views led to his imprisonment and ultimately to his exile from France.

Was Parseval a member of the French Academy of Sciences?

Despite being nominated five times between 1796 and 1828, Marc-Antoine Parseval des Chênes was never elected as a member of the French Academy of Sciences. This highlights a disconnect between his significant mathematical contributions and formal institutional recognition.

When was Parseval's Theorem first published?

While Parseval first presented the theorem in his 1799 memoir, it was actually first printed in 1800 as part of Sylvestre François Lacroix's book, Traité des différences et des séries, on page 377, predating Parseval's own collected publication of his memoirs in 1806.