Approximates an original function by sines. If the function is "well behaved enough", the approximation is to arbitrary precision.

Fourier's original motivation, and a key application, is solving partial differential equations with the Fourier series.

Can only be used to approximate for periodic functions (obviously from its definition!). The Fourier transform however overcomes that restriction:

The Fourier series behaves really nicely in $L^2$, where it always exists and converges pointwise to the function: Carleson's theorem.

The history of computer science is a vast and intricate narrative that traces the evolution of computing from ancient tools to the sophisticated technologies we use today. Here's an overview of key milestones and developments in the history of computer science: ### Ancient Foundations - **Abacus (circa 2400 BC)**: One of the earliest known devices for performing arithmetic calculations. - **Algorithms**: The concept of algorithms dates back to ancient civilizations; for example, Euclid's algorithm for finding the greatest common divisor.

The term "19th century in computing" typically refers to the foundational ideas and early mechanical devices that laid the groundwork for the field of computing as we know it today.

The 1940s were a pivotal decade in the history of computing, marking the transition from mechanical computing devices to electronic computers. Here are some key developments and milestones from that era: 1. **ENIAC (Electronic Numerical Integrator and Computer)**: Completed in 1945, ENIAC is often considered the first general-purpose electronic digital computer. It was developed by John W. Mauchly and J. Presper Eckert at the University of Pennsylvania.

20th-century software refers to computer programs and applications that were developed during the 20th century, primarily between the 1940s and the end of the century in 1999. This period witnessed the evolution of software from early machine code to more sophisticated high-level programming languages and operating systems.

Ancient Greek logicians were philosophers and thinkers in ancient Greece who studied the principles of reasoning and argumentation. This intellectual tradition primarily emerged in the 5th and 4th centuries BCE and laid the groundwork for formal logic as we understand it today. The most notable figure in Ancient Greek logic is Aristotle (384–322 BCE), who is often considered the father of formal logic. He developed the syllogism, a form of deductive reasoning that involves drawing conclusions from premises.

Diæresis (sometimes written as "diaeresis") is a diacritical mark that consists of two dots placed over a vowel. In English, it is often used to indicate that two adjacent vowels should be pronounced separately rather than as a single sound. For example, in the word "naïve," the diæresis over the "i" indicates that it should be pronounced distinctly from the "a" rather than creating a diphthong.

"Dictum de omni et nullo" is a Latin phrase that translates to "the saying about all and none." It is a principle from medieval scholastic philosophy and logic, particularly associated with the works of Peter Abelard and later in discussions of categorical logic. The principle addresses the scope of quantification in logical statements and can be understood as dealing with the relationships between universal affirmative (all) and universal negative (none) statements.

"Logos" is a term with multiple meanings and uses, primarily found in philosophy, rhetoric, and theology. Here are some of the key contexts in which "logos" is significant: 1. **Philosophy**: In ancient Greek philosophy, particularly in the work of Heraclitus, "logos" referred to the principle of order and knowledge. It is often interpreted as the rational structure of the universe, suggesting that there is a logical reason behind the cosmos's existence and functioning.