What Is the Fourier Transform? By Shalma Wegsman, September 3, 2025 --- Overview The Fourier transform is a fundamental mathematical operation that decomposes any function into basic wave components (frequencies). This technique helps analyze complex signals by expressing them as sums of sine and cosine waves. Originally developed by Jean-Baptiste Joseph Fourier in the early 19th century, this concept underpins much of modern mathematics, physics, and technology. --- Historical Context Fourier's Life: Born in 1768 in prerevolutionary France. Orphaned early and trained initially in a convent. Participated in the French Revolution, was imprisoned during the Reign of Terror, narrowly escaping execution. Later became a scientific advisor to Napoleon and worked on heat conduction problems during the Egyptian campaign. Development of the Fourier Transform: Fourier sought to understand heat conduction in solids. Proposed that heat distribution could be represented as sums of simple waves (sine and cosine functions). Faced skepticism at first; renowned mathematicians found his ideas "impossible," especially for representing sharp changes like discontinuities. Eventually, these ideas laid the foundation for the mathematical field of harmonic analysis. --- What is the Fourier Transform? It is a function that transforms a given function (e.g., a sound wave or image data) into its constituent frequencies. The original function can be perfectly reconstructed by summing these frequencies. Uses both sine and cosine waves (or complex exponentials) to analyze the function. Helps convert complicated functions into simpler representations. Analogy: Similar to how human ears separate sounds into distinct pitches or how a perfume's ingredients are identified. --- How It Works – Simple Example Consider a function (signal). Multiply the function by sine waves of different frequencies to see which frequencies contribute. If the product averages to a significant value, that frequency is present. If it averages to zero, that frequency does not significantly contribute. The process is repeated for cosine waves as well. The collection of these contributions forms the Fourier transform. --- Applications Pure Mathematics: Harmonic analysis studies these decompositions, linking to fields like number theory and differential equations. Physics: Applies in heat conduction, quantum mechanics (e.g., uncertainty principle), wave phenomena, and gravitational wave detection. Technology: Signal processing: Audio enhancement, noise reduction. Image processing: JPEG compression by decomposing images into frequencies and removing high-frequency details mildly perceptible to humans. Algorithms: The fast Fourier transform (FFT) by Cooley and Tukey enables efficient computation. --- Higher Dimensions The Fourier transform extends to multi-dimensional data, like images. It breaks down images into 2D frequency components forming patterns (stripes, checkerboards). This decomposition can compress images by discarding less important high-frequency data. --- Legacy Fourier's work revolutionized math and science, providing a universal language for waves and functions. The Fourier transform is used daily in diverse fields, from theoretical math to practical computing. Its influence spans disciplines, making it indispensable in understanding and manipulating signals and data. --- Related Articles ‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture How Wavelets Allow Researchers to Transform, and Understand, Data Mathematicians Discover the Perfect Way to Multiply --- Note: This content is from an editorially independent publication supported by the Simons Foundation.