The book "Exploring Mathematical Analysis, Approximation Theory, and Optimization" celebrates the 270th anniversary of A.-M. Legendre's birth by examining the impact of his work on these fields. It showcases how his contributions, such as the development of Legendre polynomials and the application of analysis to number theory, have influenced modern mathematical research. The book includes historical perspectives on Legendre's geometry and trigonometry, highlighting his foundational role in Greece's higher education. Through various research and survey papers, the book demonstrates the lasting legacy of Legendre's ideas in optimization, approximation theory, and mathematical analysis.