ON THE STABILITY OF THE LINEAR FUNCTIONAL EQUATION*

ON THE STABILITY OF THE LINEAR FUNCTIONAL EQUATION*

February 18, 1941 | D. H. HYERS
The chapter discusses the theory of $q$-difference equations, highlighting its completeness in areas such as the convergence and divergence of formal series, the determination of essential transcendental invariants, the inverse Riemann theory, integral representations of solutions, and the definition of $q$-sigma periodic matrices. The material is extensive and not fully covered here. The following section, authored by D. H. Hyers, explores the stability of the linear functional equation \( f(x + y) = f(x) + f(y) \). The problem, proposed by Dr. S. Ulam, asks whether there exists a linear function that approximates an approximately linear function. Hyers proves that for each \(\epsilon > 0\), there exists a \(\delta > 0\) such that every \(\delta\)-linear transformation \( f(x) \) can be approximated by a linear transformation \( l(x) \) with \(\|f(x) - l(x)\| \leq \epsilon\). The proof involves showing that the sequence \( \{f(2^n x)/2^n\} \) is Cauchy and thus converges to a linear transformation \( l(x) \), which is unique and satisfies the inequality. Hyers also investigates the continuity of \( l(x) \) under the assumption that \( f(x) \) is continuous at a single point, proving that \( l(x) \) is continuous everywhere. Additionally, if \( f(tx) \) is continuous for all \( t \), then \( l(x) \) is homogeneous of degree one. The chapter concludes by mentioning potential generalizations of the problem to topological groups.The chapter discusses the theory of $q$-difference equations, highlighting its completeness in areas such as the convergence and divergence of formal series, the determination of essential transcendental invariants, the inverse Riemann theory, integral representations of solutions, and the definition of $q$-sigma periodic matrices. The material is extensive and not fully covered here. The following section, authored by D. H. Hyers, explores the stability of the linear functional equation \( f(x + y) = f(x) + f(y) \). The problem, proposed by Dr. S. Ulam, asks whether there exists a linear function that approximates an approximately linear function. Hyers proves that for each \(\epsilon > 0\), there exists a \(\delta > 0\) such that every \(\delta\)-linear transformation \( f(x) \) can be approximated by a linear transformation \( l(x) \) with \(\|f(x) - l(x)\| \leq \epsilon\). The proof involves showing that the sequence \( \{f(2^n x)/2^n\} \) is Cauchy and thus converges to a linear transformation \( l(x) \), which is unique and satisfies the inequality. Hyers also investigates the continuity of \( l(x) \) under the assumption that \( f(x) \) is continuous at a single point, proving that \( l(x) \) is continuous everywhere. Additionally, if \( f(tx) \) is continuous for all \( t \), then \( l(x) \) is homogeneous of degree one. The chapter concludes by mentioning potential generalizations of the problem to topological groups.
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