| Summary: | The linear differential operator with constant coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>D</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>y</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mi>y</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><mi>y</mi><mo>,</mo><mspace width="1.em"></mspace><mi>y</mi><mo>∈</mo><msup><mi mathvariant="script">C</mi><mi>n</mi></msup><mrow><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> acting in a Banach space <i>X</i> is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of <i>D</i> has distinct roots <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>r</mi><mi>k</mi></msub></semantics></math></inline-formula> satisfying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">Re</mo><msub><mi>r</mi><mi>k</mi></msub><mo>></mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>,</mo></mrow></semantics></math></inline-formula> then the best Ulam constant of <i>D</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>K</mi><mi>D</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn>0</mn></mrow><mo>∞</mo></msubsup><mo>|</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>k</mi></msup><msub><mi>V</mi><mi>k</mi></msub><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>r</mi><mi>k</mi></msub><mi>x</mi></mrow></msup><mo>|</mo><mi>d</mi><mi>x</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo>(</mo><msub><mi>r</mi><mn>1</mn></msub><mo>,</mo><msub><mi>r</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>r</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mi>k</mi></msub><mo>=</mo><mi>V</mi><mrow><mo>(</mo><msub><mi>r</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>r</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>r</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>r</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>,</mo></mrow></semantics></math></inline-formula> are Vandermonde determinants.
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