Jump distributions on [-1,1] whose orthogonal polynomials have leading coefficients with given asymptotic behavior
Jump distributions on [-1,1] whose orthogonal polynomials have leading coefficients with given asymptotic behavior
We construct an explicit even jump distribution<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d alpha left-parenthesis x right-parenthesis"><mml:semantics><mml:mrow><mml:mi>d</mml:mi><mml:mi>α<!-- α --></mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">d\alpha (x)</mml:annotation></mml:semantics></mml:math></inline-formula>on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 1 comma 1 right-bracket"><mml:semantics><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mo>−<!-- − --></mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">[-1,1]</mml:annotation></mml:semantics></mml:math></inline-formula>for which<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma Subscript n Baseline left-parenthesis d alpha right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>γ<!-- γ --></mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mi>α<!-- α --></mml:mi><mml:mo …