<p>Let <i>X</i> be a metric space with a base point 0, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{Lip}_0(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Lip</mtext> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the Banach space of all Lipschitz functions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$f:X\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f(0)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Given a set of points <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left( (x_i,y_i)\right) _{i\in I}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close=")" open="("> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mfenced> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$x_i\ne y_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>≠</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$i\in I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation>, we study the following interpolation problem: when for each bounded set <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\left( \alpha _i\right) _{i\in I}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close=")" open="("> <msub> <mi>α</mi> <mi>i</mi> </msub> </mfenced> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> the algorithm <Equation ID="Equ1"> <EquationSource Format="TEX">$ \frac{f(x_i)-f(y_i)}{d(x_i,y_i)}=\alpha _i\qquad (i\in I) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>=</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mspace width="2em" /> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>can be implemented by a function <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$f\in \textrm{Lip}_0(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msub> <mtext>Lip</mtext> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>? Our approach involves the concept of a Beurling set of functions in <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\textrm{Lip}_0(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Lip</mtext> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\left( (x_i,y_i)\right) _{i\in I}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mfenced close=")" open="("> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mfenced> <mrow> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> which has shown to be useful in the so-called transportation problem.</p>

Lipschitz Interpolating Sequences

期刊论文
发表日期：

2025年10月30日

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A. Jiménez-Vargas,
Abraham Rueda Zoca

摘要

Let X be a metric space with a base point 0, and let $\textrm{Lip}_0(X)$ ${Lip}_{0} (X)$ be the Banach space of all Lipschitz functions $f:X\rightarrow \mathbb {R}$ $f : X \to R$ such that $f(0)=0$ $f (0) = 0$ . Given a set of points $\left( (x_i,y_i)\right) _{i\in I}$ ${((x_{i}, y_{i}))}_{i \in I}$ in $X^2$ $X^{2}$ with $x_i\ne y_i$ $x_{i} \neq y_{i}$ for all $i\in I$ $i \in I$ , we study the following interpolation problem: when for each bounded set $\left( \alpha _i\right) _{i\in I}$ ${(α_{i})}_{i \in I}$ in $\mathbb {R}$ $R$ the algorithm $ \frac{f(x_i)-f(y_i)}{d(x_i,y_i)}=\alpha _i\qquad (i\in I) $ $\frac{f (x_{i}) - f (y_{i})}{d (x_{i}, y_{i})} = α_{i} (i \in I)$ can be implemented by a function $f\in \textrm{Lip}_0(X)$ $f \in {Lip}_{0} (X)$ ? Our approach involves the concept of a Beurling set of functions in $\textrm{Lip}_0(X)$ ${Lip}_{0} (X)$ for $\left( (x_i,y_i)\right) _{i\in I}$ ${((x_{i}, y_{i}))}_{i \in I}$ which has shown to be useful in the so-called transportation problem.