We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler–Fourier and Euler–Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth’s persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated with (sub)level-sets persistence of random Gaussian filtrations.

我们介绍了可构造函数的混合变换的一般定义。这些是结合勒贝格积分和欧拉微积分的积分变换。勒贝格积分可以访问经过充分研究的内核和正则性结果，而欧拉微积分则传达拓扑信息并允许与可构造函数上的操作兼容。我们对此类变换进行了系统研究，并介绍了两个新变换：欧拉-傅里叶变换和欧拉-拉普拉斯变换。我们表明第一个具有左逆，第二个提供了 Govc 和 Hepworth 的持久量级到可构造层的令人满意的概括，特别是多参数持久模块。最后，我们证明了将一大类混合变换表示为广义欧拉积分变换的索引论公式。