{
  "id": "enwiki-001242511184",
  "slug": "progressive-function",
  "title": "Progressive function",
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  "lead": "In mathematics, a progressive function ƒ ∈ L 2 (R) is a function whose Fourier transform is supported by positive frequencies only: : \\mathop{\\rm supp}\\hat{f} \\subseteq \\mathbb{R}_+. It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if : \\mathop{\\rm supp}\\hat{f} \\subseteq \\mathbb{R}_-. The complex conjugate of a progressive function is regressive, and vice versa. The space of progressive functions is sometimes denoted H^2_+(R) , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula : f(t) = \\int_0^\\infty e^{2\\pi i st} \\hat f(s)\\, ds and hence extends to a holomorphic function on the upper half-plane \\{ t + iu: t, u \\in R, u \\geq 0 \\} by the formula : f(t+iu) = \\int_0^\\infty e^{2\\pi i s(t+iu)} \\hat f(s)\\, ds = \\int_0^\\infty e^{2\\pi i st} e^{-2\\pi su} \\hat f(s)\\, ds. Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line will arise in this manner. Regressive functions are similarly associated with the Hardy space on the lower half-plane \\{ t + iu: t, u \\in R, u \\leq 0 \\} .",
  "body": "In mathematics, a progressive function ƒ ∈ L 2 (R) is a function whose Fourier transform is supported by positive frequencies only: \\mathop{\\rm supp}\\hat{f} \\subseteq \\mathbb{R}_+. It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if \\mathop{\\rm supp}\\hat{f} \\subseteq \\mathbb{R}_-. The complex conjugate of a progressive function is regressive, and vice versa. The space of progressive functions is sometimes denoted H^2_+(R) , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula f(t) = \\int_0^\\infty e^{2\\pi i st} \\hat f(s)\\, ds and hence extends to a holomorphic function on the upper half-plane \\{ t + iu: t, u \\in R, u \\geq 0 \\} by the formula f(t+iu) = \\int_0^\\infty e^{2\\pi i s(t+iu)} \\hat f(s)\\, ds = \\int_0^\\infty e^{2\\pi i st} e^{-2\\pi su} \\hat f(s)\\, ds. Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line will arise in this manner. Regressive functions are similarly associated with the Hardy space on the lower half-plane \\{ t + iu: t, u \\in R, u \\leq 0 \\} . ==References==",
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    "title": "Progressive function",
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        "family": "Anticloud FZ LLE",
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    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/progressive-function",
    "issued": {
      "date-parts": [
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          2026,
          11,
          15
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