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- W2024719951 abstract "If B is a selfadjoint translation-invariant operator on the space ${L^2}$ of complex-valued functions on n-dimensional Euclidean space which are square-summable with respect to Lebesgue measure, then the wave equation ${d^2}F/d{t^2} + {B^2}F = 0$ has the solution $F(t) = (cos tB)f + ((sin tB)/B)g$, for f and g in ${L^2}$. In the classical case in which $- {B^2}$ is the Laplacian, this solution has finite velocity of propagation in the sense that (letting supp denote support of a function) ${text {supp}};F(t) subset ({text {supp}};f cup {text {supp}};g) + {K_t}$ for all f and g and some compact set ${K_t}$ independent of f and g. We show that a converse holds, namely, if $cos ;tB$ has finite velocity of propagation (that is, if ${text {supp}} (cos tB)f subset {text {supp}};f + {K_t}$ for all f and some compact ${K_t}$) for three values of t whose reciprocals are independent over the rationals, then ${B^2}$ must be a second order differential operator. If Euclidean space is replaced by a locally compact abelian group which does not contain the real line as a subgroup, then $cos ;tB$ has finite velocity of propagation for all t if and only if it is convolution with a distribution ${T_t}$ such that all ${T_t}$ are supported on a compact open subgroup. Problems of a similar nature are discussed for compact connected abelian groups and for the nonabelian group ${text {SL}}(2,{mathbf {R}})$." @default.
- W2024719951 created "2016-06-24" @default.
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- W2024719951 date "1974-01-01" @default.
- W2024719951 modified "2023-09-27" @default.
- W2024719951 title "Wave equations with finite velocity of propagation" @default.
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- W2024719951 doi "https://doi.org/10.1090/s0002-9947-1974-0330781-6" @default.
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