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This paper proposes a frequency/time hybrid integral-equation method for thetime dependent wave equation in two and three-dimensional spatial domains.Relying on Fourier Transformation in time, the method utilizes a fixed(time-independent) number of frequency-domain integral-equation solutions toevaluate, with superalgebraically-small errors, time domain solutions forarbitrarily long times. The approach relies on two main elements, namely, 1) Asmooth time-windowing methodology that enables accurate band-limitedrepresentations for arbitrarily-long time signals, and 2) A novel Fouriertransform approach which, in a time-parallel manner and without causingspurious periodicity effects, delivers numerically dispersionlessspectrally-accurate solutions. A similar hybrid technique can be obtained onthe basis of Laplace transforms instead of Fourier transforms, but we do notconsider the Laplace-based method in the present contribution. The algorithmcan handle dispersive media, it can tackle complex physical structures, itenables parallelization in time in a straightforward manner, and it allows fortime leaping---that is, solution sampling at any given time $T$ at$\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$,and without requirement of evaluation of the solution at intermediate times.The proposed frequency-time hybridization strategy, which generalizes to anylinear partial differential equation in the time domain for whichfrequency-domain solutions can be obtained (including e.g. the time-domainMaxwell equations), and which is applicable in a wide range of scientific andengineering contexts, provides significant advantages over other availablealternatives such as volumetric discretization, time-domain integral equations,and convolution-quadrature approaches.

High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: $\mathcal{O}(1)$ Sampling Cost via Incident-Field Windowing and Recentering