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Nonlocal quantum theory of one-component scalar field in $D$-dimensionalEuclidean spacetime is studied in representations of $\mathcal{S}$-matrixtheory for both polynomial and nonpolynomial interaction Lagrangians. Thetheory is formulated on coupling constant $g$ in the form of an infrared smoothfunction of argument $x$ for space without boundary. Nonlocality is given byevolution of Gaussian propagator for the local free theory with ultravioletform factors depending on ultraviolet length parameter $l$. By representationof the $\mathcal{S}$-matrix in terms of abstract functional integral overprimary scalar field, the $\mathcal{S}$ form of a grand canonical partitionfunction is found. And, by expression of $\mathcal{S}$-matrix in terms of thepartition function, the representation for $\mathcal{S}$ in terms of basisfunctions is obtained. Derivations are given for discrete case where basisfunctions are Hermite functions, and for continuous case where basis functionsare trigonometric functions. The obtained expressions for the$\mathcal{S}$-matrix are investigated within the framework of variationalprinciple based on Jensen inequality. Equations with separable kernelssatisfied by variational function $q$ are found and solved, yielding resultsfor both the polynomial theory $\varphi^{4}$ and the nonpolynomial sine-Gordontheory. A new definition of the $\mathcal{S}$-matrix is proposed to solveadditional divergences which arise in application of Jensen inequality for thecontinuous case. Analytical results are illustrated numerically. For simplicityof numerical calculation: the $D=1$ case is considered, and propagator for thefree theory $G$ is in the form of Gaussian function typically in theVirton-Quark model. The formulation for nonlocal QFT in momentum $k$ space ofextra dimensions with subsequent compactification into physical spacetime isdiscussed.

$\mathcal{S}$-Matrix of Nonlocal Scalar Quantum Field Theory in the Representation of Basis Functions