Properties of convolutions of aperiodic functions and their applications to the variational problems (Q2719441)

Let \(f(x)\) and \(g(x)\) be aperiodic functions on the interval \( -1\leq x\leq 1\). The author specifies the aperiodic functions convolution by the formula NEWLINE\[NEWLINE \varPhi(x) = \frac{1}{2} \int\limits_{-1}^1 f_t(x)g(t) dt,\tag{1} NEWLINE\]NEWLINE where NEWLINE\[NEWLINE f_t(x) = \frac 1\pi \int\limits_{-1}^1 f[xt+r(1-x^2)^{1/2}(1-t^2)^{1/2}](1-r^2)^{1/2} dr. NEWLINE\]NEWLINE The relationship is studied between the convolutions (1) and the classical convolutions for the periodic functions. In terms of the convolutions (1) the solvability criteria are established for the variational problems degenerating on the domain boundary.