# Integration of squared periodic Bernoulli polynomials

Let $$f\in C_{per}^{2q}[0,1]=\left\{ f:f\in C^{2q}[0,1]\mbox{ and }f^{(j)}(0)=f^{(j)}(1),\ j=0,\ldots,2q-1\right\}.$$ Let $\left\{ \cdot\right\}$ denote fractional part and $B_{2q}$ be $2q$-th Bernoulli polynomial .How does integral $$\int_{0}^{1}\left(f^{(2q)}(x)\right)^{2}B_{2q}^{2}(\{Kx\})dx$$ depend on $K\in\mathbb{N}$? I tried Fourier series, change of variables,integration by parts, but nothing is working. Any ideas?

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