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Register a new userInequalities similar to those of Bernstein for non-periodic splines in $L_2$ space
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We prove Inequalities similar to those of Bernstein for non-periodic splines in $L_2$ space.
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In this paper, we investigate the wavelet coefficients for function spaces $mathcal{A}_k^p={f: |(i omega)^khat{f}(omega)|_pleq 1}, kin N, pin(1,infty)$ using an important quantity $C_{k,p}(psi)$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines, which induces a lower bound for the quantity $C_{k,p}(psi)$ with $psi$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. Comparison of these two families is done by using the quantity $C_{k,p}(psi)$.
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Let ${mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$|f|_A := sup_{x in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A subset {Bbb R}$. Let $$V_a^b(f) := int_a^b{|f^{prime}(x)| , dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 frac nkleq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{|P(1)|}} leq c_2 left( frac nk + 1 right)$$ for all integers $n geq 1$ and $k geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac nkright)^{1/2} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{|P(1)|}} leq c_2 left(frac nk + 1right)^{1/2}$$ for all integers $n geq 1$ and $k geq 1$.
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