‎Some improvements of Young inequality and its reverse for positive‎ ‎numbers with Kantorovich constant $K(t‎, ‎2)=frac{(1+t)^2}{4t}$‎ ‎are given‎. ‎Using these inequalities some operator inequalities and‎ ‎Hilbert-Schmidt norm versions for matrices are proved‎. ‎In‎ ‎particular‎, ‎it is shown that if $a‎, ‎b$ are positive numbers and‎ ‎$0 leqslant nu leqslant 1,$ then for all integers $ kgeqsl...

We derive a novel norm that corresponds to the tightest convex relaxation of sparsity combined with an l2 penalty and can also be interpreted as a group Lasso norm with overlaps. We show that this new norm provides a tighter relaxation than the elastic net and suggest using it as a replacement for the Lasso or the elastic net in sparse prediction problems.

We derive a novel norm that corresponds to the tightest convex relaxation of sparsity combined with an `2 penalty. We show that this new k-support norm provides a tighter relaxation than the elastic net and can thus be advantageous in in sparse prediction problems. We also bound the looseness of the elastic net, thus shedding new light on it and providing justification for its use.

This note considers the problem how zeros lying on the boundary of a domain influence the norm of polynomials (under the normalization that their value is fixed at a point). It is shown that k zeros raise the norm by a factor (1 + ck/n) (where n is the degree of the polynomial), while k excessive zeros on an arc compared to n times the equilibrium measure raise the norm by a factor exp(ck/n). T...

k=1 |ak| , in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤...

k=1 |ak|, in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ ...

k=1 |ak|, in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ ...

First we introduce some of the basic notations. For any vector u = (u1, . . . , up) T ∈ R, denote by |u|q the vector `q-norm defined by |u|q = (∑p k=1 |uk| )1/q for q ≥ 1 and write |u|0 = ∑p k=1 I(uk 6= 0). For any set S, denote by S its complement. For a matrix A = (ak`) ∈ Rp×p, we denote by ‖A‖2 the spectral norm, ‖A‖F the Frobenius norm, and ‖A‖1 = ∑p k,`=1 |ak`| the elementwise `1-norm. Rec...